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## Consumption

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Outline

- Consumption – the biggest component of GDP/national expenditure; a good deal smoother than income.
- The two period model.
- Friedman’s permanent income hypothesis PIH - infinitely lived representative agent etc.
- Modligiani’s life cycle hypothesis LCH – finite life, saving for retirement, population dynamics.
- Hall’s consumption function – uncertainty, rational expectations and the consumption Euler equation.
- Euler equations versus (approx.) solved out consumption functions – pros and cons.
- Example of solved out consumption function for US.

Basic Two Period Model (1)

Diagram:

- Axes - c1’y1 on horizontal axis (the present) and c2,y2 on vertical axis (the future).
- Intertemporal preferences: Regular shaped indifference curves (as opposed to linear or L shaped ones).
- Less than perfect trade-off between c1 and c2 so want to smooth consumption over time.
- Intertemporal budget line:

c1+c2/(1+r) = y1 + y2/(1+r)

(You can add an initial endowment a0(1+r) if you want to the RHS of the budget.)

Two Period Model (2)

- Budget constraint is a straight line thru’ (y1,y2) point with slope equal to minus 1/(1+r).
- No borrowing or lending restrictions.
- Borrowing and lending rates are the same.
- Intertemporal budget constraint got by combining period 1 and period 2 budget constraints:

c1 + a1 = y1

c2 =a1(1+r) + y2

Equilibrium in Two Period Model

- Equilibrium where highest attainable indifference curve is tangential to the budget line.
- You may be a borrower (c1 > y1) or lender (c1 < y1) in period 1.
- First order condition (FOC):

slope of indifference curve

= slope of budget line

ie. marginal rate of substitution (MRS) between c1 and c2 = 1/(1 + r).

D

- Consumption today financed on credit

M

Y2

(ii)

(ii) Consumption loan repayment (including interest)

R

C2

IC3

IC2

(i)

IC1

-(1+r)

Y1

C1

B

Figure 6.3(a)Optimal consumption: borrower

Consumption tomorrow

0

Consumption today

- Saving from this period’s income

(ii) Additional consumption next period

R

C2

(ii)

(i)

-(1+r)

C1

Figure 6.3(b)Optimal consumption: lender

D

Consumption tomorrow

Y2

A

IC3

IC2

IC1

Y1

B

0

Consumption today

FOC and Euler Equation*

- Suppose preferences are additive over time so U(c1,c2) = u(c1) + u(c2) where 0 < < 1 is a discount factor.
- MRS = -dc2/dc1 holding U constant = u'(c1) / (u'(c2)), where u'(c1) is the marginal utility of c1 etc.
- Thus FOC may be re-written as:

u'(c1) = (1+r)u'(c2)

FOC and Euler Equation*

u'(c1) = (1+r)u'(c2)

- This is just a non-stochastic Euler equation!
- Note intuition – indifferent between shifting one unit of consumption between the present and the future.
- Complete smoothing of consumption (c1 = c2) when = 1/ (1+r).

CRRA Preferences*

- CRRA preferences appealing – constant savings rate & fixed allocation of wealth across assets when interest rates constant.
- u(c) = c1-γ/(1-γ) with γ positive; u'(c) = c-γso Euler equation is:

c1-γ = (1+r)c2-γ

- Take natural logs and note that ln(1+r) is approx. equal to r so:

lnc2 = (ln )/γ + r/γ

CRRA Preferences (2)*

- The elasticity of intertemporal substitution EIS is the coeff. on r in the Euler equation.
- The EIS is 1/γ, the inverse of the constant coeff. of relative risk aversion.
- The Euler equation implies that a higher interest rate increases savings (c1 falls and c2 rises).
- However, need to examine this effect in more detail. (Why? Only looking at slope of budget line not position of line).

Playing Around with the Basic Two Period Model

- Rise in permanent income (both y1 and y2 rise) – outward parallel shift in budget line. c1 and c2 both rise.
- Rise in current or future income – budget line shifts out parallel but not by as much as above. Ditto for c1 and c2.
- Current consumption is higher if future income rises even if current income is unchanged!
- A transitory rise in income may be represented by a small rise in y1 (and possibly a offsetting small fall in y2?). c1 and c2 only rise by a small amount.

Real Interest Rate Effects (1)

- Suppose r rises.
- Budget line swivels around (y1,y2) and is steeper.
- Need to look at substitution and wealth effects.
- Substitution effect given by Euler equation.
- Substitution effect on c1 is negative.

Real Interest Rate Effects (2)

- For borrower, wealth effect on c1 is also negative.
- For lender, wealth effect on c1 is positive.
- Overall, the effect of a rise in real interest rate on current consumption is not clear cut.
- Empirical consensus is that interest rate effect is small and negative.
- Size of effect depends on incidence of credit constraints and initial wealth, inter alia.

