Intertemporal choice: Consumption and Savings

Transcription

1 Econ Elements of Economics Analysis 3 (Honors Macroeconomics) Lecturer: Chanont (Big) Banternghansa TA: Jonathan J. Adams Spring 2013 Introduction Intertemporal choice: Consumption and Savings This version: May 4, 2013 The purpose of this section is to study how household allocate resources over time, ie how they make the decision about how much to consume today, vs how much to save/borrow and consume tomorrow. We start with a simple 2-period endowment economy, and move to the infinite horizon problem. A 2-period endowment economy We study a partial equilibrium model, in a 2-period endowment economy: Setup The economy is populated by a representative household that lives for 2 periods: 1 and 2. Endowment: The household is endowed with: y 1 units of good at period 1. y 2 units of good at period 2. Preferences: The household has preferences over consumption at dates 1 and 2, represented by the following utility function: where u > 0, u < 0, 0 < β < 1. U(c 1, c 2 ) = u(c 1 ) + βu(c 2 ) In particular, utility is separable across the two periods and the period utility function (small u) is increasing and concave. β : discount factor. A low β means that the household discounts (cares less) a lot the utility it gets from consuming in the second period, and is impatient. 1

2 A high β characterizes a patient household because he cares more about future consumption. Perfect Credit Market The household can borrow and/or lend between period 1 and 2, at an interest rate r. The interest rate r is given exogenously (partial equilibrium). The household enters period 1 with no debts or assets. It exits on period 2. We denote the amount of savings between 1 and 2 by s, such that { s > 0 the household saves. s < 0 the househld borrows. Period budget constraints, maximization program, and Euler equation. The period budget constraints are: The maximization program is: The Lagrangian is c 1 + s = y 1 at period 1 c 2 = y 2 + (1 + r)s at period 2 max c1,c 2,s u(c 1 ) + βu(c 2 ) s.t. c 1 + s = y 1 [λ 1 ] c 2 = y 2 + (1 + r)s [λ 2 ] L = u(c 1 ) + βu(c 2 ) + λ 1 (y 1 c 1 s) + λ 2 (y 2 + (1 + r)s c 2 ) The first order conditions are Dividing [c 1 ] by [c 2 ] we get [c 1 ] u (c 1 ) = λ 1 [c 2 ] βu (c 2 ) = λ 2 [s] λ 1 + λ 2 (1 + r) = 0 u (c 1 ) βu (c 2 ) = λ 1 λ 2 Using the FOC on [s] to get the ratio of multipliers, we have: u (c 1 ) βu (c 2 ) = 1 + r 2

3 This is known as the Euler equation (EE). It gives an expression for the marginal rate of substitution between consumption at period 1 and consumption at period 2. It summarizes the optimal intertemporal choice for the household. The equilibrium allocation (c 1, c 2, s ) is given by the euler equation and the 2 budget constraints: u (c 1 ) βu (c 2 ) = 1 + r c 1 + s = y 1 c 2 = y 2 + (1 + r)s Intertemporal Budget Constraint, Maximization program and Euler Equation Let us start with the two period budget constraints: { c1 + s = y 1 c 2 = y 2 + (1 + r)s Dividing the second budget constraint by(1 + r) and summing we have c 1 + s + c r = y 1 + y 2 (1 + r)s r 1 + r c 1 + c r = y 1 + y r This is the intertemporal budget constraint. It states that the present value of all consumption flows should be equal to the present value of all income flows. Future income and consumption flows of period 2 are discounted by (1 + r), the interest rate. Now, the maximization program is The Lagrangian is { maxc1,c 2 u(c 1 ) + βu(c 2 ) s.t. c 1 + c 2 = y 1 + y 2 [λ] L = u(c 1 ) + βu(c 2 ) + λ The first order conditions are ( y 1 + y r c 1 c ) r [c 1 ] u (c 1 ) = λ [c 2 ] βu (c 2 ) = λ 1 3

4 Dividing [c 1 ] by [c 2 ], we have u (c 1 ) βu (c 2 ) = 1 + r Note that this is the same Euler equation as before. In other words, one can collapse the set of time constraint into one constraint and get the same optimal allocation. The equilibrium allocation is given by Savings are s = y 1 c 1 = c 2 y 2. Graphically: u (c 1 ) βu (c 2 ) c 1 + c 2 = y 1 + y 2 = 1 + r The budget constraint is given by the following equation: c 2 = [(1 + r)y 1 + y 2 ] (1 + r)c 1 For a given utility level ū, the indifference curves are given by the following equation (ū ) c 2 = u 1 u(c1 ) β The equilibrium is the point on the intertemporal budget constraint, where the marginal rate of substitution between period 1 and 2 is equal to the slope of the budget constraint. 4

