Date of submission: April 15, 2021; date of acceptance: June 10, 2021. * Contact information: mustafaakan@halic.edu.tr, Halic University Sutluce Mh. Im- rahor Sk. No: 82 34445 Beyoglu/Istanbul, Turkey, phone: +905333119405; ORCID ID: https://orcid.org/0000-0002-2900-4932. ** Contact information: natalija.konovalova@riseba.lv, RISEBA University of Applied Sciences, Meza street 3, Riga, Latvia, phone: +37129215208; ORCID ID: https://orcid. org/0000-0003-4072-4479. Copernican Journal of Finance & Accounting e-ISSN 2300-3065 p-ISSN 2300-12402021, volume 10, issue 3 Akan, M., & Konovalova, N. (2021). Optimal Incentives for Economic Growth in Central European Countries: A Micro Approach. Copernican Journal of Finance & Accounting, 10(3), 9–31. http:// dx.doi.org/10.12775/CJFA.2021.009 Mustafa akan* Haliç University natalia konovalova** RISEBA University of Applied Sciences optiMal incentives for econoMic growth in central european countries: a Micro approach Keywords: optimal control theory, incentives for economic growth, f luctuating de- mand. J E L Classification: D90, F43, F63, C61. Abstract: Financial crisis of 2008 and the ongoing pandemic are continuing to have a negative impact on the economies of all countries even tough interest rates have been decreased significantly. This paper attempted to view the problem from a micro point of view to suggest more effective incentives for growth. The specific objective of the study is to determine and examine the effects of these incentives on economic growth in Central European countries. Mustafa Akan, Natalia Konovalova10 An optimal control theoretic model was employed as a method of analysis with data from countries in question. Results showed, generally, that incen- tives such as interest rates, investments in production technologies, labor pro- ductivity, and the cost of inventory were important factors to induce growth with different impact in each country. Results showed that changes in interest rates will not cause significant economic growth in Poland, Hungary, and the Czech Republic where interest rates are already low. However, countries such as Croatia and Romania where interest rates are relatively high, reducing in- terest rates may lead to economic growth. The investment in production tech- nologies will have a significant impact on economic growth in Bulgaria, Hunga- ry and Croatia. For the Czech Republic, Slovenia and Poland, which are already quite advanced in the field of production technology, the impact of this factor on economic growth will be less significant. Incentives to increase labor pro- ductivity in Hungary, Bulgaria, Estonia, Latvia, and Lithuania will have signifi- cant impact on economic growth as productivity in these countries is relatively low. Incentives regarding holding costs will be effective on sectoral basis.  Introduction Governments have taken actions to stabilize the financial sector by principally increasing money supply and other measures such as taking over some of the financial institutions affected by the crisis. Growth rates in advanced countries are low and are forecasted to remain low for several more years. The trade war between the USA and China will surely have a negative impact on the growth rates. Outbreak of a virus in China will probably have a negative impact on growth. AFP (2020) estimates that the impact could be 0.1-0.2 %. Giles, Arnold and Greely (2020) have reported that OECD lowered its growth estimates for the World from 2.9% to 2.4% for 2020. All major economies of the World have taken monetary measures to minimize the impact of financial crisis on the real sector. These policies worked well in most countries to contain the financial crisis but not so well to induce growth which is evident from the growth rates in 2009 and thereafter. The interest rates in major economies of the World are already low. They are zero in Eurozone, negative in Japan. Inf lation rates in al- most all major economies are low implying that the consumers are not bor- rowing for consumption even though the interest rates are generally very low. These two facts are good indications of the end of effectiveness of monetary (easy money, low interest rate) policies. oPtimAl incentives for economic growth… 11 A brief review of literature shows such measures have either been ineffec- tive, insignificant, or even negative. Carlianne (2014) found that governments aiding private enterprises has significant negative mid-term effect on employ- ment and growth. Osario and Florida (2017) showed that business incentives have different impacts in different areas and are usually ineffective, costly, and wasteful. Peters and Fisher (2004) have concluded that incentives are costly and do not encourage investment. Kosonen (2012), in a study on Finnish firms, have found that decreasing taxes would increase the production but the tax elasticity of production is only -0.17. Petrin (2018) concluded that incentives for R/D and innovation might have positive impact but not always. A report by DPME/DSBO (2018) could not make a conclusion on the impact of govern- ment incentives on business. Conroy (2019), in her master thesis, concluded that there is a positive impact of incentives on attracting investment to a state but it depends on the type of incentives. Prillaman and Meier (2014) concluded that state tax cuts have little to no positive on the growth of the state, job crea- tion, personal income, poverty rates, and formation of new businesses. A report by Christian, Karlin, Schaff and Tucker-Roy (2019) found a positive relation- ship between incentive spending and jobs created only if specific objectives are set, measurable goals are defined, and investment is made in staff, systems, and budget. Bundrick (2016) showed that incentives have costs and they cre- ate market distortions. In a study for center for American Progress, Schwartz (2018) concluded that subsidies fail to meet promised results with no net ben- efit to the social welfare. A report by the PEW Charitable Trusts (2019) states that incentives help create an unstable economy. Mitschell (2019) concluded that incentives lead to a slower growth. A review report by World Bank (1999) has shown that, when all other factors such as infrastructure, transport costs, and political and economic stability, are approximately equal, the taxes may have significant impact on investors’ location choices. The report also conclud- ed that this effect depends on the tax instrument. Miller and Atkinson (2014) concluded that investment in Information and Communication Technology (ICT) brings about a transformative change to organizations resulting in in- creased productivity. Buss (2001), in his review of literature report on state tax incentives, summarized his report by stating that tax studies offer little guid- ance as tools of economic development. Klemm and Parys (2012) found the evi- dence that taxes were effective attracting Foreign Direct Investment into Latin America and Caribbean but not effective on increasing gross private fixed capi- tal. Crespi, Guiliodori, Guiliodori and Rodriguez (2016) studied the impacts of Mustafa Akan, Natalia Konovalova12 promoting firm level investments in R/D in Argentina and found that elasticity of such investments is greater than 1 and effects vary on the sector and the size of the firm. The effects vary depending on the type of investment considered. Zee, Stotsky and Ley (2002) concluded that tax incentives should be directed towards rectification of market failures and that cost effectiveness of such in- centives are inconclusive. In their recent paper, Tadesse and Melaku (2019) has shown that monetary policies were less effective than fiscal policies in the long run and ineffective in the short run in Ethiopia. The review of literature above on the impact of government incentives are, by no means, exhaustive, shows that they are largely ineffective, in some cas- es detrimental to proper functioning of the economy. In any case, there is not a consensus on the effectiveness of such measures. Some studies show, proper- ly structured, incentives may have positive impacts. One purpose of this paper is to contribute to this search for proper incen- tives by studying the dynamic behavior of real sector companies from a micro point of view to determine the conditions under which they will produce more under various demand expectations. Another purpose of the paper is to ana- lyze the factors that affect the production behavior of the firms and suggest more realistic macro policies to increase growth other than reduction of bor- rowing costs (interest rates). It is not the purpose of this paper to compare the effectiveness of micro and macro policies or to search for incentives to increase consumption. In the next section, a brief review on incentives for firms will be provided. In Section III. a general dynamic model of a real sector company (nonservice) op- erating in a competitive environment facing oscillating demand and different interest rate expectations will be developed using an optimal control theoret- ic model will be presented. The examples of solutions under different assump- tions will be solved in the section IV to observe the effects of changes in some parameters such as interest rates, costs, and changes in demand structure. Re- sults, conclusions, and policy implications of the results will follow in subse- quent sections. Some further research will be suggested in the last section. oPtimAl incentives for economic growth… 13 Research Methodology The general scheduling production planning problem of a firm operating in a perfectly competitive market (where price is constant) can be modelled as a cost minimization (sum of cost of production and inventory holding cost) where: u(t): the level of production at time t, c(u): strictly convex cost function h: unit holding cost, a constant. These costs include many cost items such as rent, cost of relevant equipment necessary to keep the products in the inven- tory, and insurance premiums. r(t): level of demand that is an exogenously determined function of time l(t): the level of inventory at time t, which increases by production (u(t) and decreases by sales r(t). Therefore, the model, is ℎ: unit holding cost, a constant. These costs include many cost items such as rent, cost of rele- vant equipment necessary to keep the products in the inventory, and insurance premiums. 𝑟𝑟𝑟𝑟𝑟𝑟: level of demand that is an exogenously determined function of time 𝐼𝐼𝑟𝑟𝑟𝑟: the level of inventory at time t, which increases by production (𝑢𝑢𝑟𝑟𝑟𝑟 and decreases by sales 𝑟𝑟𝑟𝑟𝑟𝑟. Therefore, the model, is ( ) 0 max [ ( ( )) ( )] T u t c u t hI t dt Subject to the constraints: 𝐼𝐼� = 𝑢𝑢𝑟𝑟𝑟𝑟 − 𝑟𝑟𝑟𝑟𝑟𝑟 𝐼𝐼𝑟0𝑟 = 𝐼𝐼� given and the non-negativity constraints. 𝑢𝑢𝑟𝑟𝑟𝑟 ≥ 0, 𝐼𝐼𝑟𝑟𝑟𝑟 ≥ 0 Cost minimization is considered since the price is assumed to be constant and the demand, r(t), is an exogenous time dependent function making the revenues an exogenous function also thus irrelevant in decision making. The last constraint is a state variable inequality constraint which has been studied by many authors including Bryson and Ho (1975), Bryson, Denham and Dreyfus (1963), Jacobson and Lele (1969), Jacobson, Lele and Speyer (1971), McIntyre and Paiewonsky (1967), and Sprzeuzkouski (1967). Taylor (1972) studied the problem stated above with no interest rate consideration and has found that; 𝑐𝑐���𝑟𝑟𝑟𝑟𝑟𝑟�𝑟𝑟�𝑟𝑟𝑟𝑟 𝑡 ℎ is the condition which has to hold for the firm to keep I (t) =0 which implies that the firm will produce just to meet the demand. The firm will start producing not only to meet the demand but also to build up inventories when the change in demand is such that the above inequality will be violated. In this paper, we will study the same problem introducing present value factor and fluctuat- ing time dependent demand function. Then, the problem becomes: ( ) 0 [ ( ( ) ( )] T i tMin e c u t hI t dt  Subject to the constraints: Subject to the constraints: ℎ: unit holding cost, a constant. These costs include many cost items such as rent, cost of rele- vant equipment necessary to keep the products in the inventory, and insurance premiums. 𝑟𝑟𝑟𝑟𝑟𝑟: level of demand that is an exogenously determined function of time 𝐼𝐼𝑟𝑟𝑟𝑟: the level of inventory at time t, which increases by production (𝑢𝑢𝑟𝑟𝑟𝑟 and decreases by sales 𝑟𝑟𝑟𝑟𝑟𝑟. Therefore, the model, is ( ) 0 max [ ( ( )) ( )] T u t c u t hI t dt Subject to the constraints: 𝐼𝐼� = 𝑢𝑢𝑟𝑟𝑟𝑟 − 𝑟𝑟𝑟𝑟𝑟𝑟 𝐼𝐼𝑟0𝑟 = 𝐼𝐼� given and the non-negativity constraints. 𝑢𝑢𝑟𝑟𝑟𝑟 ≥ 0, 𝐼𝐼𝑟𝑟𝑟𝑟 ≥ 0 Cost minimization is considered since the price is assumed to be constant and the demand, r(t), is an exogenous time dependent function making the revenues an exogenous function also thus irrelevant in decision making. The last constraint is a state variable inequality constraint which has been studied by many authors including Bryson and Ho (1975), Bryson, Denham and Dreyfus (1963), Jacobson and Lele (1969), Jacobson, Lele and Speyer (1971), McIntyre and Paiewonsky (1967), and Sprzeuzkouski (1967). Taylor (1972) studied the problem stated above with no interest rate consideration and has found that; 𝑐𝑐���𝑟𝑟𝑟𝑟𝑟𝑟�𝑟𝑟�𝑟𝑟𝑟𝑟 𝑡 ℎ is the condition which has to hold for the firm to keep I (t) =0 which implies that the firm will produce just to meet the demand. The firm will start producing not only to meet the demand but also to build up inventories when the change in demand is such that the above inequality will be violated. In this paper, we will study the same problem introducing present value factor and fluctuat- ing time dependent demand function. Then, the problem becomes: ( ) 0 [ ( ( ) ( )] T i tMin e c u t hI t dt  Subject to the constraints: ℎ: unit holding cost, a constant. These costs include many cost items such as rent, cost of rele- vant equipment necessary to keep the products in the inventory, and insurance premiums. 𝑟𝑟𝑟𝑟𝑟𝑟: level of demand that is an exogenously determined function of time 𝐼𝐼𝑟𝑟𝑟𝑟: the level of inventory at time t, which increases by production (𝑢𝑢𝑟𝑟𝑟𝑟 and decreases by sales 𝑟𝑟𝑟𝑟𝑟𝑟. Therefore, the model, is ( ) 0 max [ ( ( )) ( )] T u t c u t hI t dt Subject to the constraints: 𝐼𝐼� = 𝑢𝑢𝑟𝑟𝑟𝑟 − 𝑟𝑟𝑟𝑟𝑟𝑟 𝐼𝐼𝑟0𝑟 = 𝐼𝐼� given and the non-negativity constraints. 𝑢𝑢𝑟𝑟𝑟𝑟 ≥ 0, 𝐼𝐼𝑟𝑟𝑟𝑟 ≥ 0 Cost minimization is considered since the price is assumed to be constant and the demand, r(t), is an exogenous time dependent function making the revenues an exogenous function also thus irrelevant in decision making. The last constraint is a state variable inequality constraint which has been studied by many authors including Bryson and Ho (1975), Bryson, Denham and Dreyfus (1963), Jacobson and Lele (1969), Jacobson, Lele and Speyer (1971), McIntyre and Paiewonsky (1967), and Sprzeuzkouski (1967). Taylor (1972) studied the problem stated above with no interest rate consideration and has found that; 𝑐𝑐���𝑟𝑟𝑟𝑟𝑟𝑟�𝑟𝑟�𝑟𝑟𝑟𝑟 𝑡 ℎ is the condition which has to hold for the firm to keep I (t) =0 which implies that the firm will produce just to meet the demand. The firm will start producing not only to meet the demand but also to build up inventories when the change in demand is such that the above inequality will be violated. In this paper, we will study the same problem introducing present value factor and fluctuat- ing time dependent demand function. Then, the problem becomes: ( ) 0 [ ( ( ) ( )] T i tMin e c u t hI t dt  Subject to the constraints: given and the non-negativity constraints. ℎ: unit holding cost, a constant. These costs include many cost items such as rent, cost of rele- vant equipment necessary to keep the products in the inventory, and insurance premiums. 𝑟𝑟𝑟𝑟𝑟𝑟: level of demand that is an exogenously determined function of time 𝐼𝐼𝑟𝑟𝑟𝑟: the level of inventory at time t, which increases by production (𝑢𝑢𝑟𝑟𝑟𝑟 and decreases by sales 𝑟𝑟𝑟𝑟𝑟𝑟. Therefore, the model, is ( ) 0 max [ ( ( )) ( )] T u t c u t hI t dt Subject to the constraints: 𝐼𝐼� = 𝑢𝑢𝑟𝑟𝑟𝑟 − 𝑟𝑟𝑟𝑟𝑟𝑟 𝐼𝐼𝑟0𝑟 = 𝐼𝐼� given and the non-negativity constraints. 