problem-set-1-solutions_sherrard

Ry an Sherrard Econ 204A Octob er 5, 2018 Problem Set 1 Problem 1.1. Consider the t w o-p erio d consumption mo del: Individuals ha v e initial assets A, earn in terest r on assets, and earn w age income ( w 1 , w 2 ). They maximize utilit y U = u ( c 1 ) + \u03b2 u ( c 2 ). (a) Assume u ( c ) = ln( c ) i. Solv e for optimal consumption and p erio d-1 asset holdings as functions of w age income, the in terest rate, and the time discoun t factor, \u03b2 . Discuss under what conditions a marginally higher in terest rate reduces consumption. Rather than solv e this problem t wice (once no w and once in part b) I\u2019ll instead solv e this prob- lem for an y arbitrary p o w er utilit y function and plug in \u03b3 = 1 later. max 1 1 \u2212 \u03b3 c 1 \u2212 \u03b3 1 + \u03b2 1 \u2212 \u03b3 c 1 \u2212 \u03b3 2 s.t. a 1 = A + w 1 \u2212 c 1 a 2 = (1 + r ) a 1 + w 2 \u2212 c 2 Using the terminal condition a 2 = 0, we can recursivley solv e for the in tertemporal budget con- strain t. c 1 + c 2 1 + r = A + w 1 + w 2 1 + r W e can no w either set-up the lagrangian or recognize that standard p o w er utilit y parameters (i.e \u03b3 > 0) imply diminishing MRS. M RS = p 1 p 2 \u21d4 \ue012 c 2 c 1 \ue013 \u03b3 1 \u03b2 = 1 + r \u21d4 c 2 = \u03b2 1 \u03b3 (1 + r ) 1 \u03b3 c 1 Plugging in to the in tertemp oral B.C. c 1 [1 + \u03b2 1 \u03b3 (1 + r ) 1 \u03b3 ] = A + w 1 + w 2 1 + r c \u2217 1 = [1 + \u03b2 1 \u03b3 (1 + r ) 1 \u2212 \u03b3 \u03b3 ] \u2212 1 [ A + w 1 + w 2 1 + r ] \u21d2 c \u2217 2 = \u03b2 1 \u03b3 (1 + r ) 1 \u03b3 1 + \u03b2 1 \u03b3 (1 + r ) 1 \u2212 \u03b3 \u03b3 [ A + w 1 + w 2 1 + r ] Plugging in \u03b3 = 1, c \u2217 1 = 1 1 + \u03b2 [ A + w 1 + w 2 1 + r ] c \u2217 2 = \u03b2 (1 + r ) 1 + \u03b2 [ A + w 1 + w 2 1 + r ] 1 Plugging in for a 1 , \u21d2 a 1 = \u03b2 1 + \u03b2 [ A + w 1 ] \u2212 w 2 (1 + \u03b2 )(1 + r ) T o determine ho w consumption in eac h p erio d 1 and 2 c hanges as w e increase the in terest rate, simply tak e the deriv ativ e with resp ect to r. \u2202 c 1 \u2202 (1 + r ) = \u2212 w 2 (1 + \u03b2 )(1 + r ) 2 \u2264 0 (with equalit y if w 2 = 0) \u2202 c 2 \u2202 (1 + r ) = \u03b2 1 + \u03b2 [ A + w 1 ] ( 0 i\ufb00 A \u2212 w 1 and > 0 i\ufb00 A > \u2212 w 1 ) In tuitiv ely , as (1 + r ) \u2191 , consumption in p erio d 2 b ecomes relativ ely less exp ensiv e. Th us, the sub- stitution e\ufb00ect decreases c 1 and increases c 2 . If w 2 = 0, then income e\ufb00ect is p ositiv e and cancels out the substitution e\ufb00ect for p erio d 1 consumption. Otherwise, the income e\ufb00ect is su\ufb03ciently small (p ossibly negativ e) so that c 1 decreases. F or p erio d 2 consumption, so long as individuals ha v e su\ufb00ucien tly small initial debt, the income e\ufb00ect