Macroeconomics. Lecture 5: Consumption. Hernán D. Seoane. Spring, 2016 MEDEG, UC3M UC3M

Transcription

1 Macroeconomics MEDEG, UC3M Lecture 5: Consumption Hernán D. Seoane UC3M Spring, 2016

2 Introduction A key component in NIPA accounts and the households budget constraint is the consumption It represents about 70% of total aggregate demand Consumption is important for both growth and the business cycle fluctuation Consumption (and investment) also introduces interesting features of financial markets. In other words, financial markets affect the economy through consumption and investment

3 Readings for this lecture Romer De Gregorio 3 Andolfatto 4

4 Keynesian consumption function A basic formulation of consumption function is C t = C + c (Y t T t ) Y t T t is the after-tax-income Remember some names, c is the marginal propensity of consumption

5 Keynesian consumption function C 1 PMC<1 0 Y-T Your consumption increases in the same way, regardless of after tax income level What matters here is period-t after tax

6 Keynesian consumption function We already discussed some issues about this representation of consumption But remember Lucas Critic! Other problems: this model is static No future involved, no expectations, tomorrow s income does not affect today s consumption, is this reasonable? Introduce an intertemporal theory of consumption

7 2 periods model Assume the economy only lasts for 2 periods Output is exogenous and assume the interest rate, r, is constant Then Y 1 = C 1 + A 2 C 2 = Y 2 + (1 + r)a 2 The intertemporal budget constraint is Y 1 + Y r = C 1 + C r Y 1 + Y r = C 1 + C r

8 2 periods model Agents maximize utility subject to the 2 period budget constraint U t = U(C 1, C 2 ) Assume a well behaved function (continuous, differentiable, concave and increasing in both arguments)

9 Utility Maximization Problem C 2 V(1+r) Y 2 Y Slope = -(1+r) C 2 U 1 Y 1 C 1 V C 1 Debt

10 What happens when r changes? Interest rate is a relative price Substitution effect Income effect Lender Borrower None

11 Increase in r (Romer Ch7) Initially not lender nor borrower, there is no income effect The individual s initial consumption bundle is still in the new budget constraint First period s consumption falls, the agent becomes saver (pure substitution effect, future consumption became cheaper)

12 Increase in r (Romer Ch7) C 1 < Y 1 initially. So initially so you are a lender The increase in r makes you richer: you would like to consume more C 1. You can strictly afford more than the initial bundle But the increase in r makes future goods cheaper, you would like to substitute today s consumption by future consumption: consume less C 1 Uncertain

13 Increase in r (Romer Ch7) C 1 > Y 1 initially. So initially so you are a borrower The increase in r makes you poorer: you would like to consume less C 1 But the increase in r makes future goods cheaper, you would like to substitute today s consumption by future consumption: consume less C 1 Decrease present consumption (savings rise)

14 What happens when r changes? Consider subject to Income effect max u(c 1 ) ρ u(c 2) Y 1 + Y r = C 1 + C r Assume: u(c) = C1 σ 1 1 σ Assume: u(c) = log(c)

15 Utility functions 1/σ is the elasticity of intertemporal substitution u(c) = log(c), is the limiting case of σ 1 if σ, the function tends to a linear one This parameter regulates how much an agent will respond to changes in the interest rate

16 Solving the 2 periods problem Note L = C1 σ C 1 σ [ λ Y 1 σ 1 + ρ Y 2 σ 1 + r C 1 C ] r FOC Or C σ 1 = λ C σ 2 = λ 1 + ρ 1 + r ( C1 C 2 ) σ = 1 + ρ 1 + r

17 Solving the 2 periods problem Using the intertemporal BC C 1 = And savings are ( Y 1 + Y ) r (1 + ρ) 1/σ (1 + r) 1 σ σ + (1 + ρ) 1/σ S = Y 1 C 1

18 The intertemporal budget constraint Let s go back to the intertemporal budgeetconstraint but now generalize the analysis to T periods Total income in period t is Y t = Y l,t + ra t

19 The intertemporal budget constraint Uses The income is used to consume, accumulate more assets or to pay taxes C t + T t + A t+1 A t A simplified version of the households budget constraint is then, Y l,t + ra t = C t + T t + A t+1 A t or A t+1 = Y l,t C t T t + (1 + r)a t Note this constraint holds for all periods t Moreover, constraints from different periods are linked by asset accumulation.

