CLASS #5: CONSUMPTION: THE INTERTEMPORAL CHOICE

"Consumption is the sole purpose of all production" (Adam Smith) (Ch. 4 and Ch. 8, sections 8.1 through 8.4) What are we trying to understand today?

• What determines aggregate consumption in a closed economy
• What determines the -real- interest rate in a closed economy
• How do fiscal policies (taxes, transfers, government spending) affect consumption

WHAT DETERMINES CONSUMPTION? INTERTEMPORAL CHOICE (Irving Fisher)

The decision to consume is intimately connected with the decision to save... and what is saving but future consumption? The bottom line of this class will be:
consumption decisions do not depend on current income (flow), but on wealth (stock) and wealth is defined in a forward looking manner Simple model. Assumptions:

• two periods
• endowment economy (no production: farmers collecting fruits from tree)
• initial wealth (W), income in period one (Y1) and two (Y2)
• the household can borrow and lend at an interest rate r

The household's problem:

$$max_{C1,C2} U(C1,C2) = max_{C1,C2} \ln (C1) + \beta \ln (C2)$$

subject to the constraint "what you save this period you can consume next period"

(1+r)(Y1+W-C1)=C2-Y2

GRAPHICAL ANALYSIS

• Optimal choice for the household, given the interest rate
• Effect of changes in wealth and income
• Effect of changes in interest rate: substitution effect, income effect

FORMAL ANALYSIS

substituting the budget constraint into the objective function we obtain:

max_C1 U(C1,Y2+(1+r)(Y1+W-C1))

from the first order conditions we obtain:

$$\frac{U_1(C1,C2) }{U_2(C1,C2)}=(1+r)$$

the household's optimal choice of C1 and C2 is such that the marginal rate of substitution between consumption today and consumption tomorrow is equal to (1+r)
the real interest rate (1+r) is the relative price of consumption today versus consumption tomorrow: INTERTEMPORAL CHOICE if we assume that utility is logarithmic, then the first order conditions become:

$$C2=C1 \beta (1+r)$$

after substituting into the budget constraint we find that the solution is:

$$C1= \frac{1}{1+ \beta}(W+ Y1+ \frac{Y2}{1+r})$$

$$C2= \frac{(1+r) \beta}{1+ \beta}(W + Y1+ \frac{Y2}{1+r})$$

if we further assume that $\beta(1+r)=1$, or $\beta=\frac1{1+r}$, we find that the solution is:

$$C1=C2= \frac{1+r}{1+2r}(W+ Y1+ \frac{Y2}{1+r})$$

and if r=0, we obtain:

$$C1=C2= \frac{1+r}{1+2r}(W+ Y1+ \frac{Y2}{1+r})$$

Consumption Smoothing!

WHAT HAVE WE LEARNED?

• the household likes to smooth consumption over time, irrespectively of what the original pattern of income is
• the household's consumption decisions depends on wealth, defined in a forward looking manner: current wealth, plus the PRESENT VALUE of present and future income: W+Y1+Y2/(1+r)
• what is a "present value"?
• PERMANENT INCOME THEORY (Milton Friedman)
• what do we do with the Keynesian consumption function: C=a+cY?

CONSUMPTION IN THE CLOSED ECONOMY

"general equilibrium": prices have to be such that in each period aggregate demand is equal to aggregate supply

C1=W+Y1, C2=Y2

if the household is the whole economy, the following must hold: the relative price of consumption tomorrow versus consumption today (interest rate) has to make the household happy to consume the available resources: Y1+W in the first period, and Y2 in the second period

THE REAL INTEREST RATE AND THE BUSINESS CYCLE

What does the theory say about how the real interest rate moves with the cycle? remember: $(1+r)= \frac{C2}{C1 \beta}$ if C1=Y1, C2=Y2:

$$(1+r)= \frac{Y2}{Y1 \beta}$$

implication: the real interest rate is high during recession (expensive to borrow: everybody wants to borrow) and low during booms (plenty of resources): "countercyclical" like taking taxis in the rain