𝑢𝑢𝑟𝑟𝑟𝑟 ≥ 0, 𝐼𝐼𝑟𝑟𝑟𝑟 ≥ 0 Cost minimization is considered since the price is assumed to be constant and the demand, r(t), is an exogenous time dependent function making the revenues an exogenous function also thus irrelevant in decision making. The last constraint is a state variable inequality constraint which has been studied by many authors including Bryson and Ho (1975), Bryson, Denham and Dreyfus (1963), Jacobson and Lele (1969), Jacobson, Lele and Speyer (1971), McIntyre and Paiewonsky (1967), and Sprzeuzkouski (1967). Taylor (1972) studied the problem stated above with no interest rate consideration and has found that; 𝑐𝑐���𝑟𝑟𝑟𝑟𝑟𝑟�𝑟𝑟�𝑟𝑟𝑟𝑟 𝑡 ℎ is the condition which has to hold for the firm to keep I (t) =0 which implies that the firm will produce just to meet the demand. The firm will start producing not only to meet the demand but also to build up inventories when the change in demand is such that the above inequality will be violated. In this paper, we will study the same problem introducing present value factor and fluctuat- ing time dependent demand function. Then, the problem becomes: ( ) 0 [ ( ( ) ( )] T i tMin e c u t hI t dt  Subject to the constraints: Cost minimization is considered since the price is assumed to be constant and the demand, r(t), is an exogenous time dependent function making the revenues an exogenous function also thus irrelevant in decision making. The last constraint is a state variable inequality constraint which has been studied by many authors including Bryson and Ho (1975), Bryson, Denham and Dreyfus (1963), Jacobson and Lele (1969), Jacobson, Lele and Speyer (1971), McIntyre and Paiewonsky (1967), and Sprzeuzkouski (1967). Taylor (1972) studied the problem stated above with no interest rate consideration and has found that; Mustafa Akan, Natalia Konovalova14 ℎ: unit holding cost, a constant. These costs include many cost items such as rent, cost of rele- vant equipment necessary to keep the products in the inventory, and insurance premiums. 𝑟𝑟𝑟𝑟𝑟𝑟: level of demand that is an exogenously determined function of time 𝐼𝐼𝑟𝑟𝑟𝑟: the level of inventory at time t, which increases by production (𝑢𝑢𝑟𝑟𝑟𝑟 and decreases by sales 𝑟𝑟𝑟𝑟𝑟𝑟. Therefore, the model, is ( ) 0 max [ ( ( )) ( )] T u t c u t hI t dt Subject to the constraints: 𝐼𝐼� = 𝑢𝑢𝑟𝑟𝑟𝑟 − 𝑟𝑟𝑟𝑟𝑟𝑟 𝐼𝐼𝑟0𝑟 = 𝐼𝐼� given and the non-negativity constraints. 𝑢𝑢𝑟𝑟𝑟𝑟 ≥ 0, 𝐼𝐼𝑟𝑟𝑟𝑟 ≥ 0 Cost minimization is considered since the price is assumed to be constant and the demand, r(t), is an exogenous time dependent function making the revenues an exogenous function also thus irrelevant in decision making. The last constraint is a state variable inequality constraint which has been studied by many authors including Bryson and Ho (1975), Bryson, Denham and Dreyfus (1963), Jacobson and Lele (1969), Jacobson, Lele and Speyer (1971), McIntyre and Paiewonsky (1967), and Sprzeuzkouski (1967). Taylor (1972) studied the problem stated above with no interest rate consideration and has found that; 𝑐𝑐���𝑟𝑟𝑟𝑟𝑟𝑟�𝑟𝑟�𝑟𝑟𝑟𝑟 𝑡 ℎ is the condition which has to hold for the firm to keep I (t) =0 which implies that the firm will produce just to meet the demand. The firm will start producing not only to meet the demand but also to build up inventories when the change in demand is such that the above inequality will be violated. In this paper, we will study the same problem introducing present value factor and fluctuat- ing time dependent demand function. Then, the problem becomes: ( ) 0 [ ( ( ) ( )] T i tMin e c u t hI t dt  Subject to the constraints: is the condition which has to hold for the firm to keep I (t) =0 which implies that the firm will produce just to meet the demand. The firm will start producing not only to meet the demand but also to build up inventories when the change in demand is such that the above inequality will be violated. In this paper, we will study the same problem introducing present value fac- tor and f luctuating time dependent demand function. Then, the problem becomes: ℎ: unit holding cost, a constant. These costs include many cost items such as rent, cost of rele- vant equipment necessary to keep the products in the inventory, and insurance premiums. 𝑟𝑟𝑟𝑟𝑟𝑟: level of demand that is an exogenously determined function of time 𝐼𝐼𝑟𝑟𝑟𝑟: the level of inventory at time t, which increases by production (𝑢𝑢𝑟𝑟𝑟𝑟 and decreases by sales 𝑟𝑟𝑟𝑟𝑟𝑟. Therefore, the model, is ( ) 0 max [ ( ( )) ( )] T u t c u t hI t dt Subject to the constraints: 𝐼𝐼� = 𝑢𝑢𝑟𝑟𝑟𝑟 − 𝑟𝑟𝑟𝑟𝑟𝑟 𝐼𝐼𝑟0𝑟 = 𝐼𝐼� given and the non-negativity constraints. 𝑢𝑢𝑟𝑟𝑟𝑟 ≥ 0, 𝐼𝐼𝑟𝑟𝑟𝑟 ≥ 0 Cost minimization is considered since the price is assumed to be constant and the demand, r(t), is an exogenous time dependent function making the revenues an exogenous function also thus irrelevant in decision making. The last constraint is a state variable inequality constraint which has been studied by many authors including Bryson and Ho (1975), Bryson, Denham and Dreyfus (1963), Jacobson and Lele (1969), Jacobson, Lele and Speyer (1971), McIntyre and Paiewonsky (1967), and Sprzeuzkouski (1967). Taylor (1972) studied the problem stated above with no interest rate consideration and has found that; 𝑐𝑐���𝑟𝑟𝑟𝑟𝑟𝑟�𝑟𝑟�𝑟𝑟𝑟𝑟 𝑡 ℎ is the condition which has to hold for the firm to keep I (t) =0 which implies that the firm will produce just to meet the demand. The firm will start producing not only to meet the demand but also to build up inventories when the change in demand is such that the above inequality will be violated. In this paper, we will study the same problem introducing present value factor and fluctuat- ing time dependent demand function. Then, the problem becomes: ( ) 0 [ ( ( ) ( )] T i tMin e c u t hI t dt  Subject to the constraints: Subject to the constraints: 𝐼𝐼� = 𝑢𝑢(𝑡𝑡) − 𝑟𝑟(𝑡𝑡) 𝐼𝐼(0) = 𝐼𝐼� a given number. 𝑢𝑢(𝑡𝑡) 𝑢 0 𝐼𝐼(𝑡𝑡) 𝑢 0 With the assumptions that the demand (𝑟𝑟 (𝑡𝑡)) is a simple sinusoidal function and the inter- est rate 𝑖𝑖(𝑡𝑡) is time dependent, i.e.: 𝑟𝑟(𝑡𝑡) = 𝐴𝐴 𝐴 𝐴𝐴𝐴𝐴𝑖𝑖𝐴𝐴(𝐴𝐴𝐴𝐴𝐴𝑡𝑡) with no growth, and, 𝑖𝑖(𝑡𝑡) = � 𝑓𝑓(𝐴𝐴)𝑑𝑑𝐴𝐴�� implying that 𝑖𝑖(𝑡𝑡) = 𝑓𝑓𝑡𝑡 if 𝑓𝑓 is constant. Notice also that 𝑖𝑖�(𝑡𝑡) = 𝑓𝑓(𝑡𝑡) where f (t) denotes time dependent interest rate. For ease of analysis we will assume 𝑓𝑓(𝑡𝑡) to be con- stant. The letter k denotes the periodicity of the sinusoidal demand function over the planning period. The only reason for the use sinusoidal demand curve is that it represents cyclicality (a measure of risk) and that it is easier to work with it. The appropriate Lagrangian for this problem is: 𝐻𝐻 = 𝐻𝐻��(�)�𝑐𝑐�𝑢𝑢(𝑡𝑡)� 𝐴 ℎ𝐼𝐼(𝑡𝑡)� 𝐴 𝜌𝜌(𝑡𝑡)�𝑢𝑢(𝑡𝑡) − 𝑟𝑟(𝑡𝑡)� 𝐴 𝜇𝜇(𝑡𝑡)�𝑟𝑟(𝑡𝑡) − 𝑢𝑢(𝑡𝑡)� Then the necessary conditions are: 𝐻𝐻 ��(�)𝑐𝑐�(𝑡𝑡) 𝐴 𝜌𝜌(𝑡𝑡) − 𝜇𝜇(𝑡𝑡) = 0 (1) 𝜌𝜌�(𝑡𝑡) = −𝐻𝐻��(�)ℎ (2) 𝐼𝐼�(𝑡𝑡) = 𝑢𝑢(𝑡𝑡) − 𝑟𝑟(𝑡𝑡) (3) Necessary conditions are also sufficient since the Hamiltonian is convex in both the control (𝑢𝑢) and state (𝐼𝐼) variables. The planning horizon (𝑇𝑇) will be assumed one. On a constraint arc on which 𝐼𝐼(𝑡𝑡) = 0 for a certain period in [0, 1] i.e. 0 < 𝑡𝑡� ≤ 𝑡𝑡 ≤ 𝑡𝑡� < 1 we have; 𝐼𝐼� = 0 = 𝑢𝑢(𝑡𝑡) − 𝑟𝑟(𝑡𝑡) implying that over this period a firm will produce just enough to meet the demand, which is assumed to be nonzero during the planning horizon. 𝐼𝐼� = 𝑢𝑢(𝑡𝑡) − 𝑟𝑟(𝑡𝑡) 𝐼𝐼(0) = 𝐼𝐼� a given number. 𝑢𝑢(𝑡𝑡) 𝑢 0 𝐼𝐼(𝑡𝑡) 𝑢 0 With the assumptions that the demand (𝑟𝑟 (𝑡𝑡)) is a simple sinusoidal function and the inter- est rate 𝑖𝑖(𝑡𝑡) is time dependent, i.e.: 𝑟𝑟(𝑡𝑡) = 𝐴𝐴 𝐴 𝐴𝐴𝐴𝐴𝑖𝑖𝐴𝐴(𝐴𝐴𝐴𝐴𝐴𝑡𝑡) with no growth, and, 𝑖𝑖(𝑡𝑡) = � 𝑓𝑓(𝐴𝐴)𝑑𝑑𝐴𝐴�� implying that 𝑖𝑖(𝑡𝑡) = 𝑓𝑓𝑡𝑡 if 𝑓𝑓 is constant. Notice also that 𝑖𝑖�(𝑡𝑡) = 𝑓𝑓(𝑡𝑡) where f (t) denotes time dependent interest rate. For ease of analysis we will assume 𝑓𝑓(𝑡𝑡) to be con- stant. The letter k denotes the periodicity of the sinusoidal demand function over the planning period. The only reason for the use sinusoidal demand curve is that it represents cyclicality (a measure of risk) and that it is easier to work with it. The appropriate Lagrangian for this problem is: 𝐻𝐻 = 𝐻𝐻��(�)�𝑐𝑐�𝑢𝑢(𝑡𝑡)� 𝐴 ℎ𝐼𝐼(𝑡𝑡)� 𝐴 𝜌𝜌(𝑡𝑡)�𝑢𝑢(𝑡𝑡) − 𝑟𝑟(𝑡𝑡)� 𝐴 𝜇𝜇(𝑡𝑡)�𝑟𝑟(𝑡𝑡) − 𝑢𝑢(𝑡𝑡)� Then the necessary conditions are: 𝐻𝐻 ��(�)𝑐𝑐�(𝑡𝑡) 𝐴 𝜌𝜌(𝑡𝑡) − 𝜇𝜇(𝑡𝑡) = 0 (1) 𝜌𝜌�(𝑡𝑡) = −𝐻𝐻��(�)ℎ (2) 𝐼𝐼�(𝑡𝑡) = 𝑢𝑢(𝑡𝑡) − 𝑟𝑟(𝑡𝑡) (3) Necessary conditions are also sufficient since the Hamiltonian is convex in both the control (𝑢𝑢) and state (𝐼𝐼) variables. The planning horizon (𝑇𝑇) will be assumed one. On a constraint arc on which 𝐼𝐼(𝑡𝑡) = 0 for a certain period in [0, 1] i.e. 0 < 𝑡𝑡� ≤ 𝑡𝑡 ≤ 𝑡𝑡� < 1 we have; 𝐼𝐼� = 0 = 𝑢𝑢(𝑡𝑡) − 𝑟𝑟(𝑡𝑡) implying that over this period a firm will produce just enough to meet the demand, which is assumed to be nonzero during the planning horizon. a given number. 𝐼𝐼� = 𝑢𝑢(𝑡𝑡) − 𝑟𝑟(𝑡𝑡) 𝐼𝐼(0) = 𝐼𝐼� a given number. 𝑢𝑢(𝑡𝑡) 𝑢 0 𝐼𝐼(𝑡𝑡) 𝑢 0 With the assumptions that the demand (𝑟𝑟 (𝑡𝑡)) is a simple sinusoidal function and the inter- est rate 𝑖𝑖(𝑡𝑡) is time dependent, i.e.: 𝑟𝑟(𝑡𝑡) = 𝐴𝐴 𝐴 𝐴𝐴𝐴𝐴𝑖𝑖𝐴𝐴(𝐴𝐴𝐴𝐴𝐴𝑡𝑡) with no growth, and, 𝑖𝑖(𝑡𝑡) = � 𝑓𝑓(𝐴𝐴)𝑑𝑑𝐴𝐴�� implying that 𝑖𝑖(𝑡𝑡) = 𝑓𝑓𝑡𝑡 if 𝑓𝑓 is constant. Notice also that 𝑖𝑖�(𝑡𝑡) = 𝑓𝑓(𝑡𝑡) where f (t) denotes time dependent interest rate. For ease of analysis we will assume 𝑓𝑓(𝑡𝑡) to be con- stant. The letter k denotes the periodicity of the sinusoidal demand function over the planning period. The only reason for the use sinusoidal demand curve is that it represents cyclicality (a measure of risk) and that it is easier to work with it. The appropriate Lagrangian for this problem is: 𝐻𝐻 = 𝐻𝐻��(�)�𝑐𝑐�𝑢𝑢(𝑡𝑡)� 𝐴 ℎ𝐼𝐼(𝑡𝑡)� 𝐴 𝜌𝜌(𝑡𝑡)�𝑢𝑢(𝑡𝑡) − 𝑟𝑟(𝑡𝑡)� 𝐴 𝜇𝜇(𝑡𝑡)�𝑟𝑟(𝑡𝑡) − 𝑢𝑢(𝑡𝑡)� Then the necessary conditions are: 𝐻𝐻 ��(�)𝑐𝑐�(𝑡𝑡) 𝐴 𝜌𝜌(𝑡𝑡) − 𝜇𝜇(𝑡𝑡) = 0 (1) 𝜌𝜌�(𝑡𝑡) = −𝐻𝐻��(�)ℎ (2) 𝐼𝐼�(𝑡𝑡) = 𝑢𝑢(𝑡𝑡) − 𝑟𝑟(𝑡𝑡) (3) Necessary conditions are also sufficient since the Hamiltonian is convex in both the control (𝑢𝑢) and state (𝐼𝐼) variables. The planning horizon (𝑇𝑇) will be assumed one. On a constraint arc on which 𝐼𝐼(𝑡𝑡) = 0 for a certain period in [0, 1] i.e. 0 < 𝑡𝑡� ≤ 𝑡𝑡 ≤ 𝑡𝑡� < 1 we have; 𝐼𝐼� = 0 = 𝑢𝑢(𝑡𝑡) − 𝑟𝑟(𝑡𝑡) implying that over this period a firm will produce just enough to meet the demand, which is assumed to be nonzero during the planning horizon. With the assumptions that the demand ( r(t) ) is a simple sinusoidal function and the interest rate i(t) is time dependent, i.e.: 𝐼𝐼� = 𝑢𝑢(𝑡𝑡) − 𝑟𝑟(𝑡𝑡) 𝐼𝐼(0) = 𝐼𝐼� a given number. 𝑢𝑢(𝑡𝑡) 𝑢 0 𝐼𝐼(𝑡𝑡) 𝑢 0 With the assumptions that the demand (𝑟𝑟 (𝑡𝑡)) is a simple sinusoidal function and the inter- est rate 𝑖𝑖(𝑡𝑡) is time dependent, i.e.: 𝑟𝑟(𝑡𝑡) = 𝐴𝐴 𝐴 𝐴𝐴𝐴𝐴𝑖𝑖𝐴𝐴(𝐴𝐴𝐴𝐴𝐴𝑡𝑡) with no growth, and, 𝑖𝑖(𝑡𝑡) = � 𝑓𝑓(𝐴𝐴)𝑑𝑑𝐴𝐴�� implying that 𝑖𝑖(𝑡𝑡) = 𝑓𝑓𝑡𝑡 if 𝑓𝑓 is constant. Notice also that 𝑖𝑖�(𝑡𝑡) = 𝑓𝑓(𝑡𝑡) where f (t) denotes time dependent interest rate. For ease of analysis we will assume 𝑓𝑓(𝑡𝑡) to be con- stant. The letter k denotes the periodicity of the sinusoidal demand function over the planning period. The only reason for the use sinusoidal demand curve is that it represents cyclicality (a measure of risk) and that it is easier to work with it. The appropriate Lagrangian for this problem is: 𝐻𝐻 = 𝐻𝐻��(�)�𝑐𝑐�𝑢𝑢(𝑡𝑡)� 𝐴 ℎ𝐼𝐼(𝑡𝑡)� 𝐴 𝜌𝜌(𝑡𝑡)�𝑢𝑢(𝑡𝑡) − 𝑟𝑟(𝑡𝑡)� 𝐴 𝜇𝜇(𝑡𝑡)�𝑟𝑟(𝑡𝑡) − 𝑢𝑢(𝑡𝑡)� Then the necessary conditions are: 𝐻𝐻 ��(�)𝑐𝑐�(𝑡𝑡) 𝐴 𝜌𝜌(𝑡𝑡) − 𝜇𝜇(𝑡𝑡) = 0 (1) 𝜌𝜌�(𝑡𝑡) = −𝐻𝐻��(�)ℎ (2) 𝐼𝐼�(𝑡𝑡) = 𝑢𝑢(𝑡𝑡) − 𝑟𝑟(𝑡𝑡) (3) Necessary conditions are also sufficient since the Hamiltonian is convex in both the control (𝑢𝑢) and state (𝐼𝐼) variables. The planning horizon (𝑇𝑇) will be assumed one. On a constraint arc on which 𝐼𝐼(𝑡𝑡) = 0 for a certain period in [0, 1] i.e. 0 < 𝑡𝑡� ≤ 𝑡𝑡 ≤ 𝑡𝑡� < 1 we have; 𝐼𝐼� = 0 = 𝑢𝑢(𝑡𝑡) − 𝑟𝑟(𝑡𝑡) implying that over this period a firm will produce just enough to meet the demand, which is assumed to be nonzero during the planning horizon. with no growth, and, 𝐼𝐼� = 𝑢𝑢(𝑡𝑡) − 𝑟𝑟(𝑡𝑡) 𝐼𝐼(0) = 𝐼𝐼� a given number. 𝑢𝑢(𝑡𝑡) 𝑢 0 𝐼𝐼(𝑡𝑡) 𝑢 0 With the assumptions that the demand (𝑟𝑟 (𝑡𝑡)) is a simple sinusoidal function and the inter- est rate 𝑖𝑖(𝑡𝑡) is time dependent, i.e.: 𝑟𝑟(𝑡𝑡) = 𝐴𝐴 𝐴 𝐴𝐴𝐴𝐴𝑖𝑖𝐴𝐴(𝐴𝐴𝐴𝐴𝐴𝑡𝑡) with no growth, and, 𝑖𝑖(𝑡𝑡) = � 𝑓𝑓(𝐴𝐴)𝑑𝑑𝐴𝐴�� implying that 𝑖𝑖(𝑡𝑡) = 𝑓𝑓𝑡𝑡 if 𝑓𝑓 is constant. Notice also that 𝑖𝑖�(𝑡𝑡) = 𝑓𝑓(𝑡𝑡) where f (t) denotes time dependent interest rate. For ease of analysis we will assume 𝑓𝑓(𝑡𝑡) to be con- stant. The letter k denotes the periodicity of the sinusoidal demand function over the planning period. The only reason for the use sinusoidal demand curve is that it represents cyclicality (a measure of risk) and that it is easier to work with it. The appropriate Lagrangian for this problem is: 𝐻𝐻 = 𝐻𝐻��(�)�𝑐𝑐�𝑢𝑢(𝑡𝑡)� 𝐴 ℎ𝐼𝐼(𝑡𝑡)� 𝐴 𝜌𝜌(𝑡𝑡)�𝑢𝑢(𝑡𝑡) − 𝑟𝑟(𝑡𝑡)� 𝐴 𝜇𝜇(𝑡𝑡)�𝑟𝑟(𝑡𝑡) − 𝑢𝑢(𝑡𝑡)� Then the necessary conditions are: 𝐻𝐻 ��(�)𝑐𝑐�(𝑡𝑡) 𝐴 𝜌𝜌(𝑡𝑡) − 𝜇𝜇(𝑡𝑡) = 0 (1) 𝜌𝜌�(𝑡𝑡) = −𝐻𝐻��(�)ℎ (2) 𝐼𝐼�(𝑡𝑡) = 𝑢𝑢(𝑡𝑡) − 𝑟𝑟(𝑡𝑡) (3) Necessary conditions are also sufficient since the Hamiltonian is convex in both the control (𝑢𝑢) and state (𝐼𝐼) variables. The planning horizon (𝑇𝑇) will be assumed one. On a constraint arc on which 𝐼𝐼(𝑡𝑡) = 0 for a certain period in [0, 1] i.e. 0 < 𝑡𝑡� ≤ 𝑡𝑡 ≤ 𝑡𝑡� < 1 we have; 𝐼𝐼� = 0 = 𝑢𝑢(𝑡𝑡) − 𝑟𝑟(𝑡𝑡) implying that over this period a firm will produce just enough to meet the demand, which is assumed to be nonzero during the planning horizon. implying that i(t) = f t if f is constant. Notice also that 𝐼𝐼� = 𝑢𝑢(𝑡𝑡) − 𝑟𝑟(𝑡𝑡) 𝐼𝐼(0) = 𝐼𝐼� a given number. 𝑢𝑢(𝑡𝑡) 𝑢 0 𝐼𝐼(𝑡𝑡) 𝑢 0 With the assumptions that the demand (𝑟𝑟 (𝑡𝑡)) is a simple sinusoidal function and the inter- est rate 𝑖𝑖(𝑡𝑡) is time dependent, i.e.: 𝑟𝑟(𝑡𝑡) = 𝐴𝐴 𝐴 𝐴𝐴𝐴𝐴𝑖𝑖𝐴𝐴(𝐴𝐴𝐴𝐴𝐴𝑡𝑡) with no growth, and, 𝑖𝑖(𝑡𝑡) = � 𝑓𝑓(𝐴𝐴)𝑑𝑑𝐴𝐴�� implying that 𝑖𝑖(𝑡𝑡) = 𝑓𝑓𝑡𝑡 if 𝑓𝑓 is constant. Notice also that 𝑖𝑖�(𝑡𝑡) = 𝑓𝑓(𝑡𝑡) where f (t) denotes time dependent interest rate. For ease of analysis we will assume 𝑓𝑓(𝑡𝑡) to be con- stant. The letter k denotes the periodicity of the sinusoidal demand function over the planning period. The only reason for the use sinusoidal demand curve is that it represents cyclicality (a measure of risk) and that it is easier to work with it. The appropriate Lagrangian for this problem is: 𝐻𝐻 = 𝐻𝐻��(�)�𝑐𝑐�𝑢𝑢(𝑡𝑡)� 𝐴 ℎ𝐼𝐼(𝑡𝑡)� 𝐴 𝜌𝜌(𝑡𝑡)�𝑢𝑢(𝑡𝑡) − 𝑟𝑟(𝑡𝑡)� 𝐴 𝜇𝜇(𝑡𝑡)�𝑟𝑟(𝑡𝑡) − 𝑢𝑢(𝑡𝑡)� Then the necessary conditions are: 𝐻𝐻 ��(�)𝑐𝑐�(𝑡𝑡) 𝐴 𝜌𝜌(𝑡𝑡) − 𝜇𝜇(𝑡𝑡) = 0 (1) 𝜌𝜌�(𝑡𝑡) = −𝐻𝐻��(�)ℎ (2) 𝐼𝐼�(𝑡𝑡) = 𝑢𝑢(𝑡𝑡) − 𝑟𝑟(𝑡𝑡) (3) Necessary conditions are also sufficient since the Hamiltonian is convex in both the control (𝑢𝑢) and state (𝐼𝐼) variables. The planning horizon (𝑇𝑇) will be assumed one. On a constraint arc on which 𝐼𝐼(𝑡𝑡) = 0 for a certain period in [0, 1] i.e. 0 < 𝑡𝑡� ≤ 𝑡𝑡 ≤ 𝑡𝑡� < 1 we have; 𝐼𝐼� = 0 = 𝑢𝑢(𝑡𝑡) − 𝑟𝑟(𝑡𝑡) implying that over this period a firm will produce just enough to meet the demand, which is assumed to be nonzero during the planning horizon. where f (t) denotes time dependent interest rate. For ease of analy- sis we will assume f(t) to be constant. The letter k denotes the periodicity of the sinusoidal demand function over the planning period. The only reason for the use sinusoidal demand curve is that it represents cyclicality (a measure of risk) and that it is easier to work with it. The appropriate Lagrangian for this problem is: 𝐼𝐼� = 𝑢𝑢(𝑡𝑡) − 𝑟𝑟(𝑡𝑡) 𝐼𝐼(0) = 𝐼𝐼� a given number. 𝑢𝑢(𝑡𝑡) 𝑢 0 𝐼𝐼(𝑡𝑡) 𝑢 0 With the assumptions that the demand (𝑟𝑟 (𝑡𝑡)) is a simple sinusoidal function and the inter- est rate 𝑖𝑖(𝑡𝑡) is time dependent, i.e.: 𝑟𝑟(𝑡𝑡) = 𝐴𝐴 𝐴 𝐴𝐴𝐴𝐴𝑖𝑖𝐴𝐴(𝐴𝐴𝐴𝐴𝐴𝑡𝑡) with no growth, and, 𝑖𝑖(𝑡𝑡) = � 𝑓𝑓(𝐴𝐴)𝑑𝑑𝐴𝐴�� implying that 𝑖𝑖(𝑡𝑡) = 𝑓𝑓𝑡𝑡 if 𝑓𝑓 is constant. Notice also that 𝑖𝑖�(𝑡𝑡) = 𝑓𝑓(𝑡𝑡) where f (t) denotes time dependent interest rate. For ease of analysis we will assume 𝑓𝑓(𝑡𝑡) to be con- stant. The letter k denotes the periodicity of the sinusoidal demand function over the planning period. The only reason for the use sinusoidal demand curve is that it represents cyclicality (a measure of risk) and that it is easier to work with it. The appropriate Lagrangian for this problem is: 𝐻𝐻 = 𝐻𝐻��(�)�𝑐𝑐�𝑢𝑢(𝑡𝑡)� 𝐴 ℎ𝐼𝐼(𝑡𝑡)� 𝐴 𝜌𝜌(𝑡𝑡)�𝑢𝑢(𝑡𝑡) − 𝑟𝑟(𝑡𝑡)� 𝐴 𝜇𝜇(𝑡𝑡)�𝑟𝑟(𝑡𝑡) − 𝑢𝑢(𝑡𝑡)� Then the necessary conditions are: 𝐻𝐻 ��(�)𝑐𝑐�(𝑡𝑡) 𝐴 𝜌𝜌(𝑡𝑡) − 𝜇𝜇(𝑡𝑡) = 0 (1) 𝜌𝜌�(𝑡𝑡) = −𝐻𝐻��(�)ℎ (2) 𝐼𝐼�(𝑡𝑡) = 𝑢𝑢(𝑡𝑡) − 𝑟𝑟(𝑡𝑡) (3) Necessary conditions are also sufficient since the Hamiltonian is convex in both the control (𝑢𝑢) and state (𝐼𝐼) variables. The planning horizon (𝑇𝑇) will be assumed one. On a constraint arc on which 𝐼𝐼(𝑡𝑡) = 0 for a certain period in [0, 1] i.e. 0 < 𝑡𝑡� ≤ 𝑡𝑡 ≤ 𝑡𝑡� < 1 we have; 𝐼𝐼� = 0 = 𝑢𝑢(𝑡𝑡) − 𝑟𝑟(𝑡𝑡) implying that over this period a firm will produce just enough to meet the demand, which is assumed to be nonzero during the planning horizon. oPtimAl incentives for economic growth… 15 Then the necessary conditions are: 𝐼𝐼� = 𝑢𝑢(𝑡𝑡) − 𝑟𝑟(𝑡𝑡) 𝐼𝐼(0) = 𝐼𝐼� a given number. 