5 Net Savers and Net borrowers Net savers: The agent is a net saver if s > 0 or c 1 < y 1. This is more likely if y 1 is large (compared to y 2 ). r is large. β is large. Net Borrowers: The agent is a net borrower if s < 0 or c 1 > y 1. This is more likely if y 1 is low (compared to y 2 ). 5

6 r is low. β is low. Comparative statics: Increase in y 1 We consider a temporary increase in endowment (y 1 > y 1). This has three effects on the equilibrium allocation: increase in c 1 increase in c 2 increase in s (increase in savings or decrease in borrowings) More precisely, we have a positive income shock, which would translate into an increase in c 1, and/or an increase in c 2. In fact, both have to increase at the same time. Indeed, if only c 1 or c 2 increased, the marginal rate of substitution between period 1 and 2 would be modified and the euler equation u (c 1 ) βu (c 2 ) = (1 + r) would be violated. In other words, the pure income shock translates into an increase in consumption in both periods. Consumption at period 1 and at period 2 are both normal goods. The agent exhibits a preference for consumption smoothing. 6

7 Comparative statics: Increase in y 2 We consider a temporary increase in endowment (y 2 > y 2). This has three effects on the equilibrium allocation: increase in c 1 increase in c 2 decrease in s (decrease in savings or increase in borrowings) Again we observe consumption smoothing. What matters for consumption today is the permanent income: an increase in future revenues results in higher consumption today. This highlights the importance of expectations in the economy. 7

8 Comparative statics: Increase in β An increase in β means that agent becomes more patient. This has three effects on the equilibrium allocation: decrease in c 1 increase in c 2 increase in s (increase in savings or decrease in borrowings) The increase in β decreases the marginal rate of substitution for any given (c 1, c 2 ) pair. Hence, for the Euler equation to be satisfied, we need c 1 to decrease (u (c 1 ) to increase) and c 2 to increase (u (c 2 ) to decrease), since the budget constraint is unchanged. 8

9 Comparative statics: Increase in r An increase in r has two effects. It makes the price of the good at period 2 to fall, in the sense that the technology that transforms goods at period 1 into goods at period 2 becomes more productive substitution. It makes total income (y 1 + y 2 fall in today s price, but rise (y 1(1 + r) + y 2 ) in tomorrow s price income effect. In all, we have a substitution and an income effect that makes the effects on c 1 and c 2 difficult to predict. In fact, it is useful to distinguish between borrowers and lenders. Net savers: c 1 c 2 Substitution Effect Income Effect Total? Lenders (think of an active worker, well endowed in current income y 1, but poorly endowed in future income y 2 (ie retirement)). The lenders face two phenomenon Substitution: c 2 becomes cheaper, so they substitute away from c 1 to c 2. Income: Since net lenders have s > 0, the increase in r increases the value of his future income (because y 1 is more efficiently transferred to period 2), which means there s a positive income effect on both c 1 and c 2. 9

10 Ambiguous effect on c 1, c 2 increases for sure. Net borrowers: c 1 c 2 Substitution Effect Income Effect Total? Borrowers (think of a poor University of Chicago student, well endowed in future income y 2, but poorly endowed in current income y 1 ). Substitution: c 2 becomes cheaper, so they substitute away from c 1 to c 2. Income: Since net borrowers have s < 0, the increase in r decreases the value of 10

11 his future income (he needs to pay a higher debt), which means there s a negative income effect on both c 1 and c 2. Ambiguous effect on c 2, c 1 decreases for sure. Zero net-supply of bonds, determination of the interest rate We now consider an economy where it is not possible for the household to save or borrow because bonds/savings instruments come in zero net supply. We have s = 0 c 1 = y 1 c 2 = y 2 In such an economy, the interest rate would be r = u ( c 1 ) βu ( c 2 ) 1 = u (y 1 ) βu (y 2 ) 1 In other words, the interest rate adjusts so it is equal to the marginal rate of substitution between consumption at period and period 2, evaluated at (y 1, y 2 ). In particular, the interest rate in equilibrium is: Decreasing in β: if agents are more patient, they are more willing to save, and we need the interest rate to decrease to bring net savings back to zero. Decreasing in y 1 : if agents have more endowments today, they are willing to smooth consumption and save part of this increase, and we need the interest rate to decrease to bring net savings back to zero. 11

