A Real Intertemporal Model with Investment
Part 1
Chapter 9
Topics in Macroeconomics 2
Economics Division
University of Southampton
April 2010
Chapter 9, Part I
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Topics in Macroeconomics
The Representative Consumer
The Representative Firm
Government, Mkt Clearing and Competitive Equilibrium
Goals in this Chapter
I
Construct a real intertemporal model that will serve as a
basis for studying money and business cycles in Chapters
10-12.
I
Understand the investment decision of the firm.
I
Show how macroeconomic shocks affect the economy.
I
Focus on the implications of future expectations for current
macroeconomic performance, and the difference between
temporary and permanent shocks.
Chapter 9, Part I
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Topics in Macroeconomics
The Representative Consumer
The Representative Firm
Government, Mkt Clearing and Competitive Equilibrium
Model Ingredients
I
Current and future periods.
I
Representative Consumer - labour/leisure choice
consumption/savings decision
I
Representative Firm - hires labour and invests in current
period, hires labour in future
I
Government - spends and taxes in present and future, and
borrows on the credit market.
Chapter 9, Part I
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Topics in Macroeconomics
The Representative Consumer
The Representative Firm
Government, Mkt Clearing and Competitive Equilibrium
Consumer’s Budget Constraint
Current labour supply
Current demand for consumption goods
Outline
The Representative Consumer
Consumer’s Budget Constraint
Current labour supply
Current demand for consumption goods
The Representative Firm
Production and Investment
Profits and current labour demand
Representative Firm’s Investment Decision
Government, Mkt Clearing and Competitive Equilibrium
Government
Market clearing
Competitive equilibrium
Chapter 9, Part I
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Topics in Macroeconomics
The Representative Consumer
The Representative Firm
Government, Mkt Clearing and Competitive Equilibrium
Consumer’s Budget Constraint
Current labour supply
Current demand for consumption goods
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 0 = w 0 (h − l 0 ) + π 0 − T 0 + (1 + r )S p
3. Consumer’s lifetime budget constraint:
C+
C0
w 0 (h − l 0 ) + π 0 − T 0
= w (h − l) + π − T +
1+r
1+r
Chapter 9, Part I
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Topics in Macroeconomics
The Representative Consumer
The Representative Firm
Government, Mkt Clearing and Competitive Equilibrium
Consumer’s Budget Constraint
Current labour supply
Current demand for consumption goods
Marginal conditions for Consumer:
1. Consumer’s current-period marginal condition:
MRSl,C = w
2. Consumer’s future-period marginal condition:
MRSl 0 ,C 0 = w 0
3. Consumer’s intertemporal marginal condition:
MRSC,C 0 = 1 + r
Chapter 9, Part I
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Topics in Macroeconomics
The Representative Consumer
The Representative Firm
Government, Mkt Clearing and Competitive Equilibrium
Consumer’s Budget Constraint
Current labour supply
Current demand for consumption goods
Marginal conditions for Consumer:
1. Consumer’s current-period marginal condition:
MRSl,C = w
2. Consumer’s future-period marginal condition:
MRSl 0 ,C 0 = w 0
3. Consumer’s intertemporal marginal condition:
MRSC,C 0 = 1 + r
Chapter 9, Part I
6/29
Topics in Macroeconomics
The Representative Consumer
The Representative Firm
Government, Mkt Clearing and Competitive Equilibrium
Consumer’s Budget Constraint
Current labour supply
Current demand for consumption goods
Outline
The Representative Consumer
Consumer’s Budget Constraint
Current labour supply
Current demand for consumption goods
The Representative Firm
Production and Investment
Profits and current labour demand
Representative Firm’s Investment Decision
Government, Mkt Clearing and Competitive Equilibrium
Government
Market clearing
Competitive equilibrium
Chapter 9, Part I
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Topics in Macroeconomics
The Representative Consumer
The Representative Firm
Government, Mkt Clearing and Competitive Equilibrium
Consumer’s Budget Constraint
Current labour supply
Current demand for consumption goods
Current labour supply
Current labour supply determined by current real wage, w ,
interest rate, r , and lifetime wealth, we
I
Current labour supplied increases with the real wage
(substitution effects are assumed to dominate income
effects).