𝑢𝑢(𝑡𝑡) 𝑢 0 𝐼𝐼(𝑡𝑡) 𝑢 0 With the assumptions that the demand (𝑟𝑟 (𝑡𝑡)) is a simple sinusoidal function and the inter- est rate 𝑖𝑖(𝑡𝑡) is time dependent, i.e.: 𝑟𝑟(𝑡𝑡) = 𝐴𝐴 𝐴 𝐴𝐴𝐴𝐴𝑖𝑖𝐴𝐴(𝐴𝐴𝐴𝐴𝐴𝑡𝑡) with no growth, and, 𝑖𝑖(𝑡𝑡) = � 𝑓𝑓(𝐴𝐴)𝑑𝑑𝐴𝐴�� implying that 𝑖𝑖(𝑡𝑡) = 𝑓𝑓𝑡𝑡 if 𝑓𝑓 is constant. Notice also that 𝑖𝑖�(𝑡𝑡) = 𝑓𝑓(𝑡𝑡) where f (t) denotes time dependent interest rate. For ease of analysis we will assume 𝑓𝑓(𝑡𝑡) to be con- stant. The letter k denotes the periodicity of the sinusoidal demand function over the planning period. The only reason for the use sinusoidal demand curve is that it represents cyclicality (a measure of risk) and that it is easier to work with it. The appropriate Lagrangian for this problem is: 𝐻𝐻 = 𝐻𝐻��(�)�𝑐𝑐�𝑢𝑢(𝑡𝑡)� 𝐴 ℎ𝐼𝐼(𝑡𝑡)� 𝐴 𝜌𝜌(𝑡𝑡)�𝑢𝑢(𝑡𝑡) − 𝑟𝑟(𝑡𝑡)� 𝐴 𝜇𝜇(𝑡𝑡)�𝑟𝑟(𝑡𝑡) − 𝑢𝑢(𝑡𝑡)� Then the necessary conditions are: 𝐻𝐻 ��(�)𝑐𝑐�(𝑡𝑡) 𝐴 𝜌𝜌(𝑡𝑡) − 𝜇𝜇(𝑡𝑡) = 0 (1) 𝜌𝜌�(𝑡𝑡) = −𝐻𝐻��(�)ℎ (2) 𝐼𝐼�(𝑡𝑡) = 𝑢𝑢(𝑡𝑡) − 𝑟𝑟(𝑡𝑡) (3) Necessary conditions are also sufficient since the Hamiltonian is convex in both the control (𝑢𝑢) and state (𝐼𝐼) variables. The planning horizon (𝑇𝑇) will be assumed one. On a constraint arc on which 𝐼𝐼(𝑡𝑡) = 0 for a certain period in [0, 1] i.e. 0 < 𝑡𝑡� ≤ 𝑡𝑡 ≤ 𝑡𝑡� < 1 we have; 𝐼𝐼� = 0 = 𝑢𝑢(𝑡𝑡) − 𝑟𝑟(𝑡𝑡) implying that over this period a firm will produce just enough to meet the demand, which is assumed to be nonzero during the planning horizon. (1) 𝐼𝐼� = 𝑢𝑢(𝑡𝑡) − 𝑟𝑟(𝑡𝑡) 𝐼𝐼(0) = 𝐼𝐼� a given number. 𝑢𝑢(𝑡𝑡) 𝑢 0 𝐼𝐼(𝑡𝑡) 𝑢 0 With the assumptions that the demand (𝑟𝑟 (𝑡𝑡)) is a simple sinusoidal function and the inter- est rate 𝑖𝑖(𝑡𝑡) is time dependent, i.e.: 𝑟𝑟(𝑡𝑡) = 𝐴𝐴 𝐴 𝐴𝐴𝐴𝐴𝑖𝑖𝐴𝐴(𝐴𝐴𝐴𝐴𝐴𝑡𝑡) with no growth, and, 𝑖𝑖(𝑡𝑡) = � 𝑓𝑓(𝐴𝐴)𝑑𝑑𝐴𝐴�� implying that 𝑖𝑖(𝑡𝑡) = 𝑓𝑓𝑡𝑡 if 𝑓𝑓 is constant. Notice also that 𝑖𝑖�(𝑡𝑡) = 𝑓𝑓(𝑡𝑡) where f (t) denotes time dependent interest rate. For ease of analysis we will assume 𝑓𝑓(𝑡𝑡) to be con- stant. The letter k denotes the periodicity of the sinusoidal demand function over the planning period. The only reason for the use sinusoidal demand curve is that it represents cyclicality (a measure of risk) and that it is easier to work with it. The appropriate Lagrangian for this problem is: 𝐻𝐻 = 𝐻𝐻��(�)�𝑐𝑐�𝑢𝑢(𝑡𝑡)� 𝐴 ℎ𝐼𝐼(𝑡𝑡)� 𝐴 𝜌𝜌(𝑡𝑡)�𝑢𝑢(𝑡𝑡) − 𝑟𝑟(𝑡𝑡)� 𝐴 𝜇𝜇(𝑡𝑡)�𝑟𝑟(𝑡𝑡) − 𝑢𝑢(𝑡𝑡)� Then the necessary conditions are: 𝐻𝐻 ��(�)𝑐𝑐�(𝑡𝑡) 𝐴 𝜌𝜌(𝑡𝑡) − 𝜇𝜇(𝑡𝑡) = 0 (1) 𝜌𝜌�(𝑡𝑡) = −𝐻𝐻��(�)ℎ (2) 𝐼𝐼�(𝑡𝑡) = 𝑢𝑢(𝑡𝑡) − 𝑟𝑟(𝑡𝑡) (3) Necessary conditions are also sufficient since the Hamiltonian is convex in both the control (𝑢𝑢) and state (𝐼𝐼) variables. The planning horizon (𝑇𝑇) will be assumed one. On a constraint arc on which 𝐼𝐼(𝑡𝑡) = 0 for a certain period in [0, 1] i.e. 0 < 𝑡𝑡� ≤ 𝑡𝑡 ≤ 𝑡𝑡� < 1 we have; 𝐼𝐼� = 0 = 𝑢𝑢(𝑡𝑡) − 𝑟𝑟(𝑡𝑡) implying that over this period a firm will produce just enough to meet the demand, which is assumed to be nonzero during the planning horizon. (2) 𝐼𝐼� = 𝑢𝑢(𝑡𝑡) − 𝑟𝑟(𝑡𝑡) 𝐼𝐼(0) = 𝐼𝐼� a given number. 𝑢𝑢(𝑡𝑡) 𝑢 0 𝐼𝐼(𝑡𝑡) 𝑢 0 With the assumptions that the demand (𝑟𝑟 (𝑡𝑡)) is a simple sinusoidal function and the inter- est rate 𝑖𝑖(𝑡𝑡) is time dependent, i.e.: 𝑟𝑟(𝑡𝑡) = 𝐴𝐴 𝐴 𝐴𝐴𝐴𝐴𝑖𝑖𝐴𝐴(𝐴𝐴𝐴𝐴𝐴𝑡𝑡) with no growth, and, 𝑖𝑖(𝑡𝑡) = � 𝑓𝑓(𝐴𝐴)𝑑𝑑𝐴𝐴�� implying that 𝑖𝑖(𝑡𝑡) = 𝑓𝑓𝑡𝑡 if 𝑓𝑓 is constant. Notice also that 𝑖𝑖�(𝑡𝑡) = 𝑓𝑓(𝑡𝑡) where f (t) denotes time dependent interest rate. For ease of analysis we will assume 𝑓𝑓(𝑡𝑡) to be con- stant. The letter k denotes the periodicity of the sinusoidal demand function over the planning period. The only reason for the use sinusoidal demand curve is that it represents cyclicality (a measure of risk) and that it is easier to work with it. The appropriate Lagrangian for this problem is: 𝐻𝐻 = 𝐻𝐻��(�)�𝑐𝑐�𝑢𝑢(𝑡𝑡)� 𝐴 ℎ𝐼𝐼(𝑡𝑡)� 𝐴 𝜌𝜌(𝑡𝑡)�𝑢𝑢(𝑡𝑡) − 𝑟𝑟(𝑡𝑡)� 𝐴 𝜇𝜇(𝑡𝑡)�𝑟𝑟(𝑡𝑡) − 𝑢𝑢(𝑡𝑡)� Then the necessary conditions are: 𝐻𝐻 ��(�)𝑐𝑐�(𝑡𝑡) 𝐴 𝜌𝜌(𝑡𝑡) − 𝜇𝜇(𝑡𝑡) = 0 (1) 𝜌𝜌�(𝑡𝑡) = −𝐻𝐻��(�)ℎ (2) 𝐼𝐼�(𝑡𝑡) = 𝑢𝑢(𝑡𝑡) − 𝑟𝑟(𝑡𝑡) (3) Necessary conditions are also sufficient since the Hamiltonian is convex in both the control (𝑢𝑢) and state (𝐼𝐼) variables. The planning horizon (𝑇𝑇) will be assumed one. On a constraint arc on which 𝐼𝐼(𝑡𝑡) = 0 for a certain period in [0, 1] i.e. 0 < 𝑡𝑡� ≤ 𝑡𝑡 ≤ 𝑡𝑡� < 1 we have; 𝐼𝐼� = 0 = 𝑢𝑢(𝑡𝑡) − 𝑟𝑟(𝑡𝑡) implying that over this period a firm will produce just enough to meet the demand, which is assumed to be nonzero during the planning horizon. (3) Necessary conditions are also sufficient since the Hamiltonian is convex in both the control (u) and state (l) variables. The planning horizon (T ) will be assumed one. On a constraint arc on which l(t) = 0 for a certain period in 𝐼𝐼� = 𝑢𝑢(𝑡𝑡) − 𝑟𝑟(𝑡𝑡) 𝐼𝐼(0) = 𝐼𝐼� a given number. 𝑢𝑢(𝑡𝑡) 𝑢 0 𝐼𝐼(𝑡𝑡) 𝑢 0 With the assumptions that the demand (𝑟𝑟 (𝑡𝑡)) is a simple sinusoidal function and the inter- est rate 𝑖𝑖(𝑡𝑡) is time dependent, i.e.: 𝑟𝑟(𝑡𝑡) = 𝐴𝐴 𝐴 𝐴𝐴𝐴𝐴𝑖𝑖𝐴𝐴(𝐴𝐴𝐴𝐴𝐴𝑡𝑡) with no growth, and, 𝑖𝑖(𝑡𝑡) = � 𝑓𝑓(𝐴𝐴)𝑑𝑑𝐴𝐴�� implying that 𝑖𝑖(𝑡𝑡) = 𝑓𝑓𝑡𝑡 if 𝑓𝑓 is constant. Notice also that 𝑖𝑖�(𝑡𝑡) = 𝑓𝑓(𝑡𝑡) where f (t) denotes time dependent interest rate. For ease of analysis we will assume 𝑓𝑓(𝑡𝑡) to be con- stant. The letter k denotes the periodicity of the sinusoidal demand function over the planning period. The only reason for the use sinusoidal demand curve is that it represents cyclicality (a measure of risk) and that it is easier to work with it. The appropriate Lagrangian for this problem is: 𝐻𝐻 = 𝐻𝐻��(�)�𝑐𝑐�𝑢𝑢(𝑡𝑡)� 𝐴 ℎ𝐼𝐼(𝑡𝑡)� 𝐴 𝜌𝜌(𝑡𝑡)�𝑢𝑢(𝑡𝑡) − 𝑟𝑟(𝑡𝑡)� 𝐴 𝜇𝜇(𝑡𝑡)�𝑟𝑟(𝑡𝑡) − 𝑢𝑢(𝑡𝑡)� Then the necessary conditions are: 𝐻𝐻 ��(�)𝑐𝑐�(𝑡𝑡) 𝐴 𝜌𝜌(𝑡𝑡) − 𝜇𝜇(𝑡𝑡) = 0 (1) 𝜌𝜌�(𝑡𝑡) = −𝐻𝐻��(�)ℎ (2) 𝐼𝐼�(𝑡𝑡) = 𝑢𝑢(𝑡𝑡) − 𝑟𝑟(𝑡𝑡) (3) Necessary conditions are also sufficient since the Hamiltonian is convex in both the control (𝑢𝑢) and state (𝐼𝐼) variables. The planning horizon (𝑇𝑇) will be assumed one. On a constraint arc on which 𝐼𝐼(𝑡𝑡) = 0 for a certain period in [0, 1] i.e. 0 < 𝑡𝑡� ≤ 𝑡𝑡 ≤ 𝑡𝑡� < 1 we have; 𝐼𝐼� = 0 = 𝑢𝑢(𝑡𝑡) − 𝑟𝑟(𝑡𝑡) implying that over this period a firm will produce just enough to meet the demand, which is assumed to be nonzero during the planning horizon. Then using 𝑢𝑢(𝑡𝑡) = 𝑟𝑟(𝑡𝑡), equation (1), and the conditions (the necessary condition to keep 𝐼𝐼(𝑡𝑡) = 0 optimal) developed by Pontryagin, Boltyanskii, Gamkrelidze and Mishechenko (1962), Hestenes (1966), Kamien and Schwartz (2012) among many others, we have; 𝜇𝜇�(𝑡𝑡) ≤ 0 for 𝑡𝑡� ≤ 𝑡𝑡 ≤ 𝑡𝑡� implies: 𝑐𝑐���𝑟𝑟(𝑡𝑡)�𝑟𝑟�(𝑡𝑡) ≤ ℎ 𝐴 𝑓𝑓(𝑡𝑡)𝑐𝑐��𝑟𝑟(𝑡𝑡)� (4) we have; 𝐼𝐼� = 𝑢𝑢(𝑡𝑡) − 𝑟𝑟(𝑡𝑡) 𝐼𝐼(0) = 𝐼𝐼� a given number. 𝑢𝑢(𝑡𝑡) 𝑢 0 𝐼𝐼(𝑡𝑡) 𝑢 0 With the assumptions that the demand (𝑟𝑟 (𝑡𝑡)) is a simple sinusoidal function and the inter- est rate 𝑖𝑖(𝑡𝑡) is time dependent, i.e.: 𝑟𝑟(𝑡𝑡) = 𝐴𝐴 𝐴 𝐴𝐴𝐴𝐴𝑖𝑖𝐴𝐴(𝐴𝐴𝐴𝐴𝐴𝑡𝑡) with no growth, and, 𝑖𝑖(𝑡𝑡) = � 𝑓𝑓(𝐴𝐴)𝑑𝑑𝐴𝐴�� implying that 𝑖𝑖(𝑡𝑡) = 𝑓𝑓𝑡𝑡 if 𝑓𝑓 is constant. Notice also that 𝑖𝑖�(𝑡𝑡) = 𝑓𝑓(𝑡𝑡) where f (t) denotes time dependent interest rate. For ease of analysis we will assume 𝑓𝑓(𝑡𝑡) to be con- stant. The letter k denotes the periodicity of the sinusoidal demand function over the planning period. The only reason for the use sinusoidal demand curve is that it represents cyclicality (a measure of risk) and that it is easier to work with it. The appropriate Lagrangian for this problem is: 𝐻𝐻 = 𝐻𝐻��(�)�𝑐𝑐�𝑢𝑢(𝑡𝑡)� 𝐴 ℎ𝐼𝐼(𝑡𝑡)� 𝐴 𝜌𝜌(𝑡𝑡)�𝑢𝑢(𝑡𝑡) − 𝑟𝑟(𝑡𝑡)� 𝐴 𝜇𝜇(𝑡𝑡)�𝑟𝑟(𝑡𝑡) − 𝑢𝑢(𝑡𝑡)� Then the necessary conditions are: 𝐻𝐻 ��(�)𝑐𝑐�(𝑡𝑡) 𝐴 𝜌𝜌(𝑡𝑡) − 𝜇𝜇(𝑡𝑡) = 0 (1) 𝜌𝜌�(𝑡𝑡) = −𝐻𝐻��(�)ℎ (2) 𝐼𝐼�(𝑡𝑡) = 𝑢𝑢(𝑡𝑡) − 𝑟𝑟(𝑡𝑡) (3) Necessary conditions are also sufficient since the Hamiltonian is convex in both the control (𝑢𝑢) and state (𝐼𝐼) variables. The planning horizon (𝑇𝑇) will be assumed one. On a constraint arc on which 𝐼𝐼(𝑡𝑡) = 0 for a certain period in [0, 1] i.e. 0 < 𝑡𝑡� ≤ 𝑡𝑡 ≤ 𝑡𝑡� < 1 we have; 𝐼𝐼� = 0 = 𝑢𝑢(𝑡𝑡) − 𝑟𝑟(𝑡𝑡) implying that over this period a firm will produce just enough to meet the demand, which is assumed to be nonzero during the planning horizon. implying that over this period a firm will produce just enough to meet the demand, which is assumed to be nonzero during the plan- ning horizon. Then using u(t) = r(t), equation (1), and the conditions (the necessary con- dition to keep I(t) = 0 optimal) developed by Pontryagin, Boltyanskii, Gamkre- lidze and Mishechenko (1962), Hestenes (1966), Kamien and Schwartz (2012) among many others, we have; 𝐼𝐼� = 𝑢𝑢(𝑡𝑡) − 𝑟𝑟(𝑡𝑡) 𝐼𝐼(0) = 𝐼𝐼� a given number. 𝑢𝑢(𝑡𝑡) 𝑢 0 𝐼𝐼(𝑡𝑡) 𝑢 0 With the assumptions that the demand (𝑟𝑟 (𝑡𝑡)) is a simple sinusoidal function and the inter- est rate 𝑖𝑖(𝑡𝑡) is time dependent, i.e.: 𝑟𝑟(𝑡𝑡) = 𝐴𝐴 𝐴 𝐴𝐴𝐴𝐴𝑖𝑖𝐴𝐴(𝐴𝐴𝐴𝐴𝐴𝑡𝑡) with no growth, and, 𝑖𝑖(𝑡𝑡) = � 𝑓𝑓(𝐴𝐴)𝑑𝑑𝐴𝐴�� implying that 𝑖𝑖(𝑡𝑡) = 𝑓𝑓𝑡𝑡 if 𝑓𝑓 is constant. Notice also that 𝑖𝑖�(𝑡𝑡) = 𝑓𝑓(𝑡𝑡) where f (t) denotes time dependent interest rate. For ease of analysis we will assume 𝑓𝑓(𝑡𝑡) to be con- stant. The letter k denotes the periodicity of the sinusoidal demand function over the planning period. The only reason for the use sinusoidal demand curve is that it represents cyclicality (a measure of risk) and that it is easier to work with it. The appropriate Lagrangian for this problem is: 𝐻𝐻 = 𝐻𝐻��(�)�𝑐𝑐�𝑢𝑢(𝑡𝑡)� 𝐴 ℎ𝐼𝐼(𝑡𝑡)� 𝐴 𝜌𝜌(𝑡𝑡)�𝑢𝑢(𝑡𝑡) − 𝑟𝑟(𝑡𝑡)� 𝐴 𝜇𝜇(𝑡𝑡)�𝑟𝑟(𝑡𝑡) − 𝑢𝑢(𝑡𝑡)� Then the necessary conditions are: 𝐻𝐻 ��(�)𝑐𝑐�(𝑡𝑡) 𝐴 𝜌𝜌(𝑡𝑡) − 𝜇𝜇(𝑡𝑡) = 0 (1) 𝜌𝜌�(𝑡𝑡) = −𝐻𝐻��(�)ℎ (2) 𝐼𝐼�(𝑡𝑡) = 𝑢𝑢(𝑡𝑡) − 𝑟𝑟(𝑡𝑡) (3) Necessary conditions are also sufficient since the Hamiltonian is convex in both the control (𝑢𝑢) and state (𝐼𝐼) variables. The planning horizon (𝑇𝑇) will be assumed one. On a constraint arc on which 𝐼𝐼(𝑡𝑡) = 0 for a certain period in [0, 1] i.e. 0 < 𝑡𝑡� ≤ 𝑡𝑡 ≤ 𝑡𝑡� < 1 we have; 𝐼𝐼� = 0 = 𝑢𝑢(𝑡𝑡) − 𝑟𝑟(𝑡𝑡) implying that over this period a firm will produce just enough to meet the demand, which is assumed to be nonzero during the planning horizon. Then using 𝑢𝑢(𝑡𝑡) = 𝑟𝑟(𝑡𝑡), equation (1), and the conditions (the necessary condition to keep 𝐼𝐼(𝑡𝑡) = 0 optimal) developed by Pontryagin, Boltyanskii, Gamkrelidze and Mishechenko (1962), Hestenes (1966), Kamien and Schwartz (2012) among many others, we have; 𝜇𝜇�(𝑡𝑡) ≤ 0 for 𝑡𝑡� ≤ 𝑡𝑡 ≤ 𝑡𝑡� implies: 𝑐𝑐���𝑟𝑟(𝑡𝑡)�𝑟𝑟�(𝑡𝑡) ≤ ℎ 𝐴 𝑓𝑓(𝑡𝑡)𝑐𝑐��𝑟𝑟(𝑡𝑡)� (4) implies: 𝐼𝐼� = 𝑢𝑢(𝑡𝑡) − 𝑟𝑟(𝑡𝑡) 𝐼𝐼(0) = 𝐼𝐼� a given number. 𝑢𝑢(𝑡𝑡) 𝑢 0 𝐼𝐼(𝑡𝑡) 𝑢 0 With the assumptions that the demand (𝑟𝑟 (𝑡𝑡)) is a simple sinusoidal function and the inter- est rate 𝑖𝑖(𝑡𝑡) is time dependent, i.e.: 𝑟𝑟(𝑡𝑡) = 𝐴𝐴 𝐴 𝐴𝐴𝐴𝐴𝑖𝑖𝐴𝐴(𝐴𝐴𝐴𝐴𝐴𝑡𝑡) with no growth, and, 𝑖𝑖(𝑡𝑡) = � 𝑓𝑓(𝐴𝐴)𝑑𝑑𝐴𝐴�� implying that 𝑖𝑖(𝑡𝑡) = 𝑓𝑓𝑡𝑡 if 𝑓𝑓 is constant. Notice also that 𝑖𝑖�(𝑡𝑡) = 𝑓𝑓(𝑡𝑡) where f (t) denotes time dependent interest rate. For ease of analysis we will assume 𝑓𝑓(𝑡𝑡) to be con- stant. The letter k denotes the periodicity of the sinusoidal demand function over the planning period. The only reason for the use sinusoidal demand curve is that it represents cyclicality (a measure of risk) and that it is easier to work with it. The appropriate Lagrangian for this problem is: 𝐻𝐻 = 𝐻𝐻��(�)�𝑐𝑐�𝑢𝑢(𝑡𝑡)� 𝐴 ℎ𝐼𝐼(𝑡𝑡)� 𝐴 𝜌𝜌(𝑡𝑡)�𝑢𝑢(𝑡𝑡) − 𝑟𝑟(𝑡𝑡)� 𝐴 𝜇𝜇(𝑡𝑡)�𝑟𝑟(𝑡𝑡) − 𝑢𝑢(𝑡𝑡)� Then the necessary conditions are: 𝐻𝐻 ��(�)𝑐𝑐�(𝑡𝑡) 𝐴 𝜌𝜌(𝑡𝑡) − 𝜇𝜇(𝑡𝑡) = 0 (1) 𝜌𝜌�(𝑡𝑡) = −𝐻𝐻��(�)ℎ (2) 𝐼𝐼�(𝑡𝑡) = 𝑢𝑢(𝑡𝑡) − 𝑟𝑟(𝑡𝑡) (3) Necessary conditions are also sufficient since the Hamiltonian is convex in both the control (𝑢𝑢) and state (𝐼𝐼) variables. The planning horizon (𝑇𝑇) will be assumed one. On a constraint arc on which 𝐼𝐼(𝑡𝑡) = 0 for a certain period in [0, 1] i.e. 0 < 𝑡𝑡� ≤ 𝑡𝑡 ≤ 𝑡𝑡� < 1 we have; 𝐼𝐼� = 0 = 𝑢𝑢(𝑡𝑡) − 𝑟𝑟(𝑡𝑡) implying that over this period a firm will produce just enough to meet the demand, which is assumed to be nonzero during the planning horizon. Then using 𝑢𝑢(𝑡𝑡) = 𝑟𝑟(𝑡𝑡), equation (1), and the conditions (the necessary condition to keep 𝐼𝐼(𝑡𝑡) = 0 optimal) developed by Pontryagin, Boltyanskii, Gamkrelidze and Mishechenko (1962), Hestenes (1966), Kamien and Schwartz (2012) among many others, we have; 𝜇𝜇�(𝑡𝑡) ≤ 0 for 𝑡𝑡� ≤ 𝑡𝑡 ≤ 𝑡𝑡� implies: 𝑐𝑐���𝑟𝑟(𝑡𝑡)�𝑟𝑟�(𝑡𝑡) ≤ ℎ 𝐴 𝑓𝑓(𝑡𝑡)𝑐𝑐��𝑟𝑟(𝑡𝑡)� (4) (4) where Then using 𝑢𝑢𝑢𝑢𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢, equation (1), and the conditions (the necessary condition to keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 optimal) developed by Pontryagin, Boltyanskii, Gamkrelidze and Mishechenko (1962), Hestenes (1966), Kamien and Schwartz (2012) among many others, we have; 𝜇𝜇�𝑢𝑢𝑢𝑢 ≤ 𝐼 for 𝑢𝑢� ≤ 𝑢𝑢 ≤ 𝑢𝑢� implies: 𝑐𝑐���𝑢𝑢𝑢𝑢𝑢𝑢�𝑢𝑢�𝑢𝑢𝑢𝑢 ≤ ℎ + 𝑓𝑓𝑢𝑢𝑢𝑢𝑐𝑐��𝑢𝑢𝑢𝑢𝑢𝑢� (4) where 𝑓𝑓𝑢𝑢𝑢𝑢 𝑢 𝑖𝑖�𝑢𝑢𝑢𝑢 denotes the interest rate. As long as the condition specified by this inequality is satisfied it will be optimal to keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 implying 𝑢𝑢𝑢𝑢𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢. This means that during the intervals where this inequality is satisfied the firm will produce just enough to satisfy the demand. So, the condition above will synthesize the periods where 𝐼𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 or 𝑢𝑢𝑢𝑢𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢 and the periods where 𝐼𝐼𝑢𝑢𝑢𝑢 𝐼 𝐼. Notice that for the condition above (equation 4) to be kept satisfied depends on ℎ (invento- ry holding cost), 𝑓𝑓𝑢𝑢𝑢𝑢 the interest rates, 𝑐𝑐𝐼𝑢𝑢𝑢𝑢, the marginal cost of production, and 𝑐𝑐𝐼𝐼𝑢𝑢𝑢𝑢 change in the marginal cost. Following conclusions can be made even without specific forms of 𝑐𝑐𝑢𝑢𝑢𝑢, 𝑓𝑓𝑢𝑢𝑢𝑢, 𝑢𝑢𝑢𝑢𝑢𝑢 and 𝑖𝑖𝑢𝑢𝑢𝑢 on the basis of equation (4): 1. The firm will keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 as long as demand is falling or constant i.e. 𝑢𝑢𝐼𝑢𝑢𝑢𝑢 0 im- plying that the firm will choose to produce just enough to meet the demand if demand is falling since both ℎ and 𝑐𝑐𝐼 are positive. 2. The firm will keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 even if demand is increasing 𝑢𝑢𝑢𝐼𝑢𝑢𝑢𝑢 𝑟 𝐼𝑢 but the term ℎ + 𝑓𝑓𝑢𝑢𝑢𝑢𝑐𝑐��𝑢𝑢𝑢𝑢𝑢𝑢� is large enough to keep inequality (4) satisfied. Decreasing interest rates will not be a remedy in this case if they are already low. Firms where 𝑐𝑐𝐼𝐼�𝑢𝑢 𝑢𝑢𝑢𝑢� is low (large and already very efficient firms) will have more difficulty to produce to build up inventories. Currently, this seems to be the case in major economies. 3. Higher interest rates and the use of inefficient technologies (large 𝑐𝑐𝐼𝑢𝑢𝑢𝑢) implies high marginal cost) will induce firms to wait longer for inventory build-up even if 𝑢𝑢𝐼𝑢𝑢𝑢𝑢 is positive. 4. The level of technology used will not affect the behavior of the firms if the interest rates are very low as they are now in many developed nations since the term 𝑓𝑓𝑢𝑢𝑢𝑢𝑐𝑐𝐼𝑢𝑢𝑢𝑢 in denotes the interest rate. As long as the condition specified by this inequality is satisfied it will be optimal to keep I(t) = 0 implying u(t) = r(t). This means that during the inter- vals where this inequality is satisfied the firm will produce just enough to sat- isfy the demand. So, the condition above will synthesize the periods where Then using 𝑢𝑢𝑢𝑢𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢, equation (1), and the conditions (the necessary condition to keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 optimal) developed by Pontryagin, Boltyanskii, Gamkrelidze and Mishechenko (1962), Hestenes (1966), Kamien and Schwartz (2012) among many others, we have; 𝜇𝜇�𝑢𝑢𝑢𝑢 ≤ 𝐼 for 𝑢𝑢� ≤ 𝑢𝑢 ≤ 𝑢𝑢� implies: 𝑐𝑐���𝑢𝑢𝑢𝑢𝑢𝑢�𝑢𝑢�𝑢𝑢𝑢𝑢 ≤ ℎ + 𝑓𝑓𝑢𝑢𝑢𝑢𝑐𝑐��𝑢𝑢𝑢𝑢𝑢𝑢� (4) where 𝑓𝑓𝑢𝑢𝑢𝑢 𝑢 𝑖𝑖�𝑢𝑢𝑢𝑢 denotes the interest rate. As long as the condition specified by this inequality is satisfied it will be optimal to keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 implying 𝑢𝑢𝑢𝑢𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢. This means that during the intervals where this inequality is satisfied the firm will produce just enough to satisfy the demand. So, the condition above will synthesize the periods where 𝐼𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 or 𝑢𝑢𝑢𝑢𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢 and the periods where 𝐼𝐼𝑢𝑢𝑢𝑢 𝐼 𝐼. Notice that for the condition above (equation 4) to be kept satisfied depends on ℎ (invento- ry holding cost), 𝑓𝑓𝑢𝑢𝑢𝑢 the interest rates, 𝑐𝑐𝐼𝑢𝑢𝑢𝑢, the marginal cost of production, and 𝑐𝑐𝐼𝐼𝑢𝑢𝑢𝑢 change in the marginal cost. Following conclusions can be made even without specific forms of 𝑐𝑐𝑢𝑢𝑢𝑢, 𝑓𝑓𝑢𝑢𝑢𝑢, 𝑢𝑢𝑢𝑢𝑢𝑢 and 𝑖𝑖𝑢𝑢𝑢𝑢 on the basis of equation (4): 1. The firm will keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 as long as demand is falling or constant i.e. 