R´

R´

B´

B´

Figure 6.9Effect of an increase in the interest rate: negative income effect for borrowers, positive for lenders

D

D

A

Consumptiontomorrow

Consumption tomorrow

R

R

A

B

B

Consumption today

Consumption today

(a) Student Crusoe(borrower)

(b) Professional athlete(lender)

Credit Constraints (1)

- Assume that representative agent cannot borrow in period 1.
- Budget line is now discontinuous at (y1,y2).
- Budget line same as before in lending region i.e. to left of (y1,y2).
- Budget line drops down to horizontal axis in borrowing region i.e. to right of (y1,y2).

Credit Constraints (2)

- Now a corner solution at (y1,y2) is a distinct possibility.
- A rise in future income y2 has no effect on current consumption if credit constrained.
- A permanent or transitory rise in current income has a large effect if credit constrained (marginal propensity to consume is one).
- Interest rate effects smaller or zero if credit constrained.

B

Figure 6.11With a credit constraint, the choice set is reduced.

C

Consumption tomorrow

A

R

D

0

Consumption today

Permanent Income & Life Cycle Hypotheses (1)

- Can generalize analysis from two periods to many or an infinite number of periods.
- Standard model often called PIH–LCH model.
- Original permanent income model of consumption uses a rational, infinitely lived, representative consumer.
- Emphasis on different response of consumption to permanent and transitory changes in income etc.

A´´R´´

Figure 6.5Temporary vs. permanent income change

D´

Temporary: R to R´

Permanent: R to R´´

D

Consumption tomorrow

Y2´

R´

Y2

A´

A=R

Y1

Y1´

B

B´

B´´

0

Consumption today

PIH and LCH (2)

- In the life cycle model, aggregate consumption derived from behaviour of individual consumers (of different ages) with finite lifespans.
- Consumption smoothing and the life cycle pattern of income mean that the young borrow, the middle aged save and the retired dis-save.
- Obviously, aggregate consumption depends positively on population and income growth.
- The level of savings also depends on length of retirement relative to length of working life.

Stochastic Income & Interest Rates

- Solved-out consumption functions useful e.g.

c1 = k(r)W

where wealth W = a0(1+r) + y1 + y2/(1 +r) +…. and k(.) is a known function of the real interest rate r.

- Difficult to derive exact results in PIH-LCH model when income and interest rates are random.
- Interest rates often assumed constant and point expectations of future income used.
- Hall’s (1978) insight – look at Euler equation.

Hall’s Consumption Equation

- The stochastic Euler equation for the infinitely lived representative consumer is:

u'(c1) = E1(1+r1)u'(c2)

where Et is the conditional expectation at time t given the information set It.

- Aside: Can rearrange Euler equation to get pricing kernel / stochastic discount factor.
- Rational expectations assumed.

When Does Consumption Follow A Random Walk?

- Under very special and unrealistic assumptions, Euler equation implies that consumption is a random walk.
- When u(c) is quadratic and = 1/(1+r), then ct = ut with Et(ut|It) = 0 so both ct and ut are innovations (unpredictable).
- Since Et(ct|It) = 0, Et(ct|It) = ct-1.

Stochastic Euler Equations (1)

- Hall’s Euler equation is only a FOC, as noted already.
- It does not tell you anything about the effects of income shocks, uncertainty etc.
- To examine these sorts of issues, you need to embed it in a bigger model!
- This is one reason why some argue that approximate solved out consumption functions are more useful.

Stochastic Euler Equations (2)

- Assuming CRRA preferences, joint normality of rt and ct etc., the best we can do is:

Etlnct = (ln )/γ + rt/γ+½(σct)2/γ

which shows that uncertainty increases savings (since ct rises and ct-1 falls as the variance of ct rises).

- Long list of assumptions – rational expectations, representative agent, no credit constraints etc.

Testing Consumption Euler Equations

- Consumption Euler equations do not fare very well empirically.
- For example, if the basic model is correct, then variables in the information set at time t-1 should not help in predicting lnct.
- A natural test of this hypothesis is to include the prediction of lnyt, using variables dated t-1. or even t-2, in the regression of lnct on a constant, rt and a proxy for (σct)2.
- Predicted income growth is always highly significant.

Solved Out Consumption Function for US

- See separate note for example of solved out consumption function for US.
- Source: Muellbauer (1994), Consumers Expenditure, Oxford Review of Economic Policy.

Summary (1)

- Rational consumers attempt to smooth consumption over time, by borrowing in bad times (or when young) and saving in good times (or in middle age) .
- Consumption is primarily driven by the present discounted value of current and future non-labour income and initial assets.
- Financial market imperfections generate credit constraints. Current income matters more for credit constrained consumers.

Summary (2)

- The effect of a change in the real interest rate is ambiguous since wealth effects differ for lenders & borrowers. Overall the effect is probably small and negative.
- Over the life cycle, consumption is smoothed by borrowing when young, saving in middle age and dis-saving when retired.
- Temporary changes in income (or other disturbances) have small effects. Permanent changes or shocks have large effects.

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