20 The intertemporal budget constraint Note for t + 2 A t+2 = Y l,t+1 C t+1 T t+1 + (1 + r)a t+1 Merging these 2 equations we can get (1 + r)a t = C t + T t Y l,t + C t+1 + T t+1 Y l,t r Or, if we repeat this process several times (1 + r)a t = J being the Judgment day J C t+s + T t+s Y l,t+s (1 + r) s=0 s + A t+j+1 (1 + r) J + A t r

21 The intertemporal budget constraint If J, then lim J A t+j+1 (1 + r) J = 0 Note J J C t+s (1 + r) s=0 s = Y l,t+s T t+s (1 + r) s=0 s + (1 + r)a t Present value of consumption equals the present value of after-tax-income and initial asset position

22 The permanent income hypothesis Simplify notation, set t = 0 and relabel s to t, suppose no taxes T = 0) and r = 0 As before, build a Lagrangian T L = t=1 T T C t = Y l,t + (1 + r)a 0 t=1 t=1 u(c t ) λ ( T t=1 ) T C t + Y l,t + (1 + r)a 0 t=1 This theory implies that (notice foc u (C t ) = λ, so C is constant) [ ] C = 1 T A T 0 + Y l,t t=1

23 The permanent income hypothesis This specification has a strongly different implication compared to the Keynesian theory Consumption is not a function of today s after tax income Consumption depends on the lifetime income Time pattern of income does not affect consumption Friedman (1957): the permanent income hypothesis Implication: a temporary tax cut will have small impact on consumption

24 The permanent income hypothesis Time pattern of income does not affect consumption, but it affects savings S t = Y t C t = [ Y t 1 T ] T Y t 1 T A 0 t=1 Savings change a lot when there s a change in current income relative to permanent income Note that savings is no more than future consumption

25 The permanent income hypothesis If r > 0, the initial period is t = 0 and the economy last forever t=0 C t (1 + r) t = t=0 As before, build a Lagrangian L = t=0 u(c t ) λ Compute consumption ( t=0 C t Y l,t (1 + r) t + (1 + r)a 0 (1 + r) t + t=0 ) Y l,t (1 + r) t + (1 + r)a 0

26 Liquidity constraints Liquidity constraints is a way of reconciling Keynesian theory with the dynamic analysis Suppose you don t have access to a debt market and you get the same profile of consumption as in the previous cases

27 Utility Maximization Problem C 2 V(1+r) Y 2 Y C 2 U 1 U 2 0 Y 1 C 1 V C 1 Debt

28 Liquidity Constraints On top of this, a 2nd effect of liquidity constraint It can raises savings. Even if they don t bind, they can reduce current consumption. Suppose 3 period model and quadratic utility function C a 2 C2 and gross R = 1 In period 3: C 3 = A 2 + Y 3 Replacing A 2 : C 3 = A 1 + Y 2 + Y 3 C 2 Let s consider the maximization problem for the last 2 periods

29 Liquidity Constraints Here U = FOC ( C 2 a ) 2 C2 2 + E 2 [(A 1 + Y 2 + Y 3 C 2 ) a 2 (A 1 + Y 2 + Y 3 C 2 ) 2] U C 2 = a (A 1 + Y 2 + E 2 (Y 3 ) 2C 2 ) = 0 Given the liquidity constraint, we know then that { } A1 + Y C 2 = min 2 + E 2 (Y 3 ), A 1 + Y 2 2 Liquidity constraint reduces consumption in period 2 if it is binding

