A Real Intertemporal Model with Investment Part 1
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1 A Real Intertemporal Model with Investment Part 1 Chapter 9 Topics in Macroeconomics 2 Economics Division University of Southampton April 2009 Chapter 9, Part I 1/29 Topics in Macroeconomics
2 Goals in this Chapter Construct a real intertemporal model that will serve as a basis for studying money and business cycles in Chapters Understand the investment decision of the firm. Show how macroeconomic shocks affect the economy. Focus on the implications of future expectations for current macroeconomic performance, and the difference between temporary and permanent shocks. Chapter 9, Part I 2/29 Topics in Macroeconomics
3 Model Ingredients Current and future periods. Representative Consumer - labour/leisure choice consumption/savings decision Representative Firm - hires labour and invests in current period, hires labour in future Government - spends and taxes in present and future, and borrows on the credit market. Chapter 9, Part I 3/29 Topics in Macroeconomics
4 Outline The Representative Consumer Government Market clearing Competitive equilibrium Chapter 9, Part I 4/29 Topics in Macroeconomics
5 Consumer s budget constraints: 1. Consumer s current-period budget constraint: C + S p = w(h l) + π T 2. Consumer s future-period budget constraint: C = w (h l ) + π T + (1 + r)s p 3. Consumer s lifetime budget constraint: C + C 1 + r = w(h l) + π T + w (h l ) + π T 1 + r Chapter 9, Part I 5/29 Topics in Macroeconomics
6 Marginal conditions for Consumer: 1. Consumer s current-period marginal condition: MRS l,c = w 2. Consumer s future-period marginal condition: MRS l,c = w 3. Consumer s intertemporal marginal condition: MRS C,C = 1 + r Chapter 9, Part I 6/29 Topics in Macroeconomics
7 Marginal conditions for Consumer: 1. Consumer s current-period marginal condition: MRS l,c = w 2. Consumer s future-period marginal condition: MRS l,c = w 3. Consumer s intertemporal marginal condition: MRS C,C = 1 + r Chapter 9, Part I 6/29 Topics in Macroeconomics
8 Outline The Representative Consumer Government Market clearing Competitive equilibrium Chapter 9, Part I 7/29 Topics in Macroeconomics
9 determined by current real wage, w, interest rate, r, and lifetime wealth, we Current labour supplied increases with the real wage (substitution effects are assumed to dominate income effects). Current labour supplied increases with the real interest rate, through intertemporal substitution of leisure. Current labour supplied decreases with lifetime wealth (e.g. taxes fall). Chapter 9, Part I 8/29 Topics in Macroeconomics
10 Representative Consumer s Current Labour Supply Chapter 9, Part I 9/29 Topics in Macroeconomics
11 An Increase in the Real Interest Rate Shifts the Current Labour Supply Curve to the Right Chapter 9, Part I 10/29 Topics in Macroeconomics
12 Outline The Representative Consumer Government Market clearing Competitive equilibrium Chapter 9, Part I 11/29 Topics in Macroeconomics
13 determined by interest rate, r, and lifetime wealth, we, which is affected by present value of current and future income and taxes as a function of current income, Y, is increasing, with a MPC less than 1. as a function of current income shifts down if the interest rate increases (assuming again that substitution effect dominates) as a function of current income shifts up if lifetime wealth increases (e.g. decrease in present value of taxes; increase in future income) Chapter 9, Part I 12/29 Topics in Macroeconomics
14 Outline The Representative Consumer Government Market clearing Competitive equilibrium Chapter 9, Part I 13/29 Topics in Macroeconomics
15 Firm s production and investment: 1. Firm s current-period production function: Y = zf(k, N) 2. Firm s future-period production function: Y = z F(K, N ) 3. Evolution of the firm s capital stock: K = (1 d)k + I Chapter 9, Part I 14/29 Topics in Macroeconomics
16 Firm s production and investment: 1. Firm s current-period production function: Y = zf(k, N) 2. Firm s future-period production function: Y = z F(K, N ) 3. Evolution of the firm s capital stock: K = (1 d)k + I Chapter 9, Part I 14/29 Topics in Macroeconomics
17 Firm s objective: present value profits 1. Firm s current-period profits: π = Y wn I 2. Firm s future-period profits: π = Y w N + (1 d)k 3. The firm maximizes the present value of profits: V = π + π 1 + r by choosing current labour demand, N, future labour demand, N, and current investment, I. Chapter 9, Part I 15/29 Topics in Macroeconomics