is su\ufb03ciently positive (though possibly still negativ e) suc h that the substitution e\ufb00ect dominates and c 2 increases. Con v ersely , so long as in- dividuals hold su\ufb03cien tly high initial debt, the income e\ufb00ect will b e negative and dominate the substitution e\ufb00ect. Hence, for consumption to decrease in both p erio ds, a su\ufb03cient condition is that households ha v e high initial debt holdings. ii. Sho w that the dep endence of p erio d-1 consumption on ( w 1 , w 2 ) can b e expressed in terms of p ermanen t income. Recall that p ermanen t income is de\ufb01ned as y p = T P t =1 (1 + r ) 1 \u2212 t w t T P t =1 (1 + r ) 1 \u2212 t Let\u2019s assume that \u03b2 \u2248 1 1+ r . Then, c 1 = 1 1 + 1 1+ r [ A + w 1 + w 2 1 + r ] \u21d4 c 1 = 1 1 + 1 1+ r A + y p 2 (b) Assume u ( c ) = c 1 \u2212 \u03b3 / (1 \u2212 \u03b3 ), where \u03b3 > 0 and \u03b3 6 = 1. Do the same as in (a). In the discussion, iden tify whic h results apply for all \u03b3 , and whic h ones only for \u03b3 greater or less than one. F rom b efore, c \u2217 1 = [1 + \u03b2 1 \u03b3 (1 + r ) 1 \u2212 \u03b3 \u03b3 ] \u2212 1 [ A + w 1 + w 2 1 + r ] \u21d2 c \u2217 2 = \u03b2 1 \u03b3 (1 + r ) 1 \u03b3 1 + \u03b2 1 \u03b3 (1 + r ) 1 \u2212 \u03b3 \u03b3 [ A + w 1 + w 2 1 + r ] Plugging in to our a 1 equation, \u21d2 a \u2217 1 = \ue010 1 \u2212 [1 + \u03b2 1 \u03b3 (1 + r ) 1 \u2212 \u03b3 \u03b3 ] \u2212 1 \ue011 [ A + w 1 ] \u2212 [1 + \u03b2 1 \u03b3 (1 + r ) 1 \u2212 \u03b3 \u03b3 ] \u2212 1 w 2 1 + r As b efore, let\u2019s determine when consumption decreases as a result of a higher in terest rate. Recall that the substitution e\ufb00ect unam biguously decreases c 1 and increases c 2 . The income e\ufb00ect has an am biguous e\ufb00ect, which depends on whether the individual is a b orrow or sa ver. Because the IE m ust b e negativ e in order to decrease c 2 , w e kno w that a necessary condition for ( c 1 , c 2 ) \u2193 is that individuals are b orro w ers. Note, ho wev er, that the IE m ust b e su\ufb03ciently negativ e to o v ercome the SE for c 2 \u2193 . Hence, the necessary and su\ufb03cient condition for c 2 to decrease is also su\ufb03cien t condition for c 1 to decrease. F or simplicit y , I\u2019ll tak e a monotonic transformation of c 2 \ufb01rst. ln c \u2217 2 = 1 \u03b3 ln \u03b2 + 1 \u03b3 ln(1 + r ) \u2212 ln \ue010 1 + \u03b2 1 \u03b3 (1 + r ) 1 \u2212 \u03b3 \u03b3 \ue011 + ln \ue012 A + w 1 + w 2 1 + r \ue013 \u21d2 \u2202 ln c \u2217 2 \u2202 (1 + r ) = 1 \u03b3 (1 + r ) \u2212 1 \u2212 \u03b3 \u03b3 \u03b2 1 \u03b3 (1 + r ) 1 \u2212 2 \u03b3 \u03b3 1 + \u03b2 1 \u03b3 (1 + r ) 1 \u2212 \u03b3 \u03b3 \u2212 w 2 (1 + r ) \u2212 2 A + w 1 + w 2 1+ r A su\ufb03cien t condition for this to b e negativ e is 1 (1 \u2212 \u03b3 ) \u03b2 1 \u03b3 (1 + r ) 1 \u2212 \u03b3 \u03b3 1 + \u03b2 1 \u03b3 (1 + r ) 1 \u2212 \u03b3 \u03b3 + \u03b3 w 2 (1 + r ) \u2212 1 A + w 1 + w 2 1+ r Note that if w e only considered when consumption in p erio d 1 w ould decrease, we see that when \u03b3 > 1 there is an am biguous a\ufb00ect on c 1 . When \u03b3 1, ho wev er, we see that c 1 is decreasing in r : \u2202 c 1 \u2202 (1 + r ) = \u2212 \ue010 1 + \u03b2 1 \u03b3 (1 + r ) 1 \u2212 \u03b3 \u03b3 \ue011 \u2212 2 \ue012 \u03b2 1 \u03b3 1 \u2212 \u03b3 \u03b3 (1 + r ) 1 \u2212 2 \u03b3 \u03b3 \ue013 \ue014 A + w 1 + w 2 1 + r \ue015 \u2212 \ue010 1 + \u03b2 1 \u03b3 (1 + r ) 1 \u2212 \u03b3 \u03b3 \ue011 \u2212 1 \ue014 w 2 (1 + r ) 2 \ue015 T o sho w that p erio d-1 consumption can b e written in terms of p ermanen t income, again assume 3 that \u03b2 \u2248 1 / (1 + r ) c \u2217 1 = w 1 + w 2 1+ r + A 1 + 1 (1+ r ) 1 /\u03b3 (1 + r ) 1 \u2212 \u03b3 \u03b3 \u21d4 c \u2217 1 = w 1 + w 2 1+ r + A 1 + 1 1+ r \u21d4 c \u2217 1 = y p + 1 1 + 1 1+ r A 4 Problem 1.3. Consider and individual with utilit y function U = ln( c 1 ) + \u03b2 ln( c 2 ) + \u03b2 2 ln( c 3 ). The p erson has lab or incomes w 1 , w 2 , w 3 . Let a t +1 = (1 + r ) a t + w t \u2212 c t denote assets carried into the next p erio d. The interest rate r is constant and assets a 1 > 0 are giv en. (a) Sp ecift the budget equations, derive the in tertemp oral B.C. and deriv e \ufb01rst-order conditions for optimalit y . The problem can b e form ulated as follo ws. max ln( c 1 ) + \u03b2 ln( c 2 ) + \u03b2 2 ln( c 3 ) s.t. a 2 = (1 + r ) a 1 + w 1 \u2212 c 1 a 3 = (1 + r ) a 2 + w 2 \u2212 c 2 a 4 = (1 + r ) a 3 + w 3 \u2212 c 3 a 4 = 0 By recursiv e substitution, w e \ufb01nd the in tertemp oral budget constrain t: c 1 + c 2 1 + r + c 3 (1 + r ) 2 = (1 + r ) a 1 + w 1 + w 2 (1 + r ) + w 3 (1 + r ) 2 ) No w, w e can set up the Lagrangian. L = ln( c 1 ) + \u03b2 ln( c 2 ) + \u03b2 2 ln( c 3 ) + \u03bb \ue012 (1 + r ) a 1 + w 1 + w 2 (1 + r ) + w 3 (1 + r ) 2 ) \u2212 c 1 + c 2 1 + r + c 3 (1 + r ) 2 \ue013 \u21d2 L c 1 = 1 c 1 = \u03bb \u21d2 L c 2 = \u03b2 c 2 = \u03bb 1 + r \u21d2 L c 3 = \u03b2 2 c 3 = \u03bb (1 + r ) 2 T ogether, these imply the follo wing optimatlit y conditions. c 2 = \u03b2 (1 + r ) c 1 and c 3 = \u03b2 2 (1 + r ) 2 c 1 5 (b) Supp ose that \u03b2 = 1 / (1 + r ), w t = w is constant. Solv e for p erio d-1 consumption as a function of lab or incomes and of in tial assets. De\ufb01ne the individual\u2019s p ermanen t income. Determine the marginal prop ensit y to consume in p erio d-1 from p erio d-1 labor income; compute the MPC v alue for \u03b2 = 0 . 