12 increasing in y 2 : if agents anticipate more endowment tomorrow, they are willing to go into debt to increase consumption today, and we need the interest rate to increase to bring net savings back to zero. In the event of a recession (fall in y 2 ) and assuming the supply of bonds is fixed, the interest rate should fall. This is quite a good prediction: periods where the interest rate is low or negative are often predictive of recessions. The infinite-horizon endowment economy Setup The economy is populated by a representative household that lives forever (from t = 0 to ) Endowments ( The household is endowed with a flow of endowments: {y t } t=0. We ) y assume that t t=0 () converges to Ȳ. t Preferences: The household has preferences over consumption at each date {c t } t=0, represented by the following utility function: where u > 0, u < 0, and 0 < β < 1. U ({y t } t=0 ) = β t u(c t ) Perfect Credit Market The household can borrow and/or lend between each period t and t + 1, at a constant rate r, taken as exogenous. t=0 We denote the amount of savings between t and t + 1 by s t such that s t > 0 means household saves at t and s t < 0 means household borrows at t. The household enters period 0 with no debt or assets, so that using our convention, s 1 = 0. The household s savings, in present value, are asymptotically 0: lim t This is called the transversality condition. s t+1 (1 + r) t+1 = 0 12

13 Period budget constraints, maximization program, and Euler equation: The budget constraint at period t is c t + s t = y t + (1 + r)s t 1 t The constraint holds at each date t. The maximization program is max ct,st t=0 βt u(c t ) s.t. c t + s t = y t + (1 + r)s t 1 t s 1 = 0 s lim t+1 t () t+1 = 0 Denoting λ t the multiplier in front of the budget constraint at date t, the Lagrangian writes: L = β t u(c t ) + λ t (y t + (1 + r)s t 1 c t s t ) t=0 t=0 The first order conditions are, for each t: [c t ] β t u (c 1 ) = λ 1 [s t ] λ t + λ t+1 (1 + r) = 0 Now, taking the FOC of [c t ] and dividing by the FOC of [c t+1 ], we have: β t u (c t ) β t+1 u (c t+1 ) = λ t λ t+1 Using the FOC of [s t ] for the ratio of multipliers, we get β t u (c t ) β t+1 u (c t+1 ) = 1 + r u (c t ) βu (c t+1 ) = 1 + r Again this is the euler equation, giving an expression for the marginal rate of substitution between consumption at date t and consumption at date t and consumption at date t

14 The equilibrium allocation {c t, s t } is given by the euler equation, the budget constraints, and the transversality condition: u (c t) βu (c t+1 ) c t + s t s 1 = 0 s lim t+1 t () t+1 = 0 = 1 + r = y t + (1 + r)s t 1 Before getting to the resuts, let us derive the intertemporal budget constraint. Intertemporal budget constraint, maximization program and euler equation: Recipe Divide the budget constraint at date t by (1 + r) t. Sum from date 0 to date t + 1 Let t + 1 tend to infinity, and use the transversality condition. This steps are shown below: Divide by (1 + r) t. Sum date 0 c 0 + s 0 = y 0 + (1 + r)s 1 /(1 + r) 0 date 1 c 1 + s 1 = y 1 + (1 + r)s 0 /(1 + r) 1 date 2 c 2 + s 2 = y 2 + (1 + r)s 1 /(1 + r) 2.. date t 1 c t 1 + s t 1 = y t 1 + (1 + r)s t 2 /(1 + r) t 1 date t c t + s t = y t + (1 + r)s t 1 /(1 + r) t date t + 1 c t+1 + s t+1 = y t+1 + (1 + r)s t /(1 + r) t+1 c 0 () 0 + c 1 () 1 + c 2 () s 0 () 0 + s 1 () 1 + s 2 = y 0 () 0 + y 1 () 1 + y 2 () () 2 + ()s 1 + ()s 0 + ()s 1 () 0 () 1 () 2 ( t+1 ) c i (1 + r) i i=0 + c t 1 + () t 1 s t 1 + () t 1 y t 1 + () t 1 s t+1 (1 + r) t+1 = ct () + c t t+1 () t ()s t 2 () t 1 ( t+1 i=0 st () + s t t+1 () t+1 yt () t + y t+1 () t+1 y i (1 + r) i + ()s t 1 () t ) + (1 + r)s 1 + ()st () t+1 14

