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**Anthony Murphy Nuffield College anthony.murphy@nuffield.ox.ac.uk**

Consumption Anthony Murphy Nuffield College

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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.

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**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.)

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**Indifference curves: Normal case**

Fig. 6.02(a) Indifference curves: Normal case Consumption tomorrow Consumption today Figure 6.2(a)

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**Indifference curves: Zero substitution**

Fig. 6.02(b) Indifference curves: Zero substitution Consumption tomorrow Consumption today Figure 6.2(b)

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**Indifference curves: Constant substitution**

Fig. 6.02(c) Indifference curves: Constant substitution Consumption tomorrow Consumption today Figure 6.2(c)

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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

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**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).

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**Optimal consumption: borrower**

Fig. 6.03 Optimal consumption: borrower IC3 B D IC2 IC1 (i) Consumption today financed on credit M Y1 Y2 (ii) (ii) Consumption loan repayment (including interest) Consumption tomorrow R C1 C2 -(1+r) Consumption today Figure 6.3(a)

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**Optimal consumption: lender**

Fig. 6.03 Optimal consumption: lender D (i) Saving from this period’s income Consumption tomorrow (ii) (ii) Additional consumption next period R C1 C2 Y2 A IC3 IC2 IC1 -(1+r) Y1 B Consumption today Figure 6.3(b)

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**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)

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**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).

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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-γ so 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

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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).

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**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.

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**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.

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**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.

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Fig. 6.09 Effect of an increase in the interest rate: negative income effect for borrowers, positive for lenders B´ B´ D D A R´ Consumption tomorrow Consumption tomorrow R R R´ A B B Consumption today Consumption today (a) Student Crusoe (borrower) (b) Professional athlete (lender) Figure 6.9

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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).

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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.

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**With a credit constraint, the choice set is reduced.**

Fig. 6.11 With a credit constraint, the choice set is reduced. C Consumption tomorrow A B R D Consumption today Figure 6.11

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**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.

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**Temporary vs. permanent income change**

Fig. 6.05 Temporary vs. permanent income change D´ Temporary: R to R´ Permanent: R to R´´ D A´´R´´ Consumption tomorrow Y2´ R´ Y2 A=R A´ Y1 Y1´ B B´ B´´ Consumption today Figure 6.5

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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.

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**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.

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**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.

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**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.

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**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.

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**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.

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**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.

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**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.

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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.

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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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Appendix 4.A A Formal Model of Consumption and Saving.

Appendix 4.A A Formal Model of Consumption and Saving.

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