Consider a firm whose objective is to maximize its two period present-value prof
ID: 1196287 • Letter: C
Question
Consider a firm whose objective is to maximize its two period present-value profits pi + pi'/1 + r Suppose the firm produces output with a production function z K alpha N^1 - alpha. The firm chooses labor demand in both periods and investment in capital for the future period to maximize lifetime profits. The initial level of capital K is given. Write down the firm's problem, stating clearly what are the choice variables for the firm and the variables that it takes as given. Solve for the optimal amount of hours worked demanded in the current period N, optimal amount of hours worked in the future period N' and for the optimal amount of next period capital K'. What happens with K' as z' increases? What happens with K' as z' increases? What happens with K' as r increases? Briefly explain. Now, consider the firm has to pay a proportional tax t (0,1) per investment unit. (it raises the cost of investment). Write down the firm's problem. Solve for the optimal amount of capital in the next period K . Is the future period capital with investment tax K' higher or smaller than what you found in (2)? Briefly explain.Explanation / Answer
The standard model of optimal growth, interpreted as a model of a market economy with infinitely long-lived agents, does not allow separation of the savings decisions of agents from the investment decisions of firms. Investment is essentially passive: the "one good" assumption leads to a perfectly elastic investment supply; the absence of installation costs for investment leads to a perfectly elastic investment demand. On the other hand, the standard model of temporary equilibrium used in macroeconomics characterizes both the savings-consumption decision and the investment decision, or, equivalently, derives a well-behaved aggregate demand which, in equilibrium, must be equal to aggregate supply. Often, however, we want to study the movement of the temporary equilibrium over time in response to a particular shock or policy. The discrepancy between the treatment of investment in the two models makes imbedding the temporary equilibrium model in the growth model difficult. This paper characterizes the dynamic behavior of the optimal growth model with adjustment costs. It shows the similarity between the temporary equilibrium of the corresponding market economy and the short-run equilibrium of standard macroeconomic models: consumption depends on wealth, investment on Tobin's q. Equilibrium is maintained by the endogenous adjustment of the term structure of interest rates. It then shows how the equivalence can be used to study the dynamic effects of policies; it considers various fiscal policies and exploits their equivalence to technological shifts in the optimal growth problem.
The standard model of optimal growth, interpreted as a model of a market economy with infinitely long—lived agents does not allow separation of the savings decisions of agents from the investment decisions of firms. Investment is essentially passive: the "one good" assumption leads to a perfectly elastic investment supply; the absence of installation costs for investment leads to a perfectly elastic investment demand. On the other hand, the standard model of temporary equilibrium used in macroeconomics, such as the Metzler [6] model for example, characterizes both the savings—consumption decision and the investment decision, or, equivalently, derives a well—behaved aggregate demand which, in equilibrium, must be equal to aggregate supply. Often, however, we want to study the movement of the temporary equilibrium over time in response to a particular shock or policy. The discrepancy between the treatment of investment in the two models makes imbedding the temporary equilibrium model in the growth model difficult. This is a particularly serious problem if the assumption of rational expectations is made, as in this case expected future events affect the current equilibrium and it becomes impossible to characterize the current equilibrium without using an intertemporal model. The obvious solution is to modify the optimal growth model by relaxing one of the two assumptions which imply passive investment behavior. This can be done either by introducing a two—sector technology which generates a well—defined investment supply function , or it can be done by introducing installation or adjustment costs which generate a well—defined investment demand function. The purpose of this paper is to characterize the dynamic behavior of the optimal growth model 2 with adjustment costs, to show the similarity between the temporary equilibrium of the corresponding market economy and the temporary equilibrium of standard short—run macroeconomic models, and finally to show how easily this model can be used to study the dynamic effects of shocks or policies.
A real intertemporal macroeconomic model, developed in this chapter, has two key elements: the output market and the labour market. 2. A competitive equilibrium in this model is characterized by a combination of output, real interest rate, real wage rate, and employment which clears both the output and labour markets. 3. The model can be used to analyze the macroeconomic effects of various shocks to the economy. 4. For example, the model predicts that a temporary increase in government purchases increases equilibrium output, employment, and the real interest rate while decreasing the real wage rate.
average consumption must equal per capita total output less per capita government spending. For a given amount of government savings, aggregate private savings is fixed. One consumer may only reallocate consumption across time if another consumer is willing to make the complementary reallocation. Borrowing and lending can improve economic outcomes only to the extent that it can help bridge differences in consumers’ allocations of income over time. In particular, if there is a single representative consumer, this consumer is stuck with consuming her gross income (net of government spending) in the current period. The investment process allows the whole economy to effectively reallocate consumption across time..