I
Current labour supplied increases with the real interest
rate, through intertemporal substitution of leisure.
I
Current labour supplied decreases with lifetime wealth
(e.g. taxes fall).
Chapter 9, Part I
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Topics in Macroeconomics
The Representative Consumer
The Representative Firm
Government, Mkt Clearing and Competitive Equilibrium
Consumer’s Budget Constraint
Current labour supply
Current demand for consumption goods
Representative Consumer’s Current Labour Supply
Chapter 9, Part I
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Topics in Macroeconomics
The Representative Consumer
The Representative Firm
Government, Mkt Clearing and Competitive Equilibrium
Consumer’s Budget Constraint
Current labour supply
Current demand for consumption goods
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
The Representative Consumer
The Representative Firm
Government, Mkt Clearing and Competitive Equilibrium
Consumer’s Budget Constraint
Current labour supply
Current demand for consumption goods
Outline
The Representative Consumer
Consumer’s Budget Constraint
Current labour supply
Current demand for consumption goods
The Representative Firm
Production and Investment
Profits and current labour demand
Representative Firm’s Investment Decision
Government, Mkt Clearing and Competitive Equilibrium
Government
Market clearing
Competitive equilibrium
Chapter 9, Part I
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Topics in Macroeconomics
The Representative Consumer
The Representative Firm
Government, Mkt Clearing and Competitive Equilibrium
Consumer’s Budget Constraint
Current labour supply
Current demand for consumption goods
Current demand for consumption goods
Current demand for consumption goods determined by interest
rate, r , and lifetime wealth, we, which is affected by present
value of current and future income and taxes
I
Current demand for consumption goods as a function of
current income, Y , is increasing, with a MPC less than 1.
I
Current demand for consumption goods as a function of
current income shifts down if the interest rate increases
(assuming again that substitution effect dominates)
I
Current demand for consumption goods 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
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Topics in Macroeconomics
The Representative Consumer
The Representative Firm
Government, Mkt Clearing and Competitive Equilibrium
Production and Investment
Profits and current labour demand
Representative Firm’s Investment Decision
Outline
The Representative Consumer
Consumer’s Budget Constraint
Current labour supply
Current demand for consumption goods
The Representative Firm
Production and Investment
Profits and current labour demand
Representative Firm’s Investment Decision
Government, Mkt Clearing and Competitive Equilibrium
Government
Market clearing
Competitive equilibrium
Chapter 9, Part I
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Topics in Macroeconomics
The Representative Consumer
The Representative Firm
Government, Mkt Clearing and Competitive Equilibrium
Production and Investment
Profits and current labour demand
Representative Firm’s Investment Decision
Firm’s production and investment:
1. Firm’s current-period production function:
Y = zF (K , N)
2. Firm’s future-period production function:
Y 0 = z 0 F (K 0 , N 0 )
3. Evolution of the firm’s capital stock:
K 0 = (1 − d )K + I
Chapter 9, Part I
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Topics in Macroeconomics
The Representative Consumer
The Representative Firm
Government, Mkt Clearing and Competitive Equilibrium
Production and Investment
Profits and current labour demand
Representative Firm’s Investment Decision
Firm’s production and investment:
1. Firm’s current-period production function:
Y = zF (K , N)
2. Firm’s future-period production function:
Y 0 = z 0 F (K 0 , N 0 )
3. Evolution of the firm’s capital stock:
K 0 = (1 − d )K + I
Chapter 9, Part I
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Topics in Macroeconomics
The Representative Consumer
The Representative Firm
Government, Mkt Clearing and Competitive Equilibrium
Production and Investment
Profits and current labour demand
Representative Firm’s Investment Decision
Firm’s objective: present value profits
1. Firm’s current-period profits:
π = Y − wN − I
2. Firm’s future-period profits:
π 0 = Y 0 − w 0 N 0 + (1 − d )K 0
3. The firm maximizes the present value of profits:
V =π+
π0
1+r
by choosing current labour demand, N, future labour
demand, N 0 , and current investment, I.