18 Firm s objective: present value profits 1. Firm s current-period profits: π = Y wn I 2. Firm s future-period profits: π = Y w N + (1 d)k 3. The firm maximizes the present value of profits: V = π + π 1 + r by choosing current labour demand, N, future labour demand, N, and current investment, I. Chapter 9, Part I 15/29 Topics in Macroeconomics
19 Outline The Representative Consumer Government Market clearing Competitive equilibrium Chapter 9, Part I 16/29 Topics in Macroeconomics
20 Intra-temporal decisions for firm: Firm s intra-temporal marginal conditions: MP N = w MP N = w Current labor demand is decreasing in the current wage. Current labor demand shifts out (right) if TFP, z, or K increase because the MP N increases for a given N. How does the firm choose K or I? Chapter 9, Part I 17/29 Topics in Macroeconomics
21 Representative Firms s Current Labour Demand Curve Chapter 9, Part I 18/29 Topics in Macroeconomics
22 Outline The Representative Consumer Government Market clearing Competitive equilibrium Chapter 9, Part I 19/29 Topics in Macroeconomics
23 Representative firm s inter-temporal decision 1. Marginal cost: MC I = 1 The firm has to give up one unit of output today for one extra unit of capital tomorrow/investment today. 2. Marginal benefit (present value): MB I = MP K + (1 d) 1 + r The firm produces MP K units of output more tomorrow for one extra unit of capital tomorrow/investment today. The firm can liquidate the remaining stock of capital. Chapter 9, Part I 20/29 Topics in Macroeconomics
24 Representative firm s inter-temporal decision 3. Firm s intertemporal marginal condition: MB I = MP K + (1 d) = 1 = MC I 1 + r Simplifying, we get net marginal product of K equals r (opportunity cost bonds) MP K d = r Chapter 9, Part I 21/29 Topics in Macroeconomics
25 Representative Firm s Investment Schedule Chapter 9, Part I 22/29 Topics in Macroeconomics
26 The Optimal Investment Schedule The optimal investment schedule shifts to the right if current capital decreases or future total factor productivity is expected to increase. (see numerical example in Williamson) Chapter 9, Part I 23/29 Topics in Macroeconomics
27 Outline Government Market clearing Competitive equilibrium The Representative Consumer Government Market clearing Competitive equilibrium Chapter 9, Part I 24/29 Topics in Macroeconomics
28 Government Market clearing Competitive equilibrium The government s present-value budget constraint: G + G 1 + r = T + T 1 + r Recall: The budget constraint in the future period implies that B = T G 1 + r Replace this expression for B in the current period budget constraint to get G = T + T G 1 + r }{{} B Rearranging gives the government present value budget constraint above. The LHS is the present value of spending, which must be equal to the present value of taxes collected on the RHS Chapter 9, Part I 25/29 Topics in Macroeconomics
29 Outline Government Market clearing Competitive equilibrium The Representative Consumer Government Market clearing Competitive equilibrium Chapter 9, Part I 26/29 Topics in Macroeconomics
30 All Markets clear Government Market clearing Competitive equilibrium Labour markets: Period 1: N = N s = N d Period 2: N = N s = N d Goods markets: Period 1: C + G + I = Y Period 2: C + G = Y + (1 d)k Credit market: S p = B + I Chapter 9, Part I 27/29 Topics in Macroeconomics
31 Outline Government Market clearing Competitive equilibrium The Representative Consumer Government Market clearing Competitive equilibrium Chapter 9, Part I 28/29 Topics in Macroeconomics
32 Government Market clearing Competitive equilibrium Definition: Competitive Equilibrium A competitive equilibrium is a set of endogenous quantities (C,C,N,N,Y,Y,T,T,B) and endogenous prices (w,w,r) such that, given exogenous variables (z,k,g,g ), the following holds: 1. For the representative consumer, given (w, w, r, T, T, π, π ), the bundle (C, C, N s, N s, S p ) maximizes the consumer s utility subject to his/her present value budget constraint 2. For the representative firm, given (w, w, r, K ), labour demand (N d, N d ) and investment (I) maximizes present value profits. 3. The government present value budget constraint holds. 4. Markets clear: Labour markets: N = N s = N d and N = N s = N d Goods markets: C + G + I = Y and C + G = Y + (1 d)k Credit market: S p = B + I Chapter 9, Part I 29/29 Topics in Macroeconomics
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