Three economic actors: 1 A representative consumer makes consumption/savings decisions and supplies its labor. 2 A representative firm hires labor and makes production and investment decisions. 3 The government engages in government spending using tax revenues and borrowed funds. Three markets: labor market, credit market, goods market.
The consumer’s budget constraints are given by C1 + S p 1 = w1(h l1) + 1 T1 C2 = w2(h l2) + 2 T2 + (1 + r)S p 1 The intertemporal budget constraint is C1 + C2 1 + r = w1(h l1) + 1 T1 + w2(h l2) + 2 T2 1 + r Consumers solve max C1,l1,C2,l2 u(C1, l1) + u(C2, l2) subject to C1 + C2 1 + r = w1(h l1) + 1 T1 + w2(h l2) + 2 T2 1 + r
Optimality conditions: Within-period decisions: ( ul (C1,l1) uC (C1,l1) = MRSC1,l1 = w1 ul (C2,l2) uC (C2,l2) = MRSC2,l2 = w2 Intertemporal decisions: uC (C1, l1) uC (C2, l2) = MRSC1,C2 = 1 + r C1 + C2 1 + r = w1(h l1) + 1 T1 + w2(h l2) + 2 T2 1 + r
Factors affecting current labor supply, N s 1 = h l1: 1 The current real wage: We will assume that the substitution effect dominates, i.e. w1 l1 N s 1 . 2 The real interest rate: Intertemporal substitution of leisure just as for consumption. Note that our existing solution conditions imply ul(C1, l1) ul(C2, l2) = MRSl1,l2 = w1(1 + r) w2 We will assume that the substitution effect dominates, i.e. r l1 N s 1 . Note that higher future wages w2 l1 N s 1 . 3 Lifetime wealth: Consumption and leisure are normal goods, so higher lifetime wealth ( , T, etc.) causes reduced labor supply.
The Representative Firm Output Y1 = z1F(K1, N1) and Y2 = z2F(K2, N2). The firm invests some of its output in capital accumulation: K2 = (1 d)K1 + I1 Profits 1 = Y1 w1N1 I1 and 2 = Y2 w2N2 + (1 d)K2. The representative consumer owns the firm and receives profits as dividend income. The firm maximizes the present value of dividend income, V = 1 + 2 1+r . The firm solves max N1,I1,N2 z1F(K1, N1) w1N1 I1 + z2F( K2 z }| { (1 d)K1 + I1, N2) w2N2 + (1 d) K2 z }| { [(1 d)K1 + I1] 1 + r Optimality conditions: Within-period decisions: z1FN(K1, N1) = MPN(K1, N1) = w1 z2FN(K2, N2) = MPN(K2, N2) = w2 Investment decision: z2FK (K2, N2) | {z } MPK (K2,N2)
Competitive Equilibrium A competitive equilibrium is prices r, w1, w2; household allocations C1, N s 1 , C2, N s 2 ; firm allocations K1, N d 1 , I1, N d 2 ; and allocations for the government G1, G2, T1, T2 such that: 1 C1, N s 1 , C2, and N s 2 solve the household’s optimization problem. 2 N d 1 , I1, and N d 2 maximize discounted profits V = 1 + 2 1+r , given K1. 3 The government’s budget constraint is satisfied: G1 + G2 1+r = T1 + T2 1+r . 4 Labor market clearing: N d 1 = N s 1 and N d 2 = N s 2 . 5 Credit market clearing: S p 1 + S g 1 = 0 S p 1 = B1 where S p 1 = w1N s 1 + 1 T1 C1 and B1 = S g 1 = G1 T1. 6 Goods market clearing: C1 + I1 + G1 = z1F(K1, N d 1 ) and C2 + G2 = z2F(K2, N d 2 ) + (1 d)K2 where K2 = (1 d)K1 + I1.
Walras’ Law Walras’ law states that we have a redundant market clearing condition, i.e. (1) – (3) + any two of (4) – (6) automatically imply the third market clearing condition. Here we show (1) – (3), (4), and (6) (5). From (6) and (2), C1 + I1 + G1 = z1F(K1, N d 1 ) C1 = z1F(K1, N d 1 ) I1 G1 C1 = w1N d 1 + z1F(K1, N d 1 ) w1N d 1 I1 | {z } 1 G1 Using N s 1 = N d 1 from (4), the household budget constraint from (1), and B1 = G1 T1 from (3) gives C1 = w1N s 1 + 1 (B1 + T1) B1 = w1N s 1 + 1 T1 C1 | {z } S p 1 B1 = S p 1
Related Questions
drjack9650@gmail.com
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.