Chapter 9, Part I
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Topics in Macroeconomics
The Representative Consumer
The Representative Firm
Government, Mkt Clearing and Competitive Equilibrium
Production and Investment
Profits and current labour demand
Representative Firm’s Investment Decision
Firm’s objective: present value profits
1. Firm’s current-period profits:
π = Y − wN − I
2. Firm’s future-period profits:
π 0 = Y 0 − w 0 N 0 + (1 − d )K 0
3. The firm maximizes the present value of profits:
V =π+
π0
1+r
by choosing current labour demand, N, future labour
demand, N 0 , and current investment, I.
Chapter 9, Part I
15/29
Topics in Macroeconomics
The Representative Consumer
The Representative Firm
Government, Mkt Clearing and Competitive Equilibrium
Production and Investment
Profits and current labour demand
Representative Firm’s Investment Decision
Outline
The Representative Consumer
Consumer’s Budget Constraint
Current labour supply
Current demand for consumption goods
The Representative Firm
Production and Investment
Profits and current labour demand
Representative Firm’s Investment Decision
Government, Mkt Clearing and Competitive Equilibrium
Government
Market clearing
Competitive equilibrium
Chapter 9, Part I
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Topics in Macroeconomics
The Representative Consumer
The Representative Firm
Government, Mkt Clearing and Competitive Equilibrium
Production and Investment
Profits and current labour demand
Representative Firm’s Investment Decision
Intra-temporal decisions for firm:
I
Firm’s intra-temporal marginal conditions:
MPN 0 = w 0
MPN = w
I
Current labor demand is decreasing in the current wage.
I
Current labor demand shifts out (right) if TFP, z, or K
increase because the MPN increases for a given N.
How does the firm choose K 0 or I?
Chapter 9, Part I
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Topics in Macroeconomics
The Representative Consumer
The Representative Firm
Government, Mkt Clearing and Competitive Equilibrium
Production and Investment
Profits and current labour demand
Representative Firm’s Investment Decision
Representative Firms’s Current Labour Demand Curve
Chapter 9, Part I
18/29
Topics in Macroeconomics
The Representative Consumer
The Representative Firm
Government, Mkt Clearing and Competitive Equilibrium
Production and Investment
Profits and current labour demand
Representative Firm’s Investment Decision
Outline
The Representative Consumer
Consumer’s Budget Constraint
Current labour supply
Current demand for consumption goods
The Representative Firm
Production and Investment
Profits and current labour demand
Representative Firm’s Investment Decision
Government, Mkt Clearing and Competitive Equilibrium
Government
Market clearing
Competitive equilibrium
Chapter 9, Part I
19/29
Topics in Macroeconomics
The Representative Consumer
The Representative Firm
Government, Mkt Clearing and Competitive Equilibrium
Production and Investment
Profits and current labour demand
Representative Firm’s Investment Decision
Representative firm’s inter-temporal decision
1. Marginal cost: MCI = 1
I
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):
MBI =
MPK0 + (1 − d )
1+r
I
The firm produces MPK0 units of output more tomorrow for
one extra unit of capital tomorrow/investment today.
I
The firm can liquidate the remaining stock of capital.