𝑢𝑢𝐼𝑢𝑢𝑢𝑢 0 im- plying that the firm will choose to produce just enough to meet the demand if demand is falling since both ℎ and 𝑐𝑐𝐼 are positive. 2. The firm will keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 even if demand is increasing 𝑢𝑢𝑢𝐼𝑢𝑢𝑢𝑢 𝑟 𝐼𝑢 but the term ℎ + 𝑓𝑓𝑢𝑢𝑢𝑢𝑐𝑐��𝑢𝑢𝑢𝑢𝑢𝑢� is large enough to keep inequality (4) satisfied. Decreasing interest rates will not be a remedy in this case if they are already low. Firms where 𝑐𝑐𝐼𝐼�𝑢𝑢 𝑢𝑢𝑢𝑢� is low (large and already very efficient firms) will have more difficulty to produce to build up inventories. Currently, this seems to be the case in major economies. 3. Higher interest rates and the use of inefficient technologies (large 𝑐𝑐𝐼𝑢𝑢𝑢𝑢) implies high marginal cost) will induce firms to wait longer for inventory build-up even if 𝑢𝑢𝐼𝑢𝑢𝑢𝑢 is positive. 4. The level of technology used will not affect the behavior of the firms if the interest rates are very low as they are now in many developed nations since the term 𝑓𝑓𝑢𝑢𝑢𝑢𝑐𝑐𝐼𝑢𝑢𝑢𝑢 in or Then using 𝑢𝑢𝑢𝑢𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢, equation (1), and the conditions (the necessary condition to keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 optimal) developed by Pontryagin, Boltyanskii, Gamkrelidze and Mishechenko (1962), Hestenes (1966), Kamien and Schwartz (2012) among many others, we have; 𝜇𝜇�𝑢𝑢𝑢𝑢 ≤ 𝐼 for 𝑢𝑢� ≤ 𝑢𝑢 ≤ 𝑢𝑢� implies: 𝑐𝑐���𝑢𝑢𝑢𝑢𝑢𝑢�𝑢𝑢�𝑢𝑢𝑢𝑢 ≤ ℎ + 𝑓𝑓𝑢𝑢𝑢𝑢𝑐𝑐��𝑢𝑢𝑢𝑢𝑢𝑢� (4) where 𝑓𝑓𝑢𝑢𝑢𝑢 𝑢 𝑖𝑖�𝑢𝑢𝑢𝑢 denotes the interest rate. As long as the condition specified by this inequality is satisfied it will be optimal to keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 implying 𝑢𝑢𝑢𝑢𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢. This means that during the intervals where this inequality is satisfied the firm will produce just enough to satisfy the demand. So, the condition above will synthesize the periods where 𝐼𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 or 𝑢𝑢𝑢𝑢𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢 and the periods where 𝐼𝐼𝑢𝑢𝑢𝑢 𝐼 𝐼. Notice that for the condition above (equation 4) to be kept satisfied depends on ℎ (invento- ry holding cost), 𝑓𝑓𝑢𝑢𝑢𝑢 the interest rates, 𝑐𝑐𝐼𝑢𝑢𝑢𝑢, the marginal cost of production, and 𝑐𝑐𝐼𝐼𝑢𝑢𝑢𝑢 change in the marginal cost. Following conclusions can be made even without specific forms of 𝑐𝑐𝑢𝑢𝑢𝑢, 𝑓𝑓𝑢𝑢𝑢𝑢, 𝑢𝑢𝑢𝑢𝑢𝑢 and 𝑖𝑖𝑢𝑢𝑢𝑢 on the basis of equation (4): 1. The firm will keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 as long as demand is falling or constant i.e. 𝑢𝑢𝐼𝑢𝑢𝑢𝑢 0 im- plying that the firm will choose to produce just enough to meet the demand if demand is falling since both ℎ and 𝑐𝑐𝐼 are positive. 2. The firm will keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 even if demand is increasing 𝑢𝑢𝑢𝐼𝑢𝑢𝑢𝑢 𝑟 𝐼𝑢 but the term ℎ + 𝑓𝑓𝑢𝑢𝑢𝑢𝑐𝑐��𝑢𝑢𝑢𝑢𝑢𝑢� is large enough to keep inequality (4) satisfied. Decreasing interest rates will not be a remedy in this case if they are already low. Firms where 𝑐𝑐𝐼𝐼�𝑢𝑢 𝑢𝑢𝑢𝑢� is low (large and already very efficient firms) will have more difficulty to produce to build up inventories. Currently, this seems to be the case in major economies. 3. Higher interest rates and the use of inefficient technologies (large 𝑐𝑐𝐼𝑢𝑢𝑢𝑢) implies high marginal cost) will induce firms to wait longer for inventory build-up even if 𝑢𝑢𝐼𝑢𝑢𝑢𝑢 is positive. 4. The level of technology used will not affect the behavior of the firms if the interest rates are very low as they are now in many developed nations since the term 𝑓𝑓𝑢𝑢𝑢𝑢𝑐𝑐𝐼𝑢𝑢𝑢𝑢 in and the periods where Then using 𝑢𝑢𝑢𝑢𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢, equation (1), and the conditions (the necessary condition to keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 optimal) developed by Pontryagin, Boltyanskii, Gamkrelidze and Mishechenko (1962), Hestenes (1966), Kamien and Schwartz (2012) among many others, we have; 𝜇𝜇�𝑢𝑢𝑢𝑢 ≤ 𝐼 for 𝑢𝑢� ≤ 𝑢𝑢 ≤ 𝑢𝑢� implies: 𝑐𝑐���𝑢𝑢𝑢𝑢𝑢𝑢�𝑢𝑢�𝑢𝑢𝑢𝑢 ≤ ℎ + 𝑓𝑓𝑢𝑢𝑢𝑢𝑐𝑐��𝑢𝑢𝑢𝑢𝑢𝑢� (4) where 𝑓𝑓𝑢𝑢𝑢𝑢 𝑢 𝑖𝑖�𝑢𝑢𝑢𝑢 denotes the interest rate. As long as the condition specified by this inequality is satisfied it will be optimal to keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 implying 𝑢𝑢𝑢𝑢𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢. This means that during the intervals where this inequality is satisfied the firm will produce just enough to satisfy the demand. So, the condition above will synthesize the periods where 𝐼𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 or 𝑢𝑢𝑢𝑢𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢 and the periods where 𝐼𝐼𝑢𝑢𝑢𝑢 𝐼 𝐼. Notice that for the condition above (equation 4) to be kept satisfied depends on ℎ (invento- ry holding cost), 𝑓𝑓𝑢𝑢𝑢𝑢 the interest rates, 𝑐𝑐𝐼𝑢𝑢𝑢𝑢, the marginal cost of production, and 𝑐𝑐𝐼𝐼𝑢𝑢𝑢𝑢 change in the marginal cost. Following conclusions can be made even without specific forms of 𝑐𝑐𝑢𝑢𝑢𝑢, 𝑓𝑓𝑢𝑢𝑢𝑢, 𝑢𝑢𝑢𝑢𝑢𝑢 and 𝑖𝑖𝑢𝑢𝑢𝑢 on the basis of equation (4): 1. The firm will keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 as long as demand is falling or constant i.e. 𝑢𝑢𝐼𝑢𝑢𝑢𝑢 0 im- plying that the firm will choose to produce just enough to meet the demand if demand is falling since both ℎ and 𝑐𝑐𝐼 are positive. 2. The firm will keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 even if demand is increasing 𝑢𝑢𝑢𝐼𝑢𝑢𝑢𝑢 𝑟 𝐼𝑢 but the term ℎ + 𝑓𝑓𝑢𝑢𝑢𝑢𝑐𝑐��𝑢𝑢𝑢𝑢𝑢𝑢� is large enough to keep inequality (4) satisfied. Decreasing interest rates will not be a remedy in this case if they are already low. Firms where 𝑐𝑐𝐼𝐼�𝑢𝑢 𝑢𝑢𝑢𝑢� is low (large and already very efficient firms) will have more difficulty to produce to build up inventories. Currently, this seems to be the case in major economies. 3. Higher interest rates and the use of inefficient technologies (large 𝑐𝑐𝐼𝑢𝑢𝑢𝑢) implies high marginal cost) will induce firms to wait longer for inventory build-up even if 𝑢𝑢𝐼𝑢𝑢𝑢𝑢 is positive. 4. The level of technology used will not affect the behavior of the firms if the interest rates are very low as they are now in many developed nations since the term 𝑓𝑓𝑢𝑢𝑢𝑢𝑐𝑐𝐼𝑢𝑢𝑢𝑢 in Notice that for the condition above (equation 4) to be kept satisfied depends on h (inventory holding cost), f(t) the interest rates, Then using 𝑢𝑢𝑢𝑢𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢, equation (1), and the conditions (the necessary condition to keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 optimal) developed by Pontryagin, Boltyanskii, Gamkrelidze and Mishechenko (1962), Hestenes (1966), Kamien and Schwartz (2012) among many others, we have; 𝜇𝜇�𝑢𝑢𝑢𝑢 ≤ 𝐼 for 𝑢𝑢� ≤ 𝑢𝑢 ≤ 𝑢𝑢� implies: 𝑐𝑐���𝑢𝑢𝑢𝑢𝑢𝑢�𝑢𝑢�𝑢𝑢𝑢𝑢 ≤ ℎ + 𝑓𝑓𝑢𝑢𝑢𝑢𝑐𝑐��𝑢𝑢𝑢𝑢𝑢𝑢� (4) where 𝑓𝑓𝑢𝑢𝑢𝑢 𝑢 𝑖𝑖�𝑢𝑢𝑢𝑢 denotes the interest rate. As long as the condition specified by this inequality is satisfied it will be optimal to keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 implying 𝑢𝑢𝑢𝑢𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢. This means that during the intervals where this inequality is satisfied the firm will produce just enough to satisfy the demand. So, the condition above will synthesize the periods where 𝐼𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 or 𝑢𝑢𝑢𝑢𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢 and the periods where 𝐼𝐼𝑢𝑢𝑢𝑢 𝐼 𝐼. Notice that for the condition above (equation 4) to be kept satisfied depends on ℎ (invento- ry holding cost), 𝑓𝑓𝑢𝑢𝑢𝑢 the interest rates, 𝑐𝑐𝐼𝑢𝑢𝑢𝑢, the marginal cost of production, and 𝑐𝑐𝐼𝐼𝑢𝑢𝑢𝑢 change in the marginal cost. Following conclusions can be made even without specific forms of 𝑐𝑐𝑢𝑢𝑢𝑢, 𝑓𝑓𝑢𝑢𝑢𝑢, 𝑢𝑢𝑢𝑢𝑢𝑢 and 𝑖𝑖𝑢𝑢𝑢𝑢 on the basis of equation (4): 1. The firm will keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 as long as demand is falling or constant i.e. 𝑢𝑢𝐼𝑢𝑢𝑢𝑢 0 im- plying that the firm will choose to produce just enough to meet the demand if demand is falling since both ℎ and 𝑐𝑐𝐼 are positive. 2. The firm will keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 even if demand is increasing 𝑢𝑢𝑢𝐼𝑢𝑢𝑢𝑢 𝑟 𝐼𝑢 but the term ℎ + 𝑓𝑓𝑢𝑢𝑢𝑢𝑐𝑐��𝑢𝑢𝑢𝑢𝑢𝑢� is large enough to keep inequality (4) satisfied. Decreasing interest rates will not be a remedy in this case if they are already low. Firms where 𝑐𝑐𝐼𝐼�𝑢𝑢 𝑢𝑢𝑢𝑢� is low (large and already very efficient firms) will have more difficulty to produce to build up inventories. Currently, this seems to be the case in major economies. 3. Higher interest rates and the use of inefficient technologies (large 𝑐𝑐𝐼𝑢𝑢𝑢𝑢) implies high marginal cost) will induce firms to wait longer for inventory build-up even if 𝑢𝑢𝐼𝑢𝑢𝑢𝑢 is positive. 4. The level of technology used will not affect the behavior of the firms if the interest rates are very low as they are now in many developed nations since the term 𝑓𝑓𝑢𝑢𝑢𝑢𝑐𝑐𝐼𝑢𝑢𝑢𝑢 in , the marginal cost of production, and Then using 𝑢𝑢𝑢𝑢𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢, equation (1), and the conditions (the necessary condition to keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 optimal) developed by Pontryagin, Boltyanskii, Gamkrelidze and Mishechenko (1962), Hestenes (1966), Kamien and Schwartz (2012) among many others, we have; 𝜇𝜇�𝑢𝑢𝑢𝑢 ≤ 𝐼 for 𝑢𝑢� ≤ 𝑢𝑢 ≤ 𝑢𝑢� implies: 𝑐𝑐���𝑢𝑢𝑢𝑢𝑢𝑢�𝑢𝑢�𝑢𝑢𝑢𝑢 ≤ ℎ + 𝑓𝑓𝑢𝑢𝑢𝑢𝑐𝑐��𝑢𝑢𝑢𝑢𝑢𝑢� (4) where 𝑓𝑓𝑢𝑢𝑢𝑢 𝑢 𝑖𝑖�𝑢𝑢𝑢𝑢 denotes the interest rate. As long as the condition specified by this inequality is satisfied it will be optimal to keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 implying 𝑢𝑢𝑢𝑢𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢. This means that during the intervals where this inequality is satisfied the firm will produce just enough to satisfy the demand. So, the condition above will synthesize the periods where 𝐼𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 or 𝑢𝑢𝑢𝑢𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢 and the periods where 𝐼𝐼𝑢𝑢𝑢𝑢 𝐼 𝐼. Notice that for the condition above (equation 4) to be kept satisfied depends on ℎ (invento- ry holding cost), 𝑓𝑓𝑢𝑢𝑢𝑢 the interest rates, 𝑐𝑐𝐼𝑢𝑢𝑢𝑢, the marginal cost of production, and 𝑐𝑐𝐼𝐼𝑢𝑢𝑢𝑢 change in the marginal cost. Following conclusions can be made even without specific forms of 𝑐𝑐𝑢𝑢𝑢𝑢, 𝑓𝑓𝑢𝑢𝑢𝑢, 𝑢𝑢𝑢𝑢𝑢𝑢 and 𝑖𝑖𝑢𝑢𝑢𝑢 on the basis of equation (4): 1. The firm will keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 as long as demand is falling or constant i.e. 𝑢𝑢𝐼𝑢𝑢𝑢𝑢 0 im- plying that the firm will choose to produce just enough to meet the demand if demand is falling since both ℎ and 𝑐𝑐𝐼 are positive. 2. The firm will keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 even if demand is increasing 𝑢𝑢𝑢𝐼𝑢𝑢𝑢𝑢 𝑟 𝐼𝑢 but the term ℎ + 𝑓𝑓𝑢𝑢𝑢𝑢𝑐𝑐��𝑢𝑢𝑢𝑢𝑢𝑢� is large enough to keep inequality (4) satisfied. Decreasing interest rates will not be a remedy in this case if they are already low. Firms where 𝑐𝑐𝐼𝐼�𝑢𝑢 𝑢𝑢𝑢𝑢� is low (large and already very efficient firms) will have more difficulty to produce to build up inventories. Currently, this seems to be the case in major economies. 3. Higher interest rates and the use of inefficient technologies (large 𝑐𝑐𝐼𝑢𝑢𝑢𝑢) implies high marginal cost) will induce firms to wait longer for inventory build-up even if 𝑢𝑢𝐼𝑢𝑢𝑢𝑢 is positive. 4. The level of technology used will not affect the behavior of the firms if the interest rates are very low as they are now in many developed nations since the term 𝑓𝑓𝑢𝑢𝑢𝑢𝑐𝑐𝐼𝑢𝑢𝑢𝑢 in change in the marginal cost. Following conclusions can be made even without specific forms of c(t), f(t), r(t) and i(t) on the basis of equation (4): Mustafa Akan, Natalia Konovalova16 1. The firm will keep l(t) = 0 as long as demand is falling or constant i.e. Then using 𝑢𝑢𝑢𝑢𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢, equation (1), and the conditions (the necessary condition to keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 optimal) developed by Pontryagin, Boltyanskii, Gamkrelidze and Mishechenko (1962), Hestenes (1966), Kamien and Schwartz (2012) among many others, we have; 𝜇𝜇�𝑢𝑢𝑢𝑢 ≤ 𝐼 for 𝑢𝑢� ≤ 𝑢𝑢 ≤ 𝑢𝑢� implies: 𝑐𝑐���𝑢𝑢𝑢𝑢𝑢𝑢�𝑢𝑢�𝑢𝑢𝑢𝑢 ≤ ℎ + 𝑓𝑓𝑢𝑢𝑢𝑢𝑐𝑐��𝑢𝑢𝑢𝑢𝑢𝑢� (4) where 𝑓𝑓𝑢𝑢𝑢𝑢 𝑢 𝑖𝑖�𝑢𝑢𝑢𝑢 denotes the interest rate. As long as the condition specified by this inequality is satisfied it will be optimal to keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 implying 𝑢𝑢𝑢𝑢𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢. This means that during the intervals where this inequality is satisfied the firm will produce just enough to satisfy the demand. So, the condition above will synthesize the periods where 𝐼𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 or 𝑢𝑢𝑢𝑢𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢 and the periods where 𝐼𝐼𝑢𝑢𝑢𝑢 𝐼 𝐼. Notice that for the condition above (equation 4) to be kept satisfied depends on ℎ (invento- ry holding cost), 𝑓𝑓𝑢𝑢𝑢𝑢 the interest rates, 𝑐𝑐𝐼𝑢𝑢𝑢𝑢, the marginal cost of production, and 𝑐𝑐𝐼𝐼𝑢𝑢𝑢𝑢 change in the marginal cost. Following conclusions can be made even without specific forms of 𝑐𝑐𝑢𝑢𝑢𝑢, 𝑓𝑓𝑢𝑢𝑢𝑢, 𝑢𝑢𝑢𝑢𝑢𝑢 and 𝑖𝑖𝑢𝑢𝑢𝑢 on the basis of equation (4): 1. The firm will keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 as long as demand is falling or constant i.e. 𝑢𝑢𝐼𝑢𝑢𝑢𝑢 0 im- plying that the firm will choose to produce just enough to meet the demand if demand is falling since both ℎ and 𝑐𝑐𝐼 are positive. 2. The firm will keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 even if demand is increasing 𝑢𝑢𝑢𝐼𝑢𝑢𝑢𝑢 𝑟 𝐼𝑢 but the term ℎ + 𝑓𝑓𝑢𝑢𝑢𝑢𝑐𝑐��𝑢𝑢𝑢𝑢𝑢𝑢� is large enough to keep inequality (4) satisfied. Decreasing interest rates will not be a remedy in this case if they are already low. Firms where 𝑐𝑐𝐼𝐼�𝑢𝑢 𝑢𝑢𝑢𝑢� is low (large and already very efficient firms) will have more difficulty to produce to build up inventories. Currently, this seems to be the case in major economies. 3. Higher interest rates and the use of inefficient technologies (large 𝑐𝑐𝐼𝑢𝑢𝑢𝑢) implies high marginal cost) will induce firms to wait longer for inventory build-up even if 𝑢𝑢𝐼𝑢𝑢𝑢𝑢 is positive. 4. The level of technology used will not affect the behavior of the firms if the interest rates are very low as they are now in many developed nations since the term 𝑓𝑓𝑢𝑢𝑢𝑢𝑐𝑐𝐼𝑢𝑢𝑢𝑢 in implying that the firm will choose to produce just enough to meet the demand if demand is falling since both Then using 𝑢𝑢𝑢𝑢𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢, equation (1), and the conditions (the necessary condition to keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 optimal) developed by Pontryagin, Boltyanskii, Gamkrelidze and Mishechenko (1962), Hestenes (1966), Kamien and Schwartz (2012) among many others, we have; 𝜇𝜇�𝑢𝑢𝑢𝑢 ≤ 𝐼 for 𝑢𝑢� ≤ 𝑢𝑢 ≤ 𝑢𝑢� implies: 𝑐𝑐���𝑢𝑢𝑢𝑢𝑢𝑢�𝑢𝑢�𝑢𝑢𝑢𝑢 ≤ ℎ + 𝑓𝑓𝑢𝑢𝑢𝑢𝑐𝑐��𝑢𝑢𝑢𝑢𝑢𝑢� (4) where 𝑓𝑓𝑢𝑢𝑢𝑢 𝑢 𝑖𝑖�𝑢𝑢𝑢𝑢 denotes the interest rate. As long as the condition specified by this inequality is satisfied it will be optimal to keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 implying 𝑢𝑢𝑢𝑢𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢. This means that during the intervals where this inequality is satisfied the firm will produce just enough to satisfy the demand. So, the condition above will synthesize the periods where 𝐼𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 or 𝑢𝑢𝑢𝑢𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢 and the periods where 𝐼𝐼𝑢𝑢𝑢𝑢 𝐼 𝐼. Notice that for the condition above (equation 4) to be kept satisfied depends on ℎ (invento- ry holding cost), 𝑓𝑓𝑢𝑢𝑢𝑢 the interest rates, 𝑐𝑐𝐼𝑢𝑢𝑢𝑢, the marginal cost of production, and 𝑐𝑐𝐼𝐼𝑢𝑢𝑢𝑢 change in the marginal cost. Following conclusions can be made even without specific forms of 𝑐𝑐𝑢𝑢𝑢𝑢, 𝑓𝑓𝑢𝑢𝑢𝑢, 𝑢𝑢𝑢𝑢𝑢𝑢 and 𝑖𝑖𝑢𝑢𝑢𝑢 on the basis of equation (4): 1. The firm will keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 as long as demand is falling or constant i.e. 𝑢𝑢𝐼𝑢𝑢𝑢𝑢 0 im- plying that the firm will choose to produce just enough to meet the demand if demand is falling since both ℎ and 𝑐𝑐𝐼 are positive. 2. The firm will keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 even if demand is increasing 𝑢𝑢𝑢𝐼𝑢𝑢𝑢𝑢 𝑟 𝐼𝑢 but the term ℎ + 𝑓𝑓𝑢𝑢𝑢𝑢𝑐𝑐��𝑢𝑢𝑢𝑢𝑢𝑢� is large enough to keep inequality (4) satisfied. Decreasing interest rates will not be a remedy in this case if they are already low. Firms where 𝑐𝑐𝐼𝐼�𝑢𝑢 𝑢𝑢𝑢𝑢� is low (large and already very efficient firms) will have more difficulty to produce to build up inventories. Currently, this seems to be the case in major economies. 