30 Liquidity Constraints This implies, that in period 1, the expectation of C 2 is smaller than the one we should have without the liquidity constraint Then, by the previous result, the budget constraint in the first period (A 1 = A 0 + Y 1 C 1 ) and the law of iterated expectations (E 1 (E 2 (Y 3 )) = E 1 (Y 3 )) or C 1 < A 0 + Y 1 + E 1 (Y 2 ) + E 2 (Y 3 ) C 1 2 C 1 < A 0 + Y 1 + E 1 (Y 2 ) + E 2 (Y 3 ) 3 This is the case even if liquidity constraints do not bind ever

31 The permanent income hypothesis Usually liquidity constrained agents are called hand to mouth or Non-Ricardian agents The world is likely to be populated by both type of guys: Ricardian and Non-Ricardian The theory implies that these guys are not going to respond in the same way to tax changes

32 The random walk hypothesis If we assume uncertainty the counterpart of the previous theory is the RWH Suppose the expected utility function is quadratic E(U) = E [ T C t a 2 C2 t t=1 ] assume r = 0 and only uncertainty is coming from income. Intertemporal BC is then T t=1 E 1 [C t ] = A 0 + T t=1 E 1 [Y t ]

33 The random walk hypothesis If the agent is maximizing, then he is setting equal all expected marginal utilities Suppose she sets optimally first period consumption and consider a reduction of C 1 equal to dc together with an equal increase at some point in the future. These changes are such that the individual is optimizing Today decrease is (1 ac 1 )dc The expected value of the increase in the future is (1 ae 1 (C t ))dc If the agent is optimizing, she should be indifferent, so marginal utilities of both changes have to be equal 1 ac 1 = E 1 [1 ac t ] which is true for all t = 2, 3,..., T

34 The random walk hypothesis A solution to this problem implies that C 1 = E 1 (C t ) But this is true for all t C t = E t (C t+1 ) This implies that changes in consumption are unpredictable. In other words C t = E t 1 (C t ) + e t or, given that C t 1 = E t 1 (C t ) C t = C t 1 + e t

35 The random walk hypothesis Hall s random walk result (Hall 1978). Intuition: if you expect consumption to change, you can do a better job in smoothing consumption Suppose consumption is expected to rise. In this case, current marginal utility of consumption is larger than the expected future marginal utility of consumption Here, you would like to increase current consumption to make both marginal utilities equal again This implies that agents adjust consumption until the point where consumption is not expected to change

36 The random walk hypothesis A solution to this problem implies that C 1 = E 1 (C t ) And C 1 = 1 T ( A 0 + T t=1 E 1 (Y t ) Moreover, if you substract 2 consecutive consumption levels, you will find C 2 C 1 = 1 T 1 ( T t=2 E 2 (Y t ) ) T t=2 E 1 (Y t ) )

37 Risky assets Consumption and risky assets imply a pricing equation Suppose an agent with access to risky assets. Optimization requires u (C t ) = 1 [ ] 1 + ρ E t (1 + r i t+1 )u (C t+1 ) Marginal utilities of consumption has to be the same regardless which asset she is saving in Note u (C t ) = 1 ( ) ( )} {E t 1 + r i t+1 E t u (C 1 + ρ t+1 ) + Cov t 1 + r i t+1, u (C t+1 ) Volatility does not matter, covariance does! In other words, you don t care how risky the asset is, but you care how it does correlate with your consumption

38 CAPM Asset return (in the general equilibrium) are endogenous. They are determined by individual asset demands Suppose a quadratic utility function, C ac 2 /2 ) = (1 + ρ)u (C t ) + acov t ( 1 + r i t+1, C t+1 ) E t (1 + r i t+1 E t u (C t+1 ) the higher the covariance with consumption the higher the expected return has to be what about the risk free asset? 1 + r t+1 = (1 + ρ)u (C t ) E t u (C t+1 )

39 CAPM Subtract E t (r i t+1 ) r t+1 = acov ( t 1 + r i t+1, C t+1 ) E t u (C t+1 ) The premium an asset has to give over the risk free asset is proportional to its covariance with consumption this is a model of determination of expected returns CAPM