15 Let t + 1 Using lim t s t+1 () t+1 i=0 = 0, we get c i (1 + r) i = y i (1 + r) i = ȳ i=0 This is the intertemporal budget constraint (IBC). It states that the present value of all consumption flows should be equal to the present value of all income flows. Now we can replace the sequence of period budget constraint by the IBC and write the maximization program: max {ct} s.t. t=0 βt u(c t ) t=0 c t () t = ȳ The Lagrangian is L = ( ) β t c t u(c t ) + λ ȳ (1 + r) t t=0 t=0 The first order condition c t is: Dividing by [c t ] by [c t+1 ] we get β t u 1 (c t ) = λ (1 + r) t β t u (c t ) β t+1 u (c t+1 ) = λ/(1 + r)t λ/(1 + r) t+1 u (c t ) βu = (1 + r) t (c t+1 ) The equilibrium allocation is given by {c t } t=0 such that u (c t ) βu (c t+1 ) = (1 + r) t t=0 c t () t = ȳ Permanent Income Note that the solution to the system of equations described above does not depend on the particular sequence of endowments. {y t } t=0, but only on the present value of all the endowment flows ȳ, known as the permanent income. 15

16 Patience and interest-rate If the agent is patient, β > 1, then the euler equation implies u (c t ) = β(1 + r)u (c t+1 ) u (c t ) > u (c t+1 ) c t < c t+1 A patient household will reduce consumption in earlier dates, save and take advantage of interest payments to increase its consumption in later periods, so that {c t } is increasing in t. If the agent is impatient, β < 1, then the euler equation implies u (c t ) = β(1 + r)u (c t+1 ) u (c t ) < u (c t+1 ) c t > c t+1 An impatient household will increase consumption in earlier dates, go into debt (or save less) and reduce consumption in later periods, so that {c t } is decreasing in t. If β = 1 then u (c t ) = β(1 + r)u (c t+1 ) u (c t ) = u (c t+1 ) c t = c t+1 = c Consumption is constant over time. Th household s impatience exactly cancels with the incentive to invest. In fact, we can solve for c. Using the IBC, we have c t t=0 () t c t=0 () c c r = ȳ = ȳ = ȳ = ȳ c = r ȳ 16

17 CRRA (constant relative risk aversion) utility Definition We consider the following utility function c σ 1 σ u(c) = σ 1 for σ > 0, σ 1 σ ln c for σ = 1 σ: intertemporal elasticity of substitution. : coefficient of relative risk aversion. 1 σ We have u (c) = c σ 1 σ 1 = c 1 σ > 0 u (c) = 1 σ c 1 σ 1 = 1 σ c 1 σ σ < 0 The name of the utility function comes from the fact that the coefficient of relative aversion (CRRA) is constant. CRRA = u (c)c u (c) CRRA = Euler equation The euler equation is 1 σ c 1/σ 1 c c 1/σ CRRA = 1 σ u (c t) u (c t+1 ) = β(1 + r) 1 c σ t 1 c σ t+1 c t c t+1 c t+1 c t = β(1 + r) = [β(1 + r)] σ = [β(1 + r)] σ If σ 0 c t+1 = [β(1 + r)] σ 1 β(1 + r) c t When the intertemporal elasticity of substitution is low, consumption flows at different dates are poorly substitutable, and the household will tend to smooth consumption perfctly, no matter what the interest rate and the discount factor are. 17

18 If σ is large c t+1 c t = [β(1 + r)] σ { σ if β > 1 0 if β < 1 When the intertemporal elasticity of substitution is high: If the household is patient (β > 1 later periods. If the household is impatient (β < 1 the very short run and go into debt. ), it will postpone its consumption to ), it will consume all its endowment in When σ is high, the household is less willing to smooth consumption. Discussion about the permanent income hypothesis and borrowing constrants Our model predicts that consumption only depends on the present value of income flows ȳ and not the particular shape of endowment flows {y t }. There is definitely some empirical counterparts to this prediction: At the aggregate level, consumption is smoother than income/gdp at the individual level, there are life-cycle patterns. 18