3. Higher interest rates and the use of inefficient technologies (large 𝑐𝑐𝐼𝑢𝑢𝑢𝑢) implies high marginal cost) will induce firms to wait longer for inventory build-up even if 𝑢𝑢𝐼𝑢𝑢𝑢𝑢 is positive. 4. The level of technology used will not affect the behavior of the firms if the interest rates are very low as they are now in many developed nations since the term 𝑓𝑓𝑢𝑢𝑢𝑢𝑐𝑐𝐼𝑢𝑢𝑢𝑢 in and Then using 𝑢𝑢𝑢𝑢𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢, equation (1), and the conditions (the necessary condition to keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 optimal) developed by Pontryagin, Boltyanskii, Gamkrelidze and Mishechenko (1962), Hestenes (1966), Kamien and Schwartz (2012) among many others, we have; 𝜇𝜇�𝑢𝑢𝑢𝑢 ≤ 𝐼 for 𝑢𝑢� ≤ 𝑢𝑢 ≤ 𝑢𝑢� implies: 𝑐𝑐���𝑢𝑢𝑢𝑢𝑢𝑢�𝑢𝑢�𝑢𝑢𝑢𝑢 ≤ ℎ + 𝑓𝑓𝑢𝑢𝑢𝑢𝑐𝑐��𝑢𝑢𝑢𝑢𝑢𝑢� (4) where 𝑓𝑓𝑢𝑢𝑢𝑢 𝑢 𝑖𝑖�𝑢𝑢𝑢𝑢 denotes the interest rate. As long as the condition specified by this inequality is satisfied it will be optimal to keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 implying 𝑢𝑢𝑢𝑢𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢. This means that during the intervals where this inequality is satisfied the firm will produce just enough to satisfy the demand. So, the condition above will synthesize the periods where 𝐼𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 or 𝑢𝑢𝑢𝑢𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢 and the periods where 𝐼𝐼𝑢𝑢𝑢𝑢 𝐼 𝐼. Notice that for the condition above (equation 4) to be kept satisfied depends on ℎ (invento- ry holding cost), 𝑓𝑓𝑢𝑢𝑢𝑢 the interest rates, 𝑐𝑐𝐼𝑢𝑢𝑢𝑢, the marginal cost of production, and 𝑐𝑐𝐼𝐼𝑢𝑢𝑢𝑢 change in the marginal cost. Following conclusions can be made even without specific forms of 𝑐𝑐𝑢𝑢𝑢𝑢, 𝑓𝑓𝑢𝑢𝑢𝑢, 𝑢𝑢𝑢𝑢𝑢𝑢 and 𝑖𝑖𝑢𝑢𝑢𝑢 on the basis of equation (4): 1. The firm will keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 as long as demand is falling or constant i.e. 𝑢𝑢𝐼𝑢𝑢𝑢𝑢 0 im- plying that the firm will choose to produce just enough to meet the demand if demand is falling since both ℎ and 𝑐𝑐𝐼 are positive. 2. The firm will keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 even if demand is increasing 𝑢𝑢𝑢𝐼𝑢𝑢𝑢𝑢 𝑟 𝐼𝑢 but the term ℎ + 𝑓𝑓𝑢𝑢𝑢𝑢𝑐𝑐��𝑢𝑢𝑢𝑢𝑢𝑢� is large enough to keep inequality (4) satisfied. Decreasing interest rates will not be a remedy in this case if they are already low. Firms where 𝑐𝑐𝐼𝐼�𝑢𝑢 𝑢𝑢𝑢𝑢� is low (large and already very efficient firms) will have more difficulty to produce to build up inventories. Currently, this seems to be the case in major economies. 3. Higher interest rates and the use of inefficient technologies (large 𝑐𝑐𝐼𝑢𝑢𝑢𝑢) implies high marginal cost) will induce firms to wait longer for inventory build-up even if 𝑢𝑢𝐼𝑢𝑢𝑢𝑢 is positive. 4. The level of technology used will not affect the behavior of the firms if the interest rates are very low as they are now in many developed nations since the term 𝑓𝑓𝑢𝑢𝑢𝑢𝑐𝑐𝐼𝑢𝑢𝑢𝑢 in are positive. 2. The firm will keep l(t) = 0 even if demand is increasing Then using 𝑢𝑢𝑢𝑢𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢, equation (1), and the conditions (the necessary condition to keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 optimal) developed by Pontryagin, Boltyanskii, Gamkrelidze and Mishechenko (1962), Hestenes (1966), Kamien and Schwartz (2012) among many others, we have; 𝜇𝜇�𝑢𝑢𝑢𝑢 ≤ 𝐼 for 𝑢𝑢� ≤ 𝑢𝑢 ≤ 𝑢𝑢� implies: 𝑐𝑐���𝑢𝑢𝑢𝑢𝑢𝑢�𝑢𝑢�𝑢𝑢𝑢𝑢 ≤ ℎ + 𝑓𝑓𝑢𝑢𝑢𝑢𝑐𝑐��𝑢𝑢𝑢𝑢𝑢𝑢� (4) where 𝑓𝑓𝑢𝑢𝑢𝑢 𝑢 𝑖𝑖�𝑢𝑢𝑢𝑢 denotes the interest rate. As long as the condition specified by this inequality is satisfied it will be optimal to keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 implying 𝑢𝑢𝑢𝑢𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢. This means that during the intervals where this inequality is satisfied the firm will produce just enough to satisfy the demand. So, the condition above will synthesize the periods where 𝐼𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 or 𝑢𝑢𝑢𝑢𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢 and the periods where 𝐼𝐼𝑢𝑢𝑢𝑢 𝐼 𝐼. Notice that for the condition above (equation 4) to be kept satisfied depends on ℎ (invento- ry holding cost), 𝑓𝑓𝑢𝑢𝑢𝑢 the interest rates, 𝑐𝑐𝐼𝑢𝑢𝑢𝑢, the marginal cost of production, and 𝑐𝑐𝐼𝐼𝑢𝑢𝑢𝑢 change in the marginal cost. Following conclusions can be made even without specific forms of 𝑐𝑐𝑢𝑢𝑢𝑢, 𝑓𝑓𝑢𝑢𝑢𝑢, 𝑢𝑢𝑢𝑢𝑢𝑢 and 𝑖𝑖𝑢𝑢𝑢𝑢 on the basis of equation (4): 1. The firm will keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 as long as demand is falling or constant i.e. 𝑢𝑢𝐼𝑢𝑢𝑢𝑢 0 im- plying that the firm will choose to produce just enough to meet the demand if demand is falling since both ℎ and 𝑐𝑐𝐼 are positive. 2. The firm will keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 even if demand is increasing 𝑢𝑢𝑢𝐼𝑢𝑢𝑢𝑢 𝑟 𝐼𝑢 but the term ℎ + 𝑓𝑓𝑢𝑢𝑢𝑢𝑐𝑐��𝑢𝑢𝑢𝑢𝑢𝑢� is large enough to keep inequality (4) satisfied. Decreasing interest rates will not be a remedy in this case if they are already low. Firms where 𝑐𝑐𝐼𝐼�𝑢𝑢 𝑢𝑢𝑢𝑢� is low (large and already very efficient firms) will have more difficulty to produce to build up inventories. Currently, this seems to be the case in major economies. 3. Higher interest rates and the use of inefficient technologies (large 𝑐𝑐𝐼𝑢𝑢𝑢𝑢) implies high marginal cost) will induce firms to wait longer for inventory build-up even if 𝑢𝑢𝐼𝑢𝑢𝑢𝑢 is positive. 4. The level of technology used will not affect the behavior of the firms if the interest rates are very low as they are now in many developed nations since the term 𝑓𝑓𝑢𝑢𝑢𝑢𝑐𝑐𝐼𝑢𝑢𝑢𝑢 in but the term Then using 𝑢𝑢𝑢𝑢𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢, equation (1), and the conditions (the necessary condition to keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 optimal) developed by Pontryagin, Boltyanskii, Gamkrelidze and Mishechenko (1962), Hestenes (1966), Kamien and Schwartz (2012) among many others, we have; 𝜇𝜇�𝑢𝑢𝑢𝑢 ≤ 𝐼 for 𝑢𝑢� ≤ 𝑢𝑢 ≤ 𝑢𝑢� implies: 𝑐𝑐���𝑢𝑢𝑢𝑢𝑢𝑢�𝑢𝑢�𝑢𝑢𝑢𝑢 ≤ ℎ + 𝑓𝑓𝑢𝑢𝑢𝑢𝑐𝑐��𝑢𝑢𝑢𝑢𝑢𝑢� (4) where 𝑓𝑓𝑢𝑢𝑢𝑢 𝑢 𝑖𝑖�𝑢𝑢𝑢𝑢 denotes the interest rate. As long as the condition specified by this inequality is satisfied it will be optimal to keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 implying 𝑢𝑢𝑢𝑢𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢. This means that during the intervals where this inequality is satisfied the firm will produce just enough to satisfy the demand. So, the condition above will synthesize the periods where 𝐼𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 or 𝑢𝑢𝑢𝑢𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢 and the periods where 𝐼𝐼𝑢𝑢𝑢𝑢 𝐼 𝐼. Notice that for the condition above (equation 4) to be kept satisfied depends on ℎ (invento- ry holding cost), 𝑓𝑓𝑢𝑢𝑢𝑢 the interest rates, 𝑐𝑐𝐼𝑢𝑢𝑢𝑢, the marginal cost of production, and 𝑐𝑐𝐼𝐼𝑢𝑢𝑢𝑢 change in the marginal cost. Following conclusions can be made even without specific forms of 𝑐𝑐𝑢𝑢𝑢𝑢, 𝑓𝑓𝑢𝑢𝑢𝑢, 𝑢𝑢𝑢𝑢𝑢𝑢 and 𝑖𝑖𝑢𝑢𝑢𝑢 on the basis of equation (4): 1. The firm will keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 as long as demand is falling or constant i.e. 𝑢𝑢𝐼𝑢𝑢𝑢𝑢 0 im- plying that the firm will choose to produce just enough to meet the demand if demand is falling since both ℎ and 𝑐𝑐𝐼 are positive. 2. The firm will keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 even if demand is increasing 𝑢𝑢𝑢𝐼𝑢𝑢𝑢𝑢 𝑟 𝐼𝑢 but the term ℎ + 𝑓𝑓𝑢𝑢𝑢𝑢𝑐𝑐��𝑢𝑢𝑢𝑢𝑢𝑢� is large enough to keep inequality (4) satisfied. Decreasing interest rates will not be a remedy in this case if they are already low. Firms where 𝑐𝑐𝐼𝐼�𝑢𝑢 𝑢𝑢𝑢𝑢� is low (large and already very efficient firms) will have more difficulty to produce to build up inventories. Currently, this seems to be the case in major economies. 3. Higher interest rates and the use of inefficient technologies (large 𝑐𝑐𝐼𝑢𝑢𝑢𝑢) implies high marginal cost) will induce firms to wait longer for inventory build-up even if 𝑢𝑢𝐼𝑢𝑢𝑢𝑢 is positive. 4. The level of technology used will not affect the behavior of the firms if the interest rates are very low as they are now in many developed nations since the term 𝑓𝑓𝑢𝑢𝑢𝑢𝑐𝑐𝐼𝑢𝑢𝑢𝑢 in is large enough to keep inequality (4) satisfied. Decreasing interest rates will not be a remedy in this case if they are al- ready low. Firms where Then using 𝑢𝑢𝑢𝑢𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢, equation (1), and the conditions (the necessary condition to keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 optimal) developed by Pontryagin, Boltyanskii, Gamkrelidze and Mishechenko (1962), Hestenes (1966), Kamien and Schwartz (2012) among many others, we have; 𝜇𝜇�𝑢𝑢𝑢𝑢 ≤ 𝐼 for 𝑢𝑢� ≤ 𝑢𝑢 ≤ 𝑢𝑢� implies: 𝑐𝑐���𝑢𝑢𝑢𝑢𝑢𝑢�𝑢𝑢�𝑢𝑢𝑢𝑢 ≤ ℎ + 𝑓𝑓𝑢𝑢𝑢𝑢𝑐𝑐��𝑢𝑢𝑢𝑢𝑢𝑢� (4) where 𝑓𝑓𝑢𝑢𝑢𝑢 𝑢 𝑖𝑖�𝑢𝑢𝑢𝑢 denotes the interest rate. As long as the condition specified by this inequality is satisfied it will be optimal to keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 implying 𝑢𝑢𝑢𝑢𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢. This means that during the intervals where this inequality is satisfied the firm will produce just enough to satisfy the demand. So, the condition above will synthesize the periods where 𝐼𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 or 𝑢𝑢𝑢𝑢𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢 and the periods where 𝐼𝐼𝑢𝑢𝑢𝑢 𝐼 𝐼. Notice that for the condition above (equation 4) to be kept satisfied depends on ℎ (invento- ry holding cost), 𝑓𝑓𝑢𝑢𝑢𝑢 the interest rates, 𝑐𝑐𝐼𝑢𝑢𝑢𝑢, the marginal cost of production, and 𝑐𝑐𝐼𝐼𝑢𝑢𝑢𝑢 change in the marginal cost. Following conclusions can be made even without specific forms of 𝑐𝑐𝑢𝑢𝑢𝑢, 𝑓𝑓𝑢𝑢𝑢𝑢, 𝑢𝑢𝑢𝑢𝑢𝑢 and 𝑖𝑖𝑢𝑢𝑢𝑢 on the basis of equation (4): 1. The firm will keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 as long as demand is falling or constant i.e. 𝑢𝑢𝐼𝑢𝑢𝑢𝑢 0 im- plying that the firm will choose to produce just enough to meet the demand if demand is falling since both ℎ and 𝑐𝑐𝐼 are positive. 2. The firm will keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 even if demand is increasing 𝑢𝑢𝑢𝐼𝑢𝑢𝑢𝑢 𝑟 𝐼𝑢 but the term ℎ + 𝑓𝑓𝑢𝑢𝑢𝑢𝑐𝑐��𝑢𝑢𝑢𝑢𝑢𝑢� is large enough to keep inequality (4) satisfied. Decreasing interest rates will not be a remedy in this case if they are already low. Firms where 𝑐𝑐𝐼𝐼�𝑢𝑢 𝑢𝑢𝑢𝑢� is low (large and already very efficient firms) will have more difficulty to produce to build up inventories. Currently, this seems to be the case in major economies. 3. Higher interest rates and the use of inefficient technologies (large 𝑐𝑐𝐼𝑢𝑢𝑢𝑢) implies high marginal cost) will induce firms to wait longer for inventory build-up even if 𝑢𝑢𝐼𝑢𝑢𝑢𝑢 is positive. 4. The level of technology used will not affect the behavior of the firms if the interest rates are very low as they are now in many developed nations since the term 𝑓𝑓𝑢𝑢𝑢𝑢𝑐𝑐𝐼𝑢𝑢𝑢𝑢 in is low (large and already very efficient firms) will have more difficulty to produce to build up inventories. Cur- rently, this seems to be the case in major economies. 3. Higher interest rates and the use of inefficient technologies (large Then using 𝑢𝑢𝑢𝑢𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢, equation (1), and the conditions (the necessary condition to keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 optimal) developed by Pontryagin, Boltyanskii, Gamkrelidze and Mishechenko (1962), Hestenes (1966), Kamien and Schwartz (2012) among many others, we have; 𝜇𝜇�𝑢𝑢𝑢𝑢 ≤ 𝐼 for 𝑢𝑢� ≤ 𝑢𝑢 ≤ 𝑢𝑢� implies: 𝑐𝑐���𝑢𝑢𝑢𝑢𝑢𝑢�𝑢𝑢�𝑢𝑢𝑢𝑢 ≤ ℎ + 𝑓𝑓𝑢𝑢𝑢𝑢𝑐𝑐��𝑢𝑢𝑢𝑢𝑢𝑢� (4) where 𝑓𝑓𝑢𝑢𝑢𝑢 𝑢 𝑖𝑖�𝑢𝑢𝑢𝑢 denotes the interest rate. As long as the condition specified by this inequality is satisfied it will be optimal to keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 implying 𝑢𝑢𝑢𝑢𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢. This means that during the intervals where this inequality is satisfied the firm will produce just enough to satisfy the demand. So, the condition above will synthesize the periods where 𝐼𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 or 𝑢𝑢𝑢𝑢𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢 and the periods where 𝐼𝐼𝑢𝑢𝑢𝑢 𝐼 𝐼. Notice that for the condition above (equation 4) to be kept satisfied depends on ℎ (invento- ry holding cost), 𝑓𝑓𝑢𝑢𝑢𝑢 the interest rates, 𝑐𝑐𝐼𝑢𝑢𝑢𝑢, the marginal cost of production, and 𝑐𝑐𝐼𝐼𝑢𝑢𝑢𝑢 change in the marginal cost. Following conclusions can be made even without specific forms of 𝑐𝑐𝑢𝑢𝑢𝑢, 𝑓𝑓𝑢𝑢𝑢𝑢, 𝑢𝑢𝑢𝑢𝑢𝑢 and 𝑖𝑖𝑢𝑢𝑢𝑢 on the basis of equation (4): 1. The firm will keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 as long as demand is falling or constant i.e. 𝑢𝑢𝐼𝑢𝑢𝑢𝑢 0 im- plying that the firm will choose to produce just enough to meet the demand if demand is falling since both ℎ and 𝑐𝑐𝐼 are positive. 2. The firm will keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 even if demand is increasing 𝑢𝑢𝑢𝐼𝑢𝑢𝑢𝑢 𝑟 𝐼𝑢 but the term ℎ + 𝑓𝑓𝑢𝑢𝑢𝑢𝑐𝑐��𝑢𝑢𝑢𝑢𝑢𝑢� is large enough to keep inequality (4) satisfied. Decreasing interest rates will not be a remedy in this case if they are already low. Firms where 𝑐𝑐𝐼𝐼�𝑢𝑢 𝑢𝑢𝑢𝑢� is low (large and already very efficient firms) will have more difficulty to produce to build up inventories. Currently, this seems to be the case in major economies. 3. Higher interest rates and the use of inefficient technologies (large 𝑐𝑐𝐼𝑢𝑢𝑢𝑢) implies high marginal cost) will induce firms to wait longer for inventory build-up even if 𝑢𝑢𝐼𝑢𝑢𝑢𝑢 is positive. 4. The level of technology used will not affect the behavior of the firms if the interest rates are very low as they are now in many developed nations since the term 𝑓𝑓𝑢𝑢𝑢𝑢𝑐𝑐𝐼𝑢𝑢𝑢𝑢 in ) implies high marginal cost) will induce firms to wait longer for inventory build-up even if Then using 𝑢𝑢𝑢𝑢𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢, equation (1), and the conditions (the necessary condition to keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 optimal) developed by Pontryagin, Boltyanskii, Gamkrelidze and Mishechenko (1962), Hestenes (1966), Kamien and Schwartz (2012) among many others, we have; 𝜇𝜇�𝑢𝑢𝑢𝑢 ≤ 𝐼 for 𝑢𝑢� ≤ 𝑢𝑢 ≤ 𝑢𝑢� implies: 𝑐𝑐���𝑢𝑢𝑢𝑢𝑢𝑢�𝑢𝑢�𝑢𝑢𝑢𝑢 ≤ ℎ + 𝑓𝑓𝑢𝑢𝑢𝑢𝑐𝑐��𝑢𝑢𝑢𝑢𝑢𝑢� (4) where 𝑓𝑓𝑢𝑢𝑢𝑢 𝑢 𝑖𝑖�𝑢𝑢𝑢𝑢 denotes the interest rate. As long as the condition specified by this inequality is satisfied it will be optimal to keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 implying 𝑢𝑢𝑢𝑢𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢. This means that during the intervals where this inequality is satisfied the firm will produce just enough to satisfy the demand. So, the condition above will synthesize the periods where 𝐼𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 or 𝑢𝑢𝑢𝑢𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢 and the periods where 𝐼𝐼𝑢𝑢𝑢𝑢 𝐼 𝐼. Notice that for the condition above (equation 4) to be kept satisfied depends on ℎ (invento- ry holding cost), 𝑓𝑓𝑢𝑢𝑢𝑢 the interest rates, 𝑐𝑐𝐼𝑢𝑢𝑢𝑢, the marginal cost of production, and 𝑐𝑐𝐼𝐼𝑢𝑢𝑢𝑢 change in the marginal cost. Following conclusions can be made even without specific forms of 𝑐𝑐𝑢𝑢𝑢𝑢, 𝑓𝑓𝑢𝑢𝑢𝑢, 𝑢𝑢𝑢𝑢𝑢𝑢 and 𝑖𝑖𝑢𝑢𝑢𝑢 on the basis of equation (4): 1. The firm will keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 as long as demand is falling or constant i.e. 𝑢𝑢𝐼𝑢𝑢𝑢𝑢 0 im- plying that the firm will choose to produce just enough to meet the demand if demand is falling since both ℎ and 𝑐𝑐𝐼 are positive. 2. The firm will keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 even if demand is increasing 𝑢𝑢𝑢𝐼𝑢𝑢𝑢𝑢 𝑟 𝐼𝑢 but the term ℎ + 𝑓𝑓𝑢𝑢𝑢𝑢𝑐𝑐��𝑢𝑢𝑢𝑢𝑢𝑢� is large enough to keep inequality (4) satisfied. Decreasing interest rates will not be a remedy in this case if they are already low. Firms where 𝑐𝑐𝐼𝐼�𝑢𝑢 𝑢𝑢𝑢𝑢� is low (large and already very efficient firms) will have more difficulty to produce to build up inventories. Currently, this seems to be the case in major economies. 3. Higher interest rates and the use of inefficient technologies (large 𝑐𝑐𝐼𝑢𝑢𝑢𝑢) implies high marginal cost) will induce firms to wait longer for inventory build-up even if 𝑢𝑢𝐼𝑢𝑢𝑢𝑢 is positive. 4. The level of technology used will not affect the behavior of the firms if the interest rates are very low as they are now in many developed nations since the term 𝑓𝑓𝑢𝑢𝑢𝑢𝑐𝑐𝐼𝑢𝑢𝑢𝑢 in is positive. 