9 Notice that from the optimalit y conditions, \u03b2 \u2248 1 / (1 + r ) \u21d2 c 1 = c 2 = c 3 . Plugging in to the budget constrain t and rearranging, c \u2217 1 = 1 1 + 1 1+ r + 1 (1+ r ) 2 A + w c \u2217 2 = 1 1 + 1 1+ r + 1 (1+ r ) 2 A + w c \u2217 3 = 1 1 + 1 1+ r + 1 (1+ r ) 2 A + w Recall the de\ufb01nition of p ermanen t income from problem 1.1. Substituting in w t = w , w e \ufb01nd that y p = w . No w, let\u2019s \ufb01nd the marginal prop ensit y to consume. c \u2217 1 = 1 1 + 1 1+ r + 1 (1+ r ) 2 \ue014 A + w 1 + w 2 1 + r + w 3 (1 + r )2 \ue015 \u21d2 \u2202 c \u2217 1 \u2202 w 1 = 1 1 + 1 1+ r + 1 (1+ r ) 2 Notice that this is less than 1. Intuitiv ely , this mak es sense due to the p ermanent income h yp othesis. That is that individuals will consume a constan t fraction of initial w ealth plus their p ermanen t income. Notice, though, that p ermanent income is composed of lab or-income from each perio d. Th us, individuals should in fact b e less resp onsiv e to 1 p erio d c hanges in income than to c hanges in their p ermanen t income. Since \u03b2 = 0 . 9 \u2248 1 / (1 + r ), we \ufb01nd that the MPC in this example is roughly 0.37. 6 (c) Again assuming \u03b2 = 1 / (1 + r ) and constan t w, show that assets a t are a declining sequence and that a t /a t \u2212 1 is also decreasing o v er time. Can y ou explain why this mak es sense economically? Use the same economic argumen t to mak e a conjecture ab out the b eha vior of a t /a t \u2212 1 in problems with more than T = 3 p erio ds, notably for T \u2192 \u221e . Plugging in to the asset holding equations, a 2 = 1 + 1 1+ r 1 + 1 1+ r + 1 (1+ r ) 2 a 1 a 3 = 1 1 + 1 1+ r + 1 (1+ r ) 2 a 1 T o sho w that assets are declining, simply sho w that a t \u2212 a t \u2212 1 0. a 2 \u2212 a 1 = \u2212 1 (1+ r ) 2 1 + 1 1+ r + 1 (1+ r ) 2 a 1 0 a 3 \u2212 a 2 = \u2212 1 1+ r 1 + 1 1+ r + 1 (1+ r ) 2 a 1 0 Th us, w e see that assets are a declining series. Next, lets sho w that a t /a t \u2212 1 is a declining series. a 2 a 1 = 1 + 1 1+ r 1 + 1 1+ r + 1 (1+ r ) 2 a 3 a 2 = 1 1 + 1 1+ r \u21d2 \u2206 a t a t \u2212 1 = \u2212 1 1+ r [1 + 1 1+ r + 1 (1+ r ) 2 ][1 + 1 1+ r ] 0 Th us, a t /a t \u2212 1 is a declining series. This mak es sense b ecause of the terminal condition. In other w ords, assets at the end of time ha v e no v alue; th us, holding assets b ecomes less and less v aluable eac h p erio d. As time go es on, they b egin sa ving less and less as result. In other words, the rate at whic h households decrease their assets will also increase. F or a \ufb01nite time intertemporal choice problem suc h as this one, y ou can imagine an asset curv e that lo oks something lik e this: 7 time assets F or in\ufb01nite time horizon problems, you can imagine the v alue of assets declines to 0 asymptotically . 8

problem-set-1-solutions_sherrard

UFV

Macroeconomia I

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