Chapter 9, Part I
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Topics in Macroeconomics
The Representative Consumer
The Representative Firm
Government, Mkt Clearing and Competitive Equilibrium
Production and Investment
Profits and current labour demand
Representative Firm’s Investment Decision
Representative firm’s inter-temporal decision
3. Firm’s intertemporal marginal condition:
MBI =
MPK0 + (1 − d )
= 1 = MCI
1+r
Simplifying, we get net marginal product of K equals r
(opportunity cost → bonds)
MPK0 − d = r
Chapter 9, Part I
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Topics in Macroeconomics
The Representative Consumer
The Representative Firm
Government, Mkt Clearing and Competitive Equilibrium
Production and Investment
Profits and current labour demand
Representative Firm’s Investment Decision
Representative Firm’s Investment Schedule
Chapter 9, Part I
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Topics in Macroeconomics
The Representative Consumer
The Representative Firm
Government, Mkt Clearing and Competitive Equilibrium
Production and Investment
Profits and current labour demand
Representative Firm’s Investment Decision
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
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Topics in Macroeconomics
The Representative Consumer
The Representative Firm
Government, Mkt Clearing and Competitive Equilibrium
Government
Market clearing
Competitive equilibrium
Outline
The Representative Consumer
Consumer’s Budget Constraint
Current labour supply
Current demand for consumption goods
The Representative Firm
Production and Investment
Profits and current labour demand
Representative Firm’s Investment Decision
Government, Mkt Clearing and Competitive Equilibrium
Government
Market clearing
Competitive equilibrium
Chapter 9, Part I
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Topics in Macroeconomics
The Representative Consumer
The Representative Firm
Government, Mkt Clearing and Competitive Equilibrium
Government
Market clearing
Competitive equilibrium
The government’s present-value budget constraint:
G+
T0
G0
=T+
1+r
1+r
Recall:
I
The budget constraint in the future period implies that
B=
I
T 0 − G0
1+r
Replace this expression for B in the current period budget constraint to get
G=T +
T 0 − G0
|
1+r
{z }
B
I
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
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Topics in Macroeconomics
The Representative Consumer
The Representative Firm
Government, Mkt Clearing and Competitive Equilibrium
Government
Market clearing
Competitive equilibrium
Outline
The Representative Consumer
Consumer’s Budget Constraint
Current labour supply
Current demand for consumption goods
The Representative Firm
Production and Investment
Profits and current labour demand
Representative Firm’s Investment Decision
Government, Mkt Clearing and Competitive Equilibrium
Government
Market clearing
Competitive equilibrium
Chapter 9, Part I
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Topics in Macroeconomics
The Representative Consumer
The Representative Firm
Government, Mkt Clearing and Competitive Equilibrium
Government
Market clearing
Competitive equilibrium
All Markets clear
Labour markets:
Period 1: N = N s = N d
Period 2: N 0 = N s0 = N d0
Goods markets:
Period 1: C + G + I = Y
Period 2: C 0 + G0 = Y 0 + (1 − d )K 0
Credit market:
Sp = B + I
Chapter 9, Part I
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Topics in Macroeconomics
The Representative Consumer
The Representative Firm
Government, Mkt Clearing and Competitive Equilibrium
Government
Market clearing
Competitive equilibrium
Outline
The Representative Consumer
Consumer’s Budget Constraint
Current labour supply
Current demand for consumption goods
The Representative Firm
Production and Investment
Profits and current labour demand
Representative Firm’s Investment Decision
Government, Mkt Clearing and Competitive Equilibrium
Government
Market clearing
Competitive equilibrium
Chapter 9, Part I
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Topics in Macroeconomics
The Representative Consumer
The Representative Firm
Government, Mkt Clearing and Competitive Equilibrium
Government
Market clearing
Competitive equilibrium
Definition: Competitive Equilibrium
A competitive equilibrium is a set of endogenous quantities
(C,C 0 ,N,N 0 ,Y ,Y 0 ,T ,T 0 ,B) and endogenous prices (w ,w 0 ,r ) such
that, given exogenous variables (z,K ,G,G0 ), the following holds:
1. For the representative consumer, given (w, w 0 , r , T , T 0 , π, π 0 ),
the bundle (C, C 0 , N s , N s0 , S p ) maximizes the consumer’s utility
subject to his/her present value budget constraint
2. For the representative firm, given (w, w 0 , r , K ), labour demand
(N d , N d0 ) 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 0 = N s0 = N d0
Goods markets: C + G + I = Y and C 0 + G0 = Y 0 + (1 − d)K 0
Credit market: S p = B + I
Chapter 9, Part I
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