4. The level of technology used will not affect the behavior of the firms if the interest rates are very low as they are now in many developed na- tions since the term Then using 𝑢𝑢𝑢𝑢𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢, equation (1), and the conditions (the necessary condition to keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 optimal) developed by Pontryagin, Boltyanskii, Gamkrelidze and Mishechenko (1962), Hestenes (1966), Kamien and Schwartz (2012) among many others, we have; 𝜇𝜇�𝑢𝑢𝑢𝑢 ≤ 𝐼 for 𝑢𝑢� ≤ 𝑢𝑢 ≤ 𝑢𝑢� implies: 𝑐𝑐���𝑢𝑢𝑢𝑢𝑢𝑢�𝑢𝑢�𝑢𝑢𝑢𝑢 ≤ ℎ + 𝑓𝑓𝑢𝑢𝑢𝑢𝑐𝑐��𝑢𝑢𝑢𝑢𝑢𝑢� (4) where 𝑓𝑓𝑢𝑢𝑢𝑢 𝑢 𝑖𝑖�𝑢𝑢𝑢𝑢 denotes the interest rate. As long as the condition specified by this inequality is satisfied it will be optimal to keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 implying 𝑢𝑢𝑢𝑢𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢. This means that during the intervals where this inequality is satisfied the firm will produce just enough to satisfy the demand. So, the condition above will synthesize the periods where 𝐼𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 or 𝑢𝑢𝑢𝑢𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢 and the periods where 𝐼𝐼𝑢𝑢𝑢𝑢 𝐼 𝐼. Notice that for the condition above (equation 4) to be kept satisfied depends on ℎ (invento- ry holding cost), 𝑓𝑓𝑢𝑢𝑢𝑢 the interest rates, 𝑐𝑐𝐼𝑢𝑢𝑢𝑢, the marginal cost of production, and 𝑐𝑐𝐼𝐼𝑢𝑢𝑢𝑢 change in the marginal cost. Following conclusions can be made even without specific forms of 𝑐𝑐𝑢𝑢𝑢𝑢, 𝑓𝑓𝑢𝑢𝑢𝑢, 𝑢𝑢𝑢𝑢𝑢𝑢 and 𝑖𝑖𝑢𝑢𝑢𝑢 on the basis of equation (4): 1. The firm will keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 as long as demand is falling or constant i.e. 𝑢𝑢𝐼𝑢𝑢𝑢𝑢 0 im- plying that the firm will choose to produce just enough to meet the demand if demand is falling since both ℎ and 𝑐𝑐𝐼 are positive. 2. The firm will keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 even if demand is increasing 𝑢𝑢𝑢𝐼𝑢𝑢𝑢𝑢 𝑟 𝐼𝑢 but the term ℎ + 𝑓𝑓𝑢𝑢𝑢𝑢𝑐𝑐��𝑢𝑢𝑢𝑢𝑢𝑢� is large enough to keep inequality (4) satisfied. Decreasing interest rates will not be a remedy in this case if they are already low. Firms where 𝑐𝑐𝐼𝐼�𝑢𝑢 𝑢𝑢𝑢𝑢� is low (large and already very efficient firms) will have more difficulty to produce to build up inventories. Currently, this seems to be the case in major economies. 3. Higher interest rates and the use of inefficient technologies (large 𝑐𝑐𝐼𝑢𝑢𝑢𝑢) implies high marginal cost) will induce firms to wait longer for inventory build-up even if 𝑢𝑢𝐼𝑢𝑢𝑢𝑢 is positive. 4. The level of technology used will not affect the behavior of the firms if the interest rates are very low as they are now in many developed nations since the term 𝑓𝑓𝑢𝑢𝑢𝑢𝑐𝑐𝐼𝑢𝑢𝑢𝑢 in in equation will be low even if Then using 𝑢𝑢𝑢𝑢𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢, equation (1), and the conditions (the necessary condition to keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 optimal) developed by Pontryagin, Boltyanskii, Gamkrelidze and Mishechenko (1962), Hestenes (1966), Kamien and Schwartz (2012) among many others, we have; 𝜇𝜇�𝑢𝑢𝑢𝑢 ≤ 𝐼 for 𝑢𝑢� ≤ 𝑢𝑢 ≤ 𝑢𝑢� implies: 𝑐𝑐���𝑢𝑢𝑢𝑢𝑢𝑢�𝑢𝑢�𝑢𝑢𝑢𝑢 ≤ ℎ + 𝑓𝑓𝑢𝑢𝑢𝑢𝑐𝑐��𝑢𝑢𝑢𝑢𝑢𝑢� (4) where 𝑓𝑓𝑢𝑢𝑢𝑢 𝑢 𝑖𝑖�𝑢𝑢𝑢𝑢 denotes the interest rate. As long as the condition specified by this inequality is satisfied it will be optimal to keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 implying 𝑢𝑢𝑢𝑢𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢. This means that during the intervals where this inequality is satisfied the firm will produce just enough to satisfy the demand. So, the condition above will synthesize the periods where 𝐼𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 or 𝑢𝑢𝑢𝑢𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢 and the periods where 𝐼𝐼𝑢𝑢𝑢𝑢 𝐼 𝐼. Notice that for the condition above (equation 4) to be kept satisfied depends on ℎ (invento- ry holding cost), 𝑓𝑓𝑢𝑢𝑢𝑢 the interest rates, 𝑐𝑐𝐼𝑢𝑢𝑢𝑢, the marginal cost of production, and 𝑐𝑐𝐼𝐼𝑢𝑢𝑢𝑢 change in the marginal cost. Following conclusions can be made even without specific forms of 𝑐𝑐𝑢𝑢𝑢𝑢, 𝑓𝑓𝑢𝑢𝑢𝑢, 𝑢𝑢𝑢𝑢𝑢𝑢 and 𝑖𝑖𝑢𝑢𝑢𝑢 on the basis of equation (4): 1. The firm will keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 as long as demand is falling or constant i.e. 𝑢𝑢𝐼𝑢𝑢𝑢𝑢 0 im- plying that the firm will choose to produce just enough to meet the demand if demand is falling since both ℎ and 𝑐𝑐𝐼 are positive. 2. The firm will keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 even if demand is increasing 𝑢𝑢𝑢𝐼𝑢𝑢𝑢𝑢 𝑟 𝐼𝑢 but the term ℎ + 𝑓𝑓𝑢𝑢𝑢𝑢𝑐𝑐��𝑢𝑢𝑢𝑢𝑢𝑢� is large enough to keep inequality (4) satisfied. Decreasing interest rates will not be a remedy in this case if they are already low. Firms where 𝑐𝑐𝐼𝐼�𝑢𝑢 𝑢𝑢𝑢𝑢� is low (large and already very efficient firms) will have more difficulty to produce to build up inventories. Currently, this seems to be the case in major economies. 3. Higher interest rates and the use of inefficient technologies (large 𝑐𝑐𝐼𝑢𝑢𝑢𝑢) implies high marginal cost) will induce firms to wait longer for inventory build-up even if 𝑢𝑢𝐼𝑢𝑢𝑢𝑢 is positive. 4. The level of technology used will not affect the behavior of the firms if the interest rates are very low as they are now in many developed nations since the term 𝑓𝑓𝑢𝑢𝑢𝑢𝑐𝑐𝐼𝑢𝑢𝑢𝑢 in is high (a firm technologically inefficient.). In countries with high interest rates, the importance of efficient technologies is evident because both terms ( f(t) and Then using 𝑢𝑢𝑢𝑢𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢, equation (1), and the conditions (the necessary condition to keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 optimal) developed by Pontryagin, Boltyanskii, Gamkrelidze and Mishechenko (1962), Hestenes (1966), Kamien and Schwartz (2012) among many others, we have; 𝜇𝜇�𝑢𝑢𝑢𝑢 ≤ 𝐼 for 𝑢𝑢� ≤ 𝑢𝑢 ≤ 𝑢𝑢� implies: 𝑐𝑐���𝑢𝑢𝑢𝑢𝑢𝑢�𝑢𝑢�𝑢𝑢𝑢𝑢 ≤ ℎ + 𝑓𝑓𝑢𝑢𝑢𝑢𝑐𝑐��𝑢𝑢𝑢𝑢𝑢𝑢� (4) where 𝑓𝑓𝑢𝑢𝑢𝑢 𝑢 𝑖𝑖�𝑢𝑢𝑢𝑢 denotes the interest rate. As long as the condition specified by this inequality is satisfied it will be optimal to keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 implying 𝑢𝑢𝑢𝑢𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢. This means that during the intervals where this inequality is satisfied the firm will produce just enough to satisfy the demand. So, the condition above will synthesize the periods where 𝐼𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 or 𝑢𝑢𝑢𝑢𝑢𝑢 𝑢 𝑢𝑢𝑢𝑢𝑢𝑢 and the periods where 𝐼𝐼𝑢𝑢𝑢𝑢 𝐼 𝐼. Notice that for the condition above (equation 4) to be kept satisfied depends on ℎ (invento- ry holding cost), 𝑓𝑓𝑢𝑢𝑢𝑢 the interest rates, 𝑐𝑐𝐼𝑢𝑢𝑢𝑢, the marginal cost of production, and 𝑐𝑐𝐼𝐼𝑢𝑢𝑢𝑢 change in the marginal cost. Following conclusions can be made even without specific forms of 𝑐𝑐𝑢𝑢𝑢𝑢, 𝑓𝑓𝑢𝑢𝑢𝑢, 𝑢𝑢𝑢𝑢𝑢𝑢 and 𝑖𝑖𝑢𝑢𝑢𝑢 on the basis of equation (4): 1. The firm will keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 as long as demand is falling or constant i.e. 𝑢𝑢𝐼𝑢𝑢𝑢𝑢 0 im- plying that the firm will choose to produce just enough to meet the demand if demand is falling since both ℎ and 𝑐𝑐𝐼 are positive. 2. The firm will keep 𝐼𝐼𝑢𝑢𝑢𝑢 𝑢 𝐼 even if demand is increasing 𝑢𝑢𝑢𝐼𝑢𝑢𝑢𝑢 𝑟 𝐼𝑢 but the term ℎ + 𝑓𝑓𝑢𝑢𝑢𝑢𝑐𝑐��𝑢𝑢𝑢𝑢𝑢𝑢� is large enough to keep inequality (4) satisfied. Decreasing interest rates will not be a remedy in this case if they are already low. Firms where 𝑐𝑐𝐼𝐼�𝑢𝑢 𝑢𝑢𝑢𝑢� is low (large and already very efficient firms) will have more difficulty to produce to build up inventories. Currently, this seems to be the case in major economies. 3. Higher interest rates and the use of inefficient technologies (large 𝑐𝑐𝐼𝑢𝑢𝑢𝑢) implies high marginal cost) will induce firms to wait longer for inventory build-up even if 𝑢𝑢𝐼𝑢𝑢𝑢𝑢 is positive. 4. The level of technology used will not affect the behavior of the firms if the interest rates are very low as they are now in many developed nations since the term 𝑓𝑓𝑢𝑢𝑢𝑢𝑐𝑐𝐼𝑢𝑢𝑢𝑢 in are) large. 5. The cyclicality of demand and the time dependency of interest rates can affect the producers. Producers, facing cyclical demand with very high periodicity will have extreme difficulty in planning and hence may not produce for inventory buildup at all. Cyclicality in the following exam- ples is denoted by the parameter k. The value of k will be assumed to be 1 in all cases for simplicity. For higher values of k, trigonometric func- tions will complete 360-degree cycle twice in the planning horizon. The results will not change except the firms will stop and start two times in the planning period indicating more instability of demand. 6. Size of the firms, expressed as a constant in the total production cost (c(r)), is irrelevant in production decisions since it will not appear in equ- ation (4). In the next section, the impact of holding cost, technology, and interest rates on the production behavior of firms is analyzed with different assumptions on interest rates and a specific form of cost function (a cubic function of level of production). Graphs are produced to visualize the impact of assumptions about interest rates, parameters of demand function, and holding cost. oPtimAl incentives for economic growth… 17 Research Process The specific cost function, in this example, will be assumed to have the more realistic form of: equation will be low even if 𝑐𝑐𝑐𝑐𝑐𝑐𝑐 is high (a firm technologically inefficient.). In coun- tries with high interest rates, the importance of efficient technologies is evident because both terms (𝑓𝑓𝑐𝑐𝑐𝑐 and 𝑐𝑐𝑐𝑐𝑐𝑐𝑐 are) large. 5. The cyclicality of demand and the time dependency of interest rates can affect the pro- ducers. Producers, facing cyclical demand with very high periodicity will have extreme difficulty in planning and hence may not produce for inventory buildup at all. Cyclicali- ty in the following examples is denoted by the parameter k. The value of k will be as- sumed to be 1 in all cases for simplicity. For higher values of k, trigonometric functions will complete 360-degree cycle twice in the planning horizon. The results will not change except the firms will stop and start two times in the planning period indicating more instability of demand. 6. Size of the firms, expressed as a constant in the total production cost (𝑐𝑐𝑐𝑐𝑐𝑐), is irrelevant in production decisions since it will not appear in equation (4). In the next section, the impact of holding cost, technology, and interest rates on the produc- tion behavior of firms is analyzed with different assumptions on interest rates and a specific form of cost function (a cubic function of level of production). Graphs are produced to visual- ize the impact of assumptions about interest rates, parameters of demand function, and holding cost. Research Process The specific cost function, in this example, will be assumed to have the more realistic form of: 𝑐𝑐�𝑐𝑐𝑐𝑐𝑐𝑐� = 𝑎𝑎𝑐𝑐� + 𝑏𝑏𝑐𝑐� + 𝑐𝑐𝑐𝑐 + 𝑐𝑐 and 𝑐𝑐𝑐𝑐𝑐𝑐 = 𝐴𝐴 + 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝑐𝑐 as defined previously where 𝐴𝐴 represents the periodicity of the demand function. In the examples below, we will assume that 𝑎𝑎 = 𝑎𝑎𝑎𝑎 𝐴𝐴 = 𝑎𝑎 𝐴𝐴 = 𝑎𝑎 𝑎 = 𝑎𝐴𝐴𝑎 𝑏𝑏 = 𝑎𝑎𝑎 𝑐𝑐 = 𝑎 for simplicity. Case I: 𝐴𝐴 = 𝑎 (one cycle of the demand curve in the planning horizon of one and interest rate 𝑓𝑓𝑐𝑐𝑐𝑐 = 𝑎.) Then the inequality (4) becomes: 𝑐𝑎𝑎𝑎 ∗ 3𝑐𝑎 + 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝑐𝑐𝑐 𝑎 𝑎𝑐𝑐𝑐𝑐𝑐𝐴𝐴𝐴𝐴𝐴𝑐𝑐 𝑐 𝐴𝑎𝑎 (5) and equation will be low even if 𝑐𝑐𝑐𝑐𝑐𝑐𝑐 is high (a firm technologically inefficient.). In coun- tries with high interest rates, the importance of efficient technologies is evident because both terms (𝑓𝑓𝑐𝑐𝑐𝑐 and 𝑐𝑐𝑐𝑐𝑐𝑐𝑐 are) large. 5. The cyclicality of demand and the time dependency of interest rates can affect the pro- ducers. Producers, facing cyclical demand with very high periodicity will have extreme difficulty in planning and hence may not produce for inventory buildup at all. Cyclicali- ty in the following examples is denoted by the parameter k. The value of k will be as- sumed to be 1 in all cases for simplicity. For higher values of k, trigonometric functions will complete 360-degree cycle twice in the planning horizon. The results will not change except the firms will stop and start two times in the planning period indicating more instability of demand. 6. Size of the firms, expressed as a constant in the total production cost (𝑐𝑐𝑐𝑐𝑐𝑐), is irrelevant in production decisions since it will not appear in equation (4). In the next section, the impact of holding cost, technology, and interest rates on the produc- tion behavior of firms is analyzed with different assumptions on interest rates and a specific form of cost function (a cubic function of level of production). Graphs are produced to visual- ize the impact of assumptions about interest rates, parameters of demand function, and holding cost. Research Process The specific cost function, in this example, will be assumed to have the more realistic form of: 𝑐𝑐�𝑐𝑐𝑐𝑐𝑐𝑐� = 𝑎𝑎𝑐𝑐� + 𝑏𝑏𝑐𝑐� + 𝑐𝑐𝑐𝑐 + 𝑐𝑐 and 𝑐𝑐𝑐𝑐𝑐𝑐 = 𝐴𝐴 + 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝑐𝑐 as defined previously where 𝐴𝐴 represents the periodicity of the demand function. In the examples below, we will assume that 𝑎𝑎 = 𝑎𝑎𝑎𝑎 𝐴𝐴 = 𝑎𝑎 𝐴𝐴 = 𝑎𝑎 𝑎 = 𝑎𝐴𝐴𝑎 𝑏𝑏 = 𝑎𝑎𝑎 𝑐𝑐 = 𝑎 for simplicity. Case I: 𝐴𝐴 = 𝑎 (one cycle of the demand curve in the planning horizon of one and interest rate 𝑓𝑓𝑐𝑐𝑐𝑐 = 𝑎.) Then the inequality (4) becomes: 𝑐𝑎𝑎𝑎 ∗ 3𝑐𝑎 + 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝑐𝑐𝑐 𝑎 𝑎𝑐𝑐𝑐𝑐𝑐𝐴𝐴𝐴𝐴𝐴𝑐𝑐 𝑐 𝐴𝑎𝑎 (5) as defined previously where k represents the perio- dicity of the demand function. In the examples below, we will assume that equation will be low even if 𝑐𝑐𝑐𝑐𝑐𝑐𝑐 is high (a firm technologically inefficient.). In coun- tries with high interest rates, the importance of efficient technologies is evident because both terms (𝑓𝑓𝑐𝑐𝑐𝑐 and 𝑐𝑐𝑐𝑐𝑐𝑐𝑐 are) large. 5. The cyclicality of demand and the time dependency of interest rates can affect the pro- ducers. Producers, facing cyclical demand with very high periodicity will have extreme difficulty in planning and hence may not produce for inventory buildup at all. Cyclicali- ty in the following examples is denoted by the parameter k. The value of k will be as- sumed to be 1 in all cases for simplicity. For higher values of k, trigonometric functions will complete 360-degree cycle twice in the planning horizon. The results will not change except the firms will stop and start two times in the planning period indicating more instability of demand. 6. Size of the firms, expressed as a constant in the total production cost (𝑐𝑐𝑐𝑐𝑐𝑐), is irrelevant in production decisions since it will not appear in equation (4). In the next section, the impact of holding cost, technology, and interest rates on the produc- tion behavior of firms is analyzed with different assumptions on interest rates and a specific form of cost function (a cubic function of level of production). Graphs are produced to visual- ize the impact of assumptions about interest rates, parameters of demand function, and holding cost. Research Process The specific cost function, in this example, will be assumed to have the more realistic form of: 𝑐𝑐�𝑐𝑐𝑐𝑐𝑐𝑐� = 𝑎𝑎𝑐𝑐� + 𝑏𝑏𝑐𝑐� + 𝑐𝑐𝑐𝑐 + 𝑐𝑐 and 𝑐𝑐𝑐𝑐𝑐𝑐 = 𝐴𝐴 + 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝑐𝑐 as defined previously where 𝐴𝐴 represents the periodicity of the demand function. In the examples below, we will assume that 𝑎𝑎 = 𝑎𝑎𝑎𝑎 𝐴𝐴 = 𝑎𝑎 𝐴𝐴 = 𝑎𝑎 𝑎 = 𝑎𝐴𝐴𝑎 𝑏𝑏 = 𝑎𝑎𝑎 𝑐𝑐 = 𝑎 for simplicity. Case I: 𝐴𝐴 = 𝑎 (one cycle of the demand curve in the planning horizon of one and interest rate 𝑓𝑓𝑐𝑐𝑐𝑐 = 𝑎.) Then the inequality (4) becomes: 𝑐𝑎𝑎𝑎 ∗ 3𝑐𝑎 + 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝑐𝑐𝑐 𝑎 𝑎𝑐𝑐𝑐𝑐𝑐𝐴𝐴𝐴𝐴𝐴𝑐𝑐 𝑐 𝐴𝑎𝑎 (5) equation will be low even if 𝑐𝑐𝑐𝑐𝑐𝑐𝑐 is high (a firm technologically inefficient.). In coun- tries with high interest rates, the importance of efficient technologies is evident because both terms (𝑓𝑓𝑐𝑐𝑐𝑐 and 𝑐𝑐𝑐𝑐𝑐𝑐𝑐 are) large. 5. The cyclicality of demand and the time dependency of interest rates can affect the pro- ducers. Producers, facing cyclical demand with very high periodicity will have extreme difficulty in planning and hence may not produce for inventory buildup at all. Cyclicali- ty in the following examples is denoted by the parameter k. The value of k will be as- sumed to be 1 in all cases for simplicity. For higher values of k, trigonometric functions will complete 360-degree cycle twice in the planning horizon. The results will not change except the firms will stop and start two times in the planning period indicating more instability of demand. 