Real Intertemporal Closed Economy Model: Consumption, Labour, Investment, and Equilibrium (Chapter 11 Study Notes)

Real Intertemporal Closed Economy Model

Introduction

The real intertemporal closed economy model is a foundational framework in macroeconomics for analyzing production, investment, consumption, and equilibrium over multiple periods. It is essential for understanding how macroeconomic shocks affect labour and goods markets, and forms the basis for studying money, business cycles, and inflation.

• Closed economy: No international trade; all transactions occur within the domestic economy.

• Intertemporal: Decisions span current and future periods, allowing analysis of savings, investment, and consumption smoothing.

Main Agents in the Model

• Representative Consumer: Makes consumption/leisure choices within a period (intratemporal) and consumption/savings choices across periods (intertemporal).

• Government: Spends and taxes in current and future periods; borrows in the credit market.

• Representative Firm: Hires labour in current and future periods; invests in the current period to increase future capital stock.

Consumer Budget Constraints

Current-Period Budget Constraint

The consumer's current disposable income is determined by wages, leisure, and dividends:

• Equation: $C + S = w(h - l) + \pi - T$ where:

  • $C$ = current consumption

  • $S$ = savings

  • $w$ = real wage

  • $h$ = total hours available

  • $l$ = leisure hours

  • $\pi$ = dividends (profits from firms)

  • $T$ = taxes

Future-Period Budget Constraint

• Equation: $C' = w'(h - l') + \pi' - T' + (1 + r)S$ where:

  • $C'$ = future consumption

  • $w'$ = future real wage

  • $l'$ = future leisure

  • $\pi'$ = future dividends

  • $T'$ = future taxes

  • $r$ = real interest rate

  • $S$ = savings from current period

Lifetime Budget Constraint

By combining the two period constraints and eliminating savings, the consumer's lifetime budget constraint is:

• Equation: $C + \frac{C'}{1 + r} = w(h - l) + \pi - T + \frac{w'(h - l') + \pi' - T'}{1 + r}$ This expresses the present value of lifetime consumption equals the present value of lifetime disposable income.

Consumer Optimization

Intratemporal Choice: Consumption vs. Leisure

Consumers choose the optimal combination of consumption and leisure by equating the marginal rate of substitution (MRS) between consumption and leisure to the real wage:

• Condition: $MRS_{C,l} = w$ where $MRS_{C,l}$ is the rate at which the consumer is willing to trade leisure for consumption.

• Graphical Representation: The optimal bundle is where the highest attainable indifference curve is tangent to the budget constraint.

Intertemporal Choice: Consumption Today vs. Tomorrow

• Condition: $MRS_{C,C'} = 1 + r$ where $MRS_{C,C'}$ is the rate at which the consumer is willing to trade current consumption for future consumption, and $r$ is the real interest rate.

Labour Market

Labour Supply

The consumer's labour supply depends on the real wage and the real interest rate:

• Substitution Effect: Higher real wage increases incentive to work (less leisure).

• Income Effect: Higher real wage increases income, allowing more leisure (less work).

• Standard Assumption: Substitution effect dominates, so higher real wage increases hours worked.

Labour Demand

The firm's labour demand is determined by the marginal product of labour (MPL):

• Condition: $MPL = w$ Firms hire labour up to the point where the value of the marginal product equals the real wage.

• Labour Demand Curve: Downward sloping due to diminishing returns to labour.

Production Function

General Form

• Equation: $Y = zF(K, N)$ where:

  • $Y$ = output

  • $z$ = total factor productivity (TFP)

  • $K$ = capital stock

  • $N$ = labour input

• Properties: Increasing $K$ or $N$ increases output, but with diminishing marginal returns.

Firm Investment Decision

Capital Accumulation

• Equation: $K' = (1 - d)K + I$ where:

  • $K'$ = future capital stock

  • $d$ = depreciation rate

  • $I$ = investment

Optimal Investment Rule

• Condition: $MPK' = r + d$ where $MPK'$ is the marginal product of future capital, $r$ is the real interest rate, and $d$ is depreciation.

• Firms invest until the net marginal product of capital equals the opportunity cost (real interest rate plus depreciation).

Government Budget Constraints

• Current-Period: $G = T + B$

  • $G$ = government spending

  • $T$ = taxes

  • $B$ = government borrowing

• Future-Period: $G' = T' + (1 + r)B$

• Present-Value Constraint: $G + \frac{G'}{1 + r} = T + \frac{T'}{1 + r}$

Equilibrium in the Model

Labour Market Equilibrium

• Determined by intersection of labour supply and demand curves.

• Equilibrium real wage and employment set aggregate output.

Goods Market Equilibrium

• Determined by intersection of output supply and output demand curves.

• Equilibrium real interest rate and aggregate output are set.

Output Supply and Demand Curves

Output Supply Curve

Shows relationship between output and the real interest rate, holding other factors constant.

• Shifts right with increases in taxes, total factor productivity ($z$), or capital stock ($K$).

Output Demand Curve

Shows relationship between aggregate demand and the real interest rate.

• Shifts right with increases in government spending ($G$), decreases in taxes ($T$), increases in $z$, or increases in expected future productivity.

Effects of Macroeconomic Shocks

Temporary Increase in Government Spending ($G$)

• Shifts output demand curve right.

• Financed by higher taxes, reducing lifetime wealth and consumption.

• Multiplier effect: Total increase in output is less than the initial increase in $G$ due to crowding out of consumption and investment.

Increase in Total Factor Productivity ($z$)

• Shifts production function and output supply curve right.

• Increases marginal product of labour and capital, raising output, employment, and real wage.

Increase in House Prices

• Increases household wealth, relaxing borrowing constraints.

• Shifts output demand curve right; increases consumption and residential investment.

• Raises real interest rate, employment, and aggregate output.

Table: Summary of Shocks and Their Effects

Shock	Output Supply Curve	Output Demand Curve	Aggregate Output	Real Interest Rate
Increase in $G$	No shift	Shifts right	Increases	Increases
Increase in $z$	Shifts right	Shifts right	Increases	Decreases
Increase in house prices	No shift	Shifts right	Increases	Increases
Increase in expected future productivity	No shift	Shifts right	Increases	Increases

Key Terms and Definitions

• Marginal Propensity to Consume (MPC): The fraction of additional income that is spent on consumption.

• Marginal Product of Labour (MPL): The additional output produced by one more unit of labour.

• Marginal Product of Capital (MPK): The additional output produced by one more unit of capital.

• Intertemporal Substitution: The process of shifting consumption or leisure between periods in response to changes in the real interest rate.

• Multiplier Effect: The total change in output resulting from an initial change in autonomous spending (e.g., government purchases).

Example: Consumer Optimization

• Suppose the real wage increases. The substitution effect leads the consumer to work more (less leisure), while the income effect allows for more leisure. If the substitution effect dominates, hours worked increase.

• Graphically, the optimal point is where the highest indifference curve is tangent to the budget constraint.

Additional info:

• Some equations and diagrams were inferred from standard macroeconomic models and textbook conventions.

• Table entries and shock effects are based on typical results from the real intertemporal model.