Roberto Perotti November 20 Problem set - Solutions Exercise Suppose the process for income is y t = y + ε t + βε t () Using the permanent income model studied in class, find the expression for c t c t as function of ε t. Answer From class, we know that From the process for y t we know that c t c t = c t E t c t (2) r E t y t+i E t y t+i = () () i s=0 E t y t E t y t = ε t (4) Hence E t y t+ E t y t+ = βε t (5) E t y t+i E t y t+i = 0 i 2 (6) c t c t = r ε t( + β ) (7) Exercise 2 Use the permanent income model studied in class, and the stochastic process y t+ = λy t + ε t+ that we also used in class. Show that the consumption function (linking c t to its determinants y t and A t ) is c t = ra t + r λ y t (8)
What happens if λ = or λ = 0? c t = ra t + r = ra t + r i=0 i=0 = ra t + r y t = ra t + For the effects of λ = 0 and λ =, see Exercise. y t+i () i (9) λ i y t () i (0) λ +r () r λ y t (2) Exercise Consider the two-period model max U = u(c ) + c,c 2 + ρ u(c 2) () s.t. The utility function is c + c 2 = y + y 2 (4) u(c) = c σ σ for σ 0, σ (5) u(c) = log c for σ = (6) a) Write the Lagrangean of the problem and show that λ (the Lagrange multplier on the intertemporal budget constraint) is equal to the marginal utility of consumption in period. The Lagrangean is: L = c σ σ + + ρ c2 σ [ σ + λ y + y 2 c c ] 2 (7) 2
Differentiating with respect to c and c 2 we get the first order conditions b) Show that the Euler equation is c σ = λ (8) ( ) + ρ c σ 2 = λ (9) ( c c 2 ) σ = + ρ (20) Combining (8) and (9) we get the result. c) Show that the elasticity of intertempral substitution log(c /c 2 ) log() (2) is equal to /σ. Interpret the intertemporal elasticity of substitution. Take the logs of both sides of (20) and then differentiate. To interpret the intertemporal elasticity of substitution, note that can be interpreted as the price of consumption in relative to consumption in 2: the higher the interest rate, the more of c 2 I give up if I consume one extra unit of c. hence the intertemporal elasticity of substitution gives the percentage change in (c /c 2 ) in response to an increase in the interest rate by one percentage point (recall that d log() dr). d) Now find the consumption function for c, relating the latter with y and y 2 Just use the intertemporal budget constraint (4) to replace c 2 with y 2 + ( + r)(y c ), to get ( c = y + y ) [ ] 2 ( + ρ) σ () σ σ + ( + ρ) σ (22) e) Show the effect on savings in period of a change in the interest rate. Assume first that y 2 = 0, then show how the conclusions are influenced when y 2 is positive. If y 2 = 0, from the equation above we have [ ] c = y ( + ρ) σ () σ σ + ( + ρ) σ (2)
and s = y c (24) An increase in r increases c and reduces savings if σ >, otherwise if σ <. The intuition is the following. When y 2 = 0, a change in r has two effects on c. The first is a substitution effect: when r increases, the relative price of c (relative to c 2 ) increases; this indiuces the individual to reduce c and increases savings. The second is an income effect: when r increases, the individual can get the same c 2 with lower savings; this pushes c up and savings down. The substitution effect is stronger, the higher the intertemporal elasticity of substitution, i.e. the lower σ is. Hence, if σ <, the substitution effect prevails, and savings increases. Now assume that y 2 > 0: there is now a third effect, the wealth effect: an increase in r reduces wealth, given y 2 : this induces the consumer to reduce c, hence to increase savings. The wealth effect then reinforces the substitution effect. Exercise 4 Consider a consumer who lives for two periods, with incomes y and y 2. The consumer can borrow at rate r D, while she can save at the rate r A, with r D > r A. a) Show the budget constraint of the consumer in the space (c, c 2 ). See Figure b) Determine the necessary and suffi cient condition for the optimal consumption to be precisely (y, y 2 ). These conditions are in the form of inequalities of the derivatives of U(c, c 2 ) evaluated at (y, y 2 ). The condition is r A U C (C, C 2 ) U C2 (C, C 2 ) r D (25) c) Suppose that the inequalities in b) are satisfied. Show graphically that if y increases by a small amount y, then c / y = and c 2 / y = 0, which is closer to the keynesian consumption function. Explain. See Figure. The consumer wanted to move closer to consumption smoothing but, if the inequality above was satisfied, it was too costly. The rise in y allows the consumer to get closer to consumption smoothing. d) Now suppose that the consumer cannot borrow at all, i.e. that r D =. Explain with a graph. 4
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See Figure. Exercise 5 Consider an individual who lives three periods ("young", "middle-aged" and "old"). Her incomes in the three periods are Y = Y, Y 2 = ( + γ)y, and Y = 0. The individual wants to do perfect consumption smoothing, i.e. C = C 2 = C. The interest rate is 0. a) Compute savings in each periods. ( ) 2 + γ C = C = C 2 = C = Y (26) Hence ( ) γ S = Y C = Y ( ) + 2γ S 2 = Y ( + γ) C = Y ( ) 2 + γ S = C = Y (27) (28) (29) b) Suppose that there is no population growth, i.e. at each point in time one third of the population alive belongs to the three ages, i.e. one third earns Y, one third Y ( + γ), and one third 0. What is aggregate savings in the economy? Let S indicate aggregate savings. S = S + S 2 + S = 0 (0) c) Suppose that a pension system is introduced whereby each young and middleaged is forced to save A, and gets 2A when old. What happens to the savings of the individuals? It depends. If A < S, the individual will continue to save S ; if instead A > S, her savings will increase. The same happens for the middle-aged. d) Now suppose that population grows at rate n each period. Compute aggregate savings of the economy. Show how aggregate savings change when γ increases. Explain, and compare with your answer to b). 7
Let S indicate aggregate savings by the young, and similarly for S 2 and S. At each point in time, if we normalize the population of the old at, there are + n middle-aged and ( + n) 2 yopung. Hence S = S + S 2 + S () = ( + n) 2 S + ( + n)s 2 + S (2) [ ] n 2 ( γ) + n = Y > 0 () When γ increases, aggergate savings falls. The reason is that the wealth of the young increases (they anticipate that their income when middle aged wil increase), hence their consumption increases; given their income, their savings when young falls. A similar reasoning applies to the middle aged. Because the two groups that save aremore numerous, aggergate savings increases.. e) What is the rate of growth of aggregate income in this economy? Y t = ( + n) 2 Y + ( + n)( + γ)y (4) Hence Y t+ = ( + n) Y + ( + n) 2 ( + γ)y (5) Ŷ t+ Y t+ Y t Y t (6) = + n (7) f) Does your answer support the often-heard statement that "in order to grow more a country has to save more"? This statement would be correct outside the steady-state in a Solow growth model. Here, aggregate savings is 0 if there is no population growth. If there is population growth, both growth and aggregate savings are positive, but both are a consequence of population grotwh: it is not the case that higher savings causes higher growth. 8