6. Size of the firms, expressed as a constant in the total production cost (𝑐𝑐𝑐𝑐𝑐𝑐), is irrelevant in production decisions since it will not appear in equation (4). In the next section, the impact of holding cost, technology, and interest rates on the produc- tion behavior of firms is analyzed with different assumptions on interest rates and a specific form of cost function (a cubic function of level of production). Graphs are produced to visual- ize the impact of assumptions about interest rates, parameters of demand function, and holding cost. Research Process The specific cost function, in this example, will be assumed to have the more realistic form of: 𝑐𝑐�𝑐𝑐𝑐𝑐𝑐𝑐� = 𝑎𝑎𝑐𝑐� + 𝑏𝑏𝑐𝑐� + 𝑐𝑐𝑐𝑐 + 𝑐𝑐 and 𝑐𝑐𝑐𝑐𝑐𝑐 = 𝐴𝐴 + 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝑐𝑐 as defined previously where 𝐴𝐴 represents the periodicity of the demand function. In the examples below, we will assume that 𝑎𝑎 = 𝑎𝑎𝑎𝑎 𝐴𝐴 = 𝑎𝑎 𝐴𝐴 = 𝑎𝑎 𝑎 = 𝑎𝐴𝐴𝑎 𝑏𝑏 = 𝑎𝑎𝑎 𝑐𝑐 = 𝑎 for simplicity. Case I: 𝐴𝐴 = 𝑎 (one cycle of the demand curve in the planning horizon of one and interest rate 𝑓𝑓𝑐𝑐𝑐𝑐 = 𝑎.) Then the inequality (4) becomes: 𝑐𝑎𝑎𝑎 ∗ 3𝑐𝑎 + 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝑐𝑐𝑐 𝑎 𝑎𝑐𝑐𝑐𝑐𝑐𝐴𝐴𝐴𝐴𝐴𝑐𝑐 𝑐 𝐴𝑎𝑎 (5) for simplicity. Case I: k = 1 (one cycle of the demand curve in the planning horizon of one and interest rate f(t) = 0.) Then the inequality (4) becomes: equation will be low even if 𝑐𝑐𝑐𝑐𝑐𝑐𝑐 is high (a firm technologically inefficient.). In coun- tries with high interest rates, the importance of efficient technologies is evident because both terms (𝑓𝑓𝑐𝑐𝑐𝑐 and 𝑐𝑐𝑐𝑐𝑐𝑐𝑐 are) large. 5. The cyclicality of demand and the time dependency of interest rates can affect the pro- ducers. Producers, facing cyclical demand with very high periodicity will have extreme difficulty in planning and hence may not produce for inventory buildup at all. Cyclicali- ty in the following examples is denoted by the parameter k. The value of k will be as- sumed to be 1 in all cases for simplicity. For higher values of k, trigonometric functions will complete 360-degree cycle twice in the planning horizon. The results will not change except the firms will stop and start two times in the planning period indicating more instability of demand. 6. Size of the firms, expressed as a constant in the total production cost (𝑐𝑐𝑐𝑐𝑐𝑐), is irrelevant in production decisions since it will not appear in equation (4). In the next section, the impact of holding cost, technology, and interest rates on the produc- tion behavior of firms is analyzed with different assumptions on interest rates and a specific form of cost function (a cubic function of level of production). Graphs are produced to visual- ize the impact of assumptions about interest rates, parameters of demand function, and holding cost. Research Process The specific cost function, in this example, will be assumed to have the more realistic form of: 𝑐𝑐�𝑐𝑐𝑐𝑐𝑐𝑐� = 𝑎𝑎𝑐𝑐� + 𝑏𝑏𝑐𝑐� + 𝑐𝑐𝑐𝑐 + 𝑐𝑐 and 𝑐𝑐𝑐𝑐𝑐𝑐 = 𝐴𝐴 + 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝑐𝑐 as defined previously where 𝐴𝐴 represents the periodicity of the demand function. In the examples below, we will assume that 𝑎𝑎 = 𝑎𝑎𝑎𝑎 𝐴𝐴 = 𝑎𝑎 𝐴𝐴 = 𝑎𝑎 𝑎 = 𝑎𝐴𝐴𝑎 𝑏𝑏 = 𝑎𝑎𝑎 𝑐𝑐 = 𝑎 for simplicity. Case I: 𝐴𝐴 = 𝑎 (one cycle of the demand curve in the planning horizon of one and interest rate 𝑓𝑓𝑐𝑐𝑐𝑐 = 𝑎.) Then the inequality (4) becomes: 𝑐𝑎𝑎𝑎 ∗ 3𝑐𝑎 + 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝑐𝑐𝑐 𝑎 𝑎𝑐𝑐𝑐𝑐𝑐𝐴𝐴𝐴𝐴𝐴𝑐𝑐 𝑐 𝐴𝑎𝑎 (5) (5) Both sides of inequality in (5) is presented in the graph below produced us- ing arbitrary numbers for relevant variables (figure 1). This graph (figure 1) implies (notice the tip of arrows in the graph shows the times production is stopped and started) that the firm will stop producing for inventories just before the demand reaches its maximum and starts just after the demand reaches its minimum (recall that the sinusoidal demand reaches its maximum at t = 0.25 and its minimum at t = 0.75). These points are indicated by arrows in all graphs (the tip of the arrow shows the point at which the left, 0.18, and right, 0.86) hand side of equation 4 becomes equal). Therefore, it is only possible for firms to stop producing later and start producing earlier is to decrease the holding costs (h) as defined above. Mustafa Akan, Natalia Konovalova18 Figure 1. Graphs of Left ( f (t)) and Right ( g(x)) hand sides of Inequality (5) and Demand Function (h(x) = r(t)) Both sides of inequality in (5) is presented in the graph below produced using arbitrary numbers for relevant variables (figure 1). This graph (figure 1) implies (notice the tip of arrows in the graph shows the times produc- tion is stopped and started) that the firm will stop producing for inventories just before the demand reaches its maximum and starts just after the demand reaches its minimum (recall that the sinusoidal demand reaches its maximum at 𝑡𝑡 𝑡 𝑡𝑡𝑡𝑡 and its minimum at 𝑡𝑡 𝑡 𝑡𝑡𝑡𝑡). These points are indicated by arrows in all graphs (the tip of the arrow shows the point at which the left, 𝑡𝑡18, and right, 𝑡𝑡86) hand side of equation 4 becomes equal). Therefore, it is only possi- ble for firms to stop producing later and start producing earlier is to decrease the holding costs (ℎ) as defined above. Figure 1. Graphs of Left (𝑓𝑓𝑓𝑡𝑡𝑓𝑓 and Right (𝑔𝑔𝑓𝑔𝑔𝑓) hand sides of Inequality (5) and Demand Function (ℎ𝑓𝑔𝑔𝑓 𝑡 𝑟𝑟𝑓𝑡𝑡𝑓) S o u r c e : constructed by authors. Case II: k = 1 and interest rate f(t) = 0 and a new technology I introduced. The impact of new technology will be analyzed in three cases: 1. Fixed costs (the term d in the cost function) is reduced. There will be no impact on production decision since the fixed costs are already shown to have on impact on the inequality (5). 2. Only the constant term (c) in the marginal cost, Source: constructed by authors. Case II: 𝑘𝑘 𝑘 𝑘 and interest rate 𝑓𝑓(𝑡𝑡) 𝑘 0 and a new technology I introduced. The impact of new technology will be analyzed in three cases: 1. Fixed costs (the term 𝑑𝑑 in the cost function) is reduced. There will be no impact on production decision since the fixed costs are already shown to have on impact on the inequality (5). 2. Only the constant term (𝑐𝑐) in the marginal cost, 𝑐𝑐𝑐 𝑘 𝑐𝑐𝑐𝑐𝑐� + 2𝑏𝑏𝑐𝑐 + 𝑐𝑐 is reduced. The left-hand side of inequality (5) will not change since 𝑐𝑐𝑐𝑐(𝑐𝑐) does not depend on the constant term 𝑐𝑐 in the production cost function. The right-hand side will be lower since 𝑐𝑐 is lower. Therefore, the decision of the firm will depend on how much the new technol- ogy reduces 𝑐𝑐. The firm will start to produce sooner if the reduction in c is large enough to violate the inequality. 3. Marginal cost function, 𝑐𝑐𝑐(𝑐𝑐), is reduced as a whole. We will consider this last case assuming that the new technology reduces the marginal cost by 20%. This implies that the left-hand side of inequality (5) will be multiplies by 0.8. Then the inequality (4) becomes: (0.5 ∗ 𝑐(4 + 𝑠𝑠𝑠𝑠𝑠𝑠2𝑠𝑠𝑡𝑡) − 𝑘)𝑐𝑐𝑐𝑐𝑠𝑠2𝑠𝑠𝑡𝑡 ∗ 0.8 𝑐 2.5 ( 6) Therefore, for the same values for the constants, the graphs of both sides of inequality above and the sales are presented below (figure 2). Inequality (6) implies the same type of results as in the previous case. However, it is evi- dent from the graphs that in the second case the firm will stop producing sooner and start pro- ducing later than in the first case. This is an important result because it indicates that introduc- tion of new technologies which reduces the marginal costs will induce firms to stop production sooner. Case III: 𝑘𝑘 𝑘 𝑘 and interest rate 𝑓𝑓(𝑡𝑡) 𝑘 0.2 The inequality (4) becomes: (0.5 ∗ 𝑐(4 + 𝑠𝑠𝑠𝑠𝑠𝑠2𝑠𝑠𝑡𝑡) − 𝑘)𝑐𝑐𝑐𝑐𝑠𝑠2𝑠𝑠𝑡𝑡 𝑐 2.5 + 0.2�0.5 ∗ 𝑐 ∗ (4 + 𝑠𝑠𝑠𝑠𝑠𝑠2𝑠𝑠𝑠𝑠)�� − 2 ∗ (4 + sin(2𝑠𝑠𝑠𝑠) + 𝑘) /2𝑠𝑠 ( 7) is reduced. The left-hand side of inequality (5) will not change since Source: constructed by authors. Case II: 𝑘𝑘 𝑘 𝑘 and interest rate 𝑓𝑓(𝑡𝑡) 𝑘 0 and a new technology I introduced. The impact of new technology will be analyzed in three cases: 1. Fixed costs (the term 𝑑𝑑 in the cost function) is reduced. There will be no impact on production decision since the fixed costs are already shown to have on impact on the inequality (5). 2. Only the constant term (𝑐𝑐) in the marginal cost, 𝑐𝑐𝑐 𝑘 𝑐𝑐𝑐𝑐𝑐� + 2𝑏𝑏𝑐𝑐 + 𝑐𝑐 is reduced. The left-hand side of inequality (5) will not change since 𝑐𝑐𝑐𝑐(𝑐𝑐) does not depend on the constant term 𝑐𝑐 in the production cost function. The right-hand side will be lower since 𝑐𝑐 is lower. Therefore, the decision of the firm will depend on how much the new technol- ogy reduces 𝑐𝑐. The firm will start to produce sooner if the reduction in c is large enough to violate the inequality. 3. Marginal cost function, 𝑐𝑐𝑐(𝑐𝑐), is reduced as a whole. We will consider this last case assuming that the new technology reduces the marginal cost by 20%. This implies that the left-hand side of inequality (5) will be multiplies by 0.8. Then the inequality (4) becomes: (0.5 ∗ 𝑐(4 + 𝑠𝑠𝑠𝑠𝑠𝑠2𝑠𝑠𝑡𝑡) − 𝑘)𝑐𝑐𝑐𝑐𝑠𝑠2𝑠𝑠𝑡𝑡 ∗ 0.8 𝑐 2.5 ( 6) Therefore, for the same values for the constants, the graphs of both sides of inequality above and the sales are presented below (figure 2). Inequality (6) implies the same type of results as in the previous case. However, it is evi- dent from the graphs that in the second case the firm will stop producing sooner and start pro- ducing later than in the first case. This is an important result because it indicates that introduc- tion of new technologies which reduces the marginal costs will induce firms to stop production sooner. Case III: 𝑘𝑘 𝑘 𝑘 and interest rate 𝑓𝑓(𝑡𝑡) 𝑘 0.2 The inequality (4) becomes: (0.5 ∗ 𝑐(4 + 𝑠𝑠𝑠𝑠𝑠𝑠2𝑠𝑠𝑡𝑡) − 𝑘)𝑐𝑐𝑐𝑐𝑠𝑠2𝑠𝑠𝑡𝑡 𝑐 2.5 + 0.2�0.5 ∗ 𝑐 ∗ (4 + 𝑠𝑠𝑠𝑠𝑠𝑠2𝑠𝑠𝑠𝑠)�� − 2 ∗ (4 + sin(2𝑠𝑠𝑠𝑠) + 𝑘) /2𝑠𝑠 ( 7) does not de- pend on the constant term c in the production cost function. The right-hand side will be lower since c is lower. Therefore, the decision of the firm will de- pend on how much the new technology reduces c. The firm will start to produce sooner if the reduction in c is large enough to violate the inequality. oPtimAl incentives for economic growth… 19 3. Marginal cost function, Source: constructed by authors. Case II: 𝑘𝑘 𝑘 𝑘 and interest rate 𝑓𝑓(𝑡𝑡) 𝑘 0 and a new technology I introduced. The impact of new technology will be analyzed in three cases: 1. Fixed costs (the term 𝑑𝑑 in the cost function) is reduced. There will be no impact on production decision since the fixed costs are already shown to have on impact on the inequality (5). 2. Only the constant term (𝑐𝑐) in the marginal cost, 𝑐𝑐𝑐 𝑘 𝑐𝑐𝑐𝑐𝑐� + 2𝑏𝑏𝑐𝑐 + 𝑐𝑐 is reduced. The left-hand side of inequality (5) will not change since 𝑐𝑐𝑐𝑐(𝑐𝑐) does not depend on the constant term 𝑐𝑐 in the production cost function. The right-hand side will be lower since 𝑐𝑐 is lower. Therefore, the decision of the firm will depend on how much the new technol- ogy reduces 𝑐𝑐. The firm will start to produce sooner if the reduction in c is large enough to violate the inequality. 3. Marginal cost function, 𝑐𝑐𝑐(𝑐𝑐), is reduced as a whole. We will consider this last case assuming that the new technology reduces the marginal cost by 20%. This implies that the left-hand side of inequality (5) will be multiplies by 0.8. Then the inequality (4) becomes: (0.5 ∗ 𝑐(4 + 𝑠𝑠𝑠𝑠𝑠𝑠2𝑠𝑠𝑡𝑡) − 𝑘)𝑐𝑐𝑐𝑐𝑠𝑠2𝑠𝑠𝑡𝑡 ∗ 0.8 𝑐 2.5 ( 6) Therefore, for the same values for the constants, the graphs of both sides of inequality above and the sales are presented below (figure 2). Inequality (6) implies the same type of results as in the previous case. However, it is evi- dent from the graphs that in the second case the firm will stop producing sooner and start pro- ducing later than in the first case. This is an important result because it indicates that introduc- tion of new technologies which reduces the marginal costs will induce firms to stop production sooner. Case III: 𝑘𝑘 𝑘 𝑘 and interest rate 𝑓𝑓(𝑡𝑡) 𝑘 0.2 The inequality (4) becomes: (0.5 ∗ 𝑐(4 + 𝑠𝑠𝑠𝑠𝑠𝑠2𝑠𝑠𝑡𝑡) − 𝑘)𝑐𝑐𝑐𝑐𝑠𝑠2𝑠𝑠𝑡𝑡 𝑐 2.5 + 0.2�0.5 ∗ 𝑐 ∗ (4 + 𝑠𝑠𝑠𝑠𝑠𝑠2𝑠𝑠𝑠𝑠)�� − 2 ∗ (4 + sin(2𝑠𝑠𝑠𝑠) + 𝑘) /2𝑠𝑠 ( 7) is reduced as a whole. We will consider this last case assuming that the new technology reduces the marginal cost by 20%. This implies that the left-hand side of inequality (5) will be multiplies by 0,8. Then the inequality (4) becomes: Source: constructed by authors. Case II: 𝑘𝑘 𝑘 𝑘 and interest rate 𝑓𝑓(𝑡𝑡) 𝑘 0 and a new technology I introduced. The impact of new technology will be analyzed in three cases: 1. Fixed costs (the term 𝑑𝑑 in the cost function) is reduced. There will be no impact on production decision since the fixed costs are already shown to have on impact on the inequality (5). 2. Only the constant term (𝑐𝑐) in the marginal cost, 𝑐𝑐𝑐 𝑘 𝑐𝑐𝑐𝑐𝑐� + 2𝑏𝑏𝑐𝑐 + 𝑐𝑐 is reduced. The left-hand side of inequality (5) will not change since 𝑐𝑐𝑐𝑐(𝑐𝑐) does not depend on the constant term 𝑐𝑐 in the production cost function. The right-hand side will be lower since 𝑐𝑐 is lower. Therefore, the decision of the firm will depend on how much the new technol- ogy reduces 𝑐𝑐. The firm will start to produce sooner if the reduction in c is large enough to violate the inequality. 3. Marginal cost function, 𝑐𝑐𝑐(𝑐𝑐), is reduced as a whole. We will consider this last case assuming that the new technology reduces the marginal cost by 20%. This implies that the left-hand side of inequality (5) will be multiplies by 0.8. Then the inequality (4) becomes: (0.5 ∗ 𝑐(4 + 𝑠𝑠𝑠𝑠𝑠𝑠2𝑠𝑠𝑡𝑡) − 𝑘)𝑐𝑐𝑐𝑐𝑠𝑠2𝑠𝑠𝑡𝑡 ∗ 0.8 𝑐 2.5 ( 6) Therefore, for the same values for the constants, the graphs of both sides of inequality above and the sales are presented below (figure 2). Inequality (6) implies the same type of results as in the previous case. However, it is evi- dent from the graphs that in the second case the firm will stop producing sooner and start pro- ducing later than in the first case. This is an important result because it indicates that introduc- tion of new technologies which reduces the marginal costs will induce firms to stop production sooner. Case III: 𝑘𝑘 𝑘 𝑘 and interest rate 𝑓𝑓(𝑡𝑡) 𝑘 0.2 The inequality (4) becomes: (0.5 ∗ 𝑐(4 + 𝑠𝑠𝑠𝑠𝑠𝑠2𝑠𝑠𝑡𝑡) − 𝑘)𝑐𝑐𝑐𝑐𝑠𝑠2𝑠𝑠𝑡𝑡 𝑐 2.5 + 0.2�0.5 ∗ 𝑐 ∗ (4 + 𝑠𝑠𝑠𝑠𝑠𝑠2𝑠𝑠𝑠𝑠)�� − 2 ∗ (4 + sin(2𝑠𝑠𝑠𝑠) + 𝑘) /2𝑠𝑠 ( 7) (6) Therefore, for the same values for the constants, the graphs of both sides of inequality above and the sales are presented below (figure 2). Inequality (6) implies the same type of results as in the previous case. How- ever, it is evident from the graphs that in the second case the firm will stop producing sooner and start producing later than in the first case. This is an im- portant result because it indicates that introduction of new technologies which reduces the marginal costs will induce firms to stop production sooner. Case III: k = 1 and interest rate f(t) = 0.2 The inequality (4) becomes: Source: constructed by authors. Case II: 𝑘𝑘 𝑘 𝑘 and interest rate 𝑓𝑓(𝑡𝑡) 𝑘 0 and a new technology I introduced. The impact of new technology will be analyzed in three cases: 1. Fixed costs (the term 𝑑𝑑 in the cost function) is reduced. There will be no impact on production decision since the fixed costs are already shown to have on impact on the inequality (5). 2. Only the constant term (𝑐𝑐) in the marginal cost, 𝑐𝑐𝑐 𝑘 𝑐𝑐𝑐𝑐𝑐� + 2𝑏𝑏𝑐𝑐 + 𝑐𝑐 is reduced. The left-hand side of inequality (5) will not change since 𝑐𝑐𝑐𝑐(𝑐𝑐) does not depend on the constant term 𝑐𝑐 in the production cost function. The right-hand side will be lower since 𝑐𝑐 is lower. Therefore, the decision of the firm will depend on how much the new technol- ogy reduces 𝑐𝑐. The firm will start to produce sooner if the reduction in c is large enough to violate the inequality. 3. Marginal cost function, 𝑐𝑐𝑐(𝑐𝑐), is reduced as a whole. We will consider this last case assuming that the new technology reduces the marginal cost by 20%. This implies that the left-hand side of inequality (5) will be multiplies by 0.8. Then the inequality (4) becomes: (0.5 ∗ 𝑐(4 + 𝑠𝑠𝑠𝑠𝑠𝑠2𝑠𝑠𝑡𝑡) − 𝑘)𝑐𝑐𝑐𝑐𝑠𝑠2𝑠𝑠𝑡𝑡 ∗ 0.8 𝑐 2.5 ( 6) Therefore, for the same values for the constants, the graphs of both sides of inequality above and the sales are presented below (figure 2). Inequality (6) implies the same type of results as in the previous case. However, it is evi- dent from the graphs that in the second case the firm will stop producing sooner and start pro- ducing later than in the first case. This is an important result because it indicates that introduc- tion of new technologies which reduces the marginal costs will induce firms to stop production sooner. Case III: 𝑘𝑘 𝑘 𝑘 and interest rate 𝑓𝑓(𝑡𝑡) 𝑘 0.2 The inequality (4) becomes: (0.5 ∗ 𝑐(4 + 𝑠𝑠𝑠𝑠𝑠𝑠2𝑠𝑠𝑡𝑡) − 𝑘)𝑐𝑐𝑐𝑐𝑠𝑠2𝑠𝑠𝑡𝑡 𝑐 2.5 + 0.2�0.5 ∗ 𝑐 ∗ (4 + 𝑠𝑠𝑠𝑠𝑠𝑠2𝑠𝑠𝑠𝑠)�� − 2 ∗ (4 + sin(2𝑠𝑠𝑠𝑠) + 𝑘) /2𝑠𝑠 ( 7) Source: constructed by authors. Case II: 𝑘𝑘 𝑘 𝑘 and interest rate 𝑓𝑓(𝑡𝑡) 𝑘 0 and a new technology I introduced. The impact of new technology will be analyzed in three cases: 1. Fixed costs (the term 𝑑𝑑 in the cost function) is reduced. There will be no impact on production decision since the fixed costs are already shown to have on impact on the inequality (5). 2. Only the constant term (𝑐𝑐) in the marginal cost, 𝑐𝑐𝑐 𝑘 𝑐𝑐𝑐𝑐𝑐� + 2𝑏𝑏𝑐𝑐 + 𝑐𝑐 is reduced. The left-hand side of inequality (5) will not change since 𝑐𝑐𝑐𝑐(𝑐𝑐) does not depend on the constant term 𝑐𝑐 in the production cost function. The right-hand side will be lower since 𝑐𝑐 is lower. Therefore, the decision of the firm will depend on how much the new technol- ogy reduces 𝑐𝑐. The firm will start to produce sooner if the reduction in c is large enough to violate the inequality. 3. Marginal cost function, 𝑐𝑐𝑐(𝑐𝑐), is reduced as a whole. We will consider this last case assuming that the new technology reduces the marginal cost by 20%. This implies that the left-hand side of inequality (5) will be multiplies by 0.8. Then the inequality (4) becomes: (0.5 ∗ 𝑐(4 + 𝑠𝑠𝑠𝑠𝑠𝑠2𝑠𝑠𝑡𝑡) − 𝑘)𝑐𝑐𝑐𝑐𝑠𝑠2𝑠𝑠𝑡𝑡 ∗ 0.8 𝑐 2.5 ( 6) Therefore, for the same values for the constants, the graphs of both sides of inequality above and the sales are presented below (figure 2). Inequality (6) implies the same type of results as in the previous case. However, it is evi- dent from the graphs that in the second case the firm will stop producing sooner and start pro- ducing later than in the first case. This is an important result because it indicates that introduc- tion of new technologies which reduces the marginal costs will induce firms to stop production sooner. Case III: 𝑘𝑘 𝑘 𝑘 and interest rate 𝑓𝑓(𝑡𝑡) 𝑘 0.2 The inequality (4) becomes: (0.5 ∗ 𝑐(4 + 𝑠𝑠𝑠𝑠𝑠𝑠2𝑠𝑠𝑡𝑡) − 𝑘)𝑐𝑐𝑐𝑐𝑠𝑠2𝑠𝑠𝑡𝑡 𝑐 2.5 + 0.2�0.5 ∗ 𝑐 ∗ (4 + 𝑠𝑠𝑠𝑠𝑠𝑠2𝑠𝑠𝑠𝑠)�� − 2 ∗ (4 + sin(2𝑠𝑠𝑠𝑠) + 𝑘) /2𝑠𝑠 ( 7) (7) The graphs of the function on both sides of this inequality above are shown in graph III below (figure 3). Figure 3 shows that introduction of interest rates causes the firms to stop production sooner but not very significantly, from t = 0.18 in graph I to about 0.16 in graph III. This implies that the firms will start to produce at about the same time as when interest rate was zero (Figure 1). This, in turn, implies that firms will stop producing later when interest rates are lowered. However, they will not start producing sooner. Case IV: k = 1 and interest rate f(t) = 0.1, a 50% decrease in interest rate. In this the inequality (5) becomes: The graphs of the function on both sides of this inequality above are shown in graph III be- low (figure 3). Figure 3 shows that introduction of interest rates causes the firms to stop production sooner but not very significantly, from 𝑡𝑡 𝑡 𝑡𝑡𝑡𝑡 in graph I to about 0.16 in graph III. This implies that the firms will start to produce at about the same time as when interest rate was zero (Fig- ure 1). This, in turn, implies that firms will stop producing later when interest rates are low- ered. However, they will not start producing sooner. Case IV: 𝑘𝑘 𝑡 𝑡 and interest rate 𝑓𝑓(𝑡𝑡) 𝑡 𝑡𝑡𝑡, a 5𝑡% decrease in interest rate. In this the inequality (5) becomes: (𝑡𝑡5 ∗ 3(4 + sin(2𝜋𝜋𝑡𝑡) − 𝑡) ∗ 𝑐𝑐𝑐𝑐𝑐𝑐2𝜋𝜋𝑡𝑡 ≤ 2𝑡5 + 𝑡𝑡𝑡𝑡 ∗ (𝑡𝑡5 ∗ 3 ∗ (4 + sin(2𝜋𝜋𝜋𝜋))� − 2 ∗ (4 + sin(2𝜋𝜋𝜋𝜋) + 𝑡)/2𝜋𝜋 ( 8) The graphs of the functions on both sides of this inequality are shown in figure 4 below. Case V: 𝑘𝑘 𝑡 𝑡𝑘 𝑓𝑓 𝑡 5%. Reducing the interest rate from 0.10 to 0.05, we get figure 5. Figure 2. Graphs of Left (𝑓𝑓(𝜋𝜋)) and Right (𝑔𝑔(𝜋𝜋)) Sides of Inequality (6) and Demand Func- tion ��(𝜋𝜋)� (8) Mustafa Akan, Natalia Konovalova20 The graphs of the functions on both sides of this inequality are shown in fig- ure 4 below. Case V: k = 1, f = 50%. Reducing the interest rate from 0.10 to 0.05, we get figure 5. Figure 2. Graphs of Left ( f (x) and Right ( g(x)) Sides of Inequality (6) and Demand Function (h(x)) Source: constructed by authors. Figure 3. Graphs of Left (𝑓𝑓𝑓𝑓𝑓𝑓𝑓 and Right (𝑔𝑔𝑓𝑓𝑓𝑓) Sides of Inequality (7) and Demand Func- tion ��𝑓𝑓𝑓𝑓� S o u r c e : constructed by authors. oPtimAl incentives for economic growth… 21 Figure 3. Graphs of Left ( f (x) and Right ( g(x)) Sides of Inequality (7) and Demand Function (h(x)) Source: constructed by authors. Figure 4. Graphs of Left (𝑓𝑓𝑓𝑓𝑓𝑓𝑓 and Right (𝑔𝑔𝑓𝑓𝑓𝑓) Sides of Inequality (8) and Demand Func- tion ��𝑓𝑓𝑓𝑓� S o u r c e : constructed by authors. Mustafa Akan, Natalia Konovalova22 Figure 4. Graphs of Left ( f (x) and Right ( g(x)) Sides of Inequality (8) and Demand Function (h(x)) Source: constructed by authors. Notice in this case that the firms stop and starts production at about the same time as in the case when interest rate is=20% (previous case) implying that reduced interest rates do not affect production significantly. Figure 5. Graphs of Left (𝑓𝑓𝑓𝑓𝑓𝑓𝑓 and Right (𝑔𝑔𝑓𝑓𝑓𝑓) Sides of Inequality (8) and Demand Func- tion ��𝑓𝑓𝑓𝑓� with 𝑓𝑓𝑓𝑡𝑡𝑓 = 5% S o u r c e : constructed by authors. Notice in this case that the firms stop and starts production at about the same time as in the case when interest rate is=20% (previous case) implying that re- duced interest rates do not affect production significantly. oPtimAl incentives for economic growth… 23 Figure 5. Graphs of Left ( f (x) and Right ( g(x)) Sides of Inequality (8) and Demand Function (h(x)) with f (t) = 5% Source: constructed by authors. Notice that the times at which production stopped and restarted are the same as in the case when interest rate was 10%. This is an important result since it helps to explain why low in- terest rates are no longer effective to induce growth. Research Results and Conclusions General Results and Conclusions: Based on the study of cases I-V and the graphs associated with these cases, following results and their policy implications can be summarized as: 1. High holding costs affect the production decisions negatively. 2. The impact of new technologies depends on the efficiencies brought about by the new technology. New technology has no impact on production decision if it reduces the fixed S o u r c e : constructed by authors. Notice that the times at which production stopped and restarted are the same as in the case when interest rate was 10%. This is an important result since it helps to explain why low interest rates are no longer effective to induce growth. Mustafa Akan, Natalia Konovalova24 Research Results and Conclusions General Results and Conclusions: Based on the study of cases I-V and the graphs associated with these cases, fol- lowing results and their policy implications can be summarized as: 1. High holding costs affect the production decisions negatively. 2. The impact of new technologies depends on the efficiencies brought abo- ut by the new technology. New technology has no impact on production decision if it reduces the fixed costs only or the fixed part of the marginal cost. The firm will stop producing sooner and start producing later if the new technology reduces the total marginal costs. This is a perplexing re- sult since common sense would dictate otherwise. 3. Changes in interest rates (with the assumed value of the parameters) do not affect production. 4. Cyclicality of demand will affect production plans. Extreme periodicity (cyclicality) of demand may result in firms never producing for inven- tories. In case of higher cyclicality, both the demand and the cost func- tions will f luctuate as many times as the cyclicality in a given time hori- zon (assumed to be 1 here) which will imply that the firm will start and stop production many times which is not a conducive situation for inve- stment. 5. Firm size (as expressed by the constant d in the expression of cost func- tion) has no impact on the production decisions. Hence, no differentia- tion should be made between small and large firms when incentives to induce firms to produce for inventories are considered. 6. Effects of changes in interest rates are especially ineffective if the inte- rest rate is already small. 7. Tax incentives have no impact on production since equation (4) does not involve any tax parameter. Therefore: Incentives for firms to lower their holding costs are an effective method to induce firms to keep production above demand. These may include lower pric- es for electricity, oil, and gas, lower property insurance rates, and lower in- surance premiums for labor to reduce holding costs. Incentives for introducing oPtimAl incentives for economic growth… 25 new technologies may be only effective if they are carefully chosen. Incentives to reduce marginal costs will not be effective. Implications for Central and Eastern European Countries The model above was developed for a company operating in a competitive sec- tor. The model was not intended to identify proper incentives for growth in ex- traordinary times like Covit-19 crisis the whole world is facing or 2008 financial crisis. The model was not intended to address incentives directed to attracting foreign investment either. Tax incentives are not addressed since a firm operat- ing in a perfectly competitive sector tries only to minimize its costs. Three important factors are relevant in the context of the paper. They are the interest rates, the production technology, and the holding cost of invento- ries. Thus, the implications for the Central and Eastern European Countries will be analyzed based on these factors. There are many classifications of Central and Eastern European Countries. Central and Eastern European Countries (CEECs) is an OECD term for the group of countries comprising Albania, Bulgaria, Croatia, the Czech Republic, Hunga- ry, Poland, Romania, the Slovak Republic, Slovenia, and the three Baltic States: Estonia, Latvia and Lithuania (OECD, 2001). According to the National Institute of Statistics and Economic Research of France (Insee, 2020), the Central Euro- pean countries defined are Bulgaria, Croatia, Estonia, Hungary, Latvia, Lith- uania, Poland, Romania, Slovenia, Slovakia, Czech Republic. Authors use both above mentioned classifications and chose the following 11 countries of Central and Eastern Europe according to In see classification for demonstration of im- plications of economic incentives on growth. The interest rates in these coun- tries are shown in table 1. Table 1. Interest Rates in Central and Eastern European Countries (%) Countries Last Previous Dates Croatia 2.50 2.50 May/20 Romania 1.75 1.75 Jun/20 Hungary 0.60 0.75 Jul/20 Czech Republic 0.25 0.25 Jun/20 Mustafa Akan, Natalia Konovalova26 Countries Last Previous Dates Poland 0.10 0.10 Jul/20 Bulgaria 0.00 0.00 Jul/20 Estonia 0.00 0.00 Jul/20 Latvia 0.00 0.00 Jul/20 Lithuania 0.00 0.00 Jul/20 Slovakia 0.00 0.00 Jul/20 Slovenia 0.00 0.00 Jul/20 S o u r c e : TR ADING ECONOMICS (2020). Thus, the first implication of the outcome of the results of this paper is that reducing interest rates will not induce growth significantly except perhaps in Croatia and Romania where interest rates are relatively high. The second factor inf luencing growth positively was the production tech- nology (production function). However, each firm operating in any economy where free market economy is practiced has a different technology. It is not possible to suggest incentives concerning technology for each firm in an econ- omy. However, it is possible to make a general statement on incentives to invest in better production technologies for a country. Results of a ranking study by Schwab (2019) on competitiveness index of Central European Countries among 141 countries are shown in the column (1) on table 2. Results of another ranking study by Getzoff (2020) on the Most Technologically Advanced Countries (among 67) are shown on column (2) on the same table. OurWorldInData.org (2021) has produced data on labor pro- ductivity in all countires on the basis of Feenstra, Inklaar and Timmer (2015). The data for Central and Eastern European Countires are produced in column 3 of table 2 also. Table 2 indicates that Estonia, Czech Republic, Slovenia, and Poland are al- ready quite advanced in production technology. Thus, incentives regarding in- vestment in production technologies in these countries will have less impact on growth than they will have in other countries in the region especially in Bul- garia, Hungary, and Croatia. Table 1. Interest… https://ourworldindata.org/economic-growth https://ourworldindata.org/economic-growth oPtimAl incentives for economic growth… 27 Incentives to increase the labor productivity in Hungary, Bulgaria, Estonia, Latvia, and Lithunia will have more impact on growth than in other countries in the region since labor productivity is relatively low in these countries. Table 2. Competitivity, Advancement, and Labor Productivity in Central and Eastern Europe Countries 1 (Competitiveness) 2 (Advanced) 3 (Productivity) (GDP$/Hour, 2017) Croatia 63 39 34.25 Romania 51 48 33.25 Hungary 47 35 25.32 Czech Republic 32 28 34.76 Poland 37 34 31.06 Bulgaria 49 51 23.26 Estonia 31 20 28.03 Latvia 41 30 28.09 Lithuania 39 27 29.46 Slovakia 42 41 32.73 Slovenia 35 33 35.63 S o u r c e : constructed by authors based on: data TR ADING ECONOMICS (2020); OurWorldInData. org (2021). Skorupinska and Torrent-Sellens (2015) analyses stage of transition to knowl- edge economy in CEE countries and shows that there is considerable gab be- tween CEE and EU countries in human capital, infrastructure, innovation ca- pacity and quality institutions. A summary of a study by Radosevic (2017) on Technology in Central and Eastern European Economies is presented below. Coupling domestic technology efforts with the import of new equipment and management practices could help promote technology upgrading. Mustafa Akan, Natalia Konovalova28 Production capability and engineering, in addition to research, are impor- tant antecedents to development and innovation. Production capability is the most significant driver of productivity growth. Innovation policy in CEE is based solely on R&D, imitating best practices in northern Europe, instead of addressing regionally specific challenges. CEE economies over-prioritize attracting foreign direct investment and do not place enough emphasis on the quality of subsidiary developments. Radosevic’s conclusions, from a macro point of view, support the same con- clusions we derived in our micro study from technology point of view. The third implication of our results was that the inventory holding costs were important in inducing growth. However, it is not possible to make gener- alizations concerning these costs with regards to a whole region other than the recommendations made in the previous section. Incentives in terms of holding costs should be based on sectors. Companies producing very high value prod- ucts such as cars, computers, planes should get better incentives since they are more likely to have higher holding costs. The general conclusions on Central European Countries are: 1. Lowering interest rates will not have an appreciable effect on growth. 2. Incentives regarding improvement of production technology will be ef- fective. 3. Incentives regarding holding costs should be instituted on sectorial basis. The incentives related to points 1 and above are relatively easy to institute. However, the incentives regarding point 2 will take a longer time to make a sig- nificant difference in spurring growth. 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