Department of Applied Eonomis Johns Hopkins University Eonomis 6 Maroeonomi Theory and Poliy Prolem Set 4 Suggested Solutions Professor Sanjay Chugh Summer Optimal Choie in the Consumption-Savings Model with Credit Constraints: A Numerial Analysis Consider our usual two-period onsumption-savings model Let preferenes of the representative onsumer e desried y the utility funtion u (, ) + β, where denotes onsumption in period one and denotes onsumption in period two The parameter β is known as the sujetive disount fator and measures the onsumer's degree of impatiene in the sense that the smaller is β, the higher the weight the onsumer assigns to present onsumption relative to future onsumption Assume that β / For this partiular utility speifiation, the marginal utility β funtions are given y u(, ) and u(, ) The representative household has initial real finanial wealth (inluding interest) of a The household earns y 5 units of goods in period one and y units in period two The real interest rate paid on assets held from period one to period two equals % (ie, r ) a Calulate the equilirium levels of onsumption in periods one and two (Hint: Set up the Lagrangian and solve) Suppose now that lenders to this onsumer impose redit onstraints on the onsumer Speifially, they impose the tightest possile redit onstraint the onsumer is not allowed to e in det at the end of period one, whih implies that the onsumer s real wealth at the end of period one must e nonnegative ( a ) (Note: here, a is defined as eing exlusive of interest, in ontrast to the definition of a aove) What is the onsumer s hoie of period-one and periodtwo onsumption under this redit onstraint? Briefly explain, either logially or graphially or oth Does the redit onstraint desried in part enhane or diminish welfare (ie, does it inrease or derease lifetime utility)? Speifially, find the level of utility under the redit onstraint and ompare it to the level of utility otained under no redit onstraint Suppose now that the onsumer experienes a temporary inrease in real inome in period one to y 9, with real inome in period two unhanged
d Calulate the effet of this positive surprise in inome on and, supposing that there is no redit onstraint on the onsumer e Finally, suppose that the redit onstraint desried in part is ak in plae Will it e inding? That is, will it affet the onsumer s hoies? Solution: a The onsumer s prolem is to maximize lifetime utility (given y u (, ) sujet to the LBC The Lagrangian for this prolem is thus y L (,, λ ) u (, ) + λ a + y+, + r + r where we must inlude the nonzero initial real wealth a The first-order onditions with respet to and are u(, ) λ λ u(, ) + r Comining these, we get the usual onsumption-savings optimality ondition, u(, ) ( + r) u(, ) (ie, the MRS equals the slope of the LBC) Using the the given utility funtion, at the optimal hoie the following ondition must e satisfied: β ( + r ) Solving this expression for as a funtion of gives ( + r) β With the speifi values given, this turns out to e Sustituting this into the lifetime udget onstraint then yields y + a + y+ + r + r Solving for gives + r y a y r + + r, + + whih, when using the values provided, yields 79 and hene 79 Note also that, although you were not asked to ompute it, you ould find the implied value for a using the period one udget onstraint + a a + y This yields that a 9, indiating that the household hooses to e a detor at the end of period one The imposition of these redit onstraints will e inding on the onsumer s ehavior That is, it will alter the hoies made y the household, as an e seen
from the fat that in the asene of the redit onstraints in part a, the onsumer hose to e in det at the end of period one Now, eing restrited to hold a nonnegative asset position at the end of period one, it will hoose that asset position losest to its unrestrited hoie ut whih also satisfies the redit onstraint that is, the onsumer will hoose a The period one udget onstraint, + a a + y then implies that 6 The household will simply onsume all it an in period one, whih is the sum of its endowment and initial assets (inlusive of interest inome on those initial assets) It remains now to solve for Examining the period two udget onstraint, + a ( + r ) a + y with the ondition a imposed and B * shows that y Extension: At this redit-onstrained hoie of onsumption, the MRS learly does not equal the slope of the LBC The slope of the LBC is the market interest rate + r, as usual However, we an define an effetive interest rate for this onsumer, whih is the interest rate that would need to prevail for the hoie 6, to e the unrestrited optimal hoie We an otain this from the ondition u(, ) ( + r) u(, ) This ondition is the same as where we started question a with, exept now, knowing values for and, we will use it to determine the onsumer s effetive interest rate Plugging the values 6, into this ondition and solving for the interest rate gives us r 4 as the effetive interest rate, the interest rate that would have made this hoie the onsumer s unrestrited optimal hoie With the values for onsumption in eah of the two periods from parts a and, the utility funtion shows that utility without redit onstraints equals u (, ) 534 and utility with redit onstraints is u (, ) 59 Utility is lower under redit onstraints, thus welfare is redued y their imposition This should strike you as sensile the onsumer wanted (rationally and with perfet information) to e in det at the end of period one, ut anks were unwilling to lend, thus the onsumer is worse off Graphially, this means that the hosen onsumption undle under redit onstraints lies on an indifferene urve lower than the hosen onsumption undle in the asene of redit onstraints (Tehnial note: You annot say something like, "welfare is not lowered y muh" eause of redit onstraints Although we did not disuss it, the numers attahed to the utility funtion themselves have no eonomi meaning all they are used for is omparing relative welfare, not for making any asolute statements aout welfare You are not responsile for knowing this tehnial detail, ut FYI) d Using exatly the same solution proedure as in part a, you get that and Implied y this hoie of onsumption is that a (using the period one udget onstraint) That is, the optimal hoie of the onsumer following the positive inome shok involves a zero asset position at the end of period one e With the redit onstraint now ak in plae (with y 9), there will e no hange in household ehavior relative to the ase without the redit onstraint That is, in part d, the optimal hoie of households already involves a hoie for 3
a that satisfies the redit onstraint Thus, the redit onstraint is not inding, and welfare is unaffeted Government in the Two-Period Eonomy Consider again our usual two-period onsumption-savings model, augmented with a government setor Eah onsumer has preferenes desried y the utility funtion u (, ) ln+ ln, where ln stands for the natural logarithm, is onsumption in period one, and is onsumption in period two The assoiated marginal utility funtions are u (, ) and u (, ) Suppose that oth households and the government start with zero initial assets (ie, A and ), and that the real interest rate is always perent Assume that government purhases in the first period are one ( g ) and in the seond period are 99 ( g 99 ) In the first period, the government levies lump-sum taxes in the amount of 8 ( t 8 ) Finally, the real inomes of the onsumer in the two periods are y 9 and y 3 Solution: a What are lump-sum taxes in period two ( t ), given the aove information? Compute the optimal level of onsumption in periods one and two, as well as national savings in period one Consider a tax ut in the first period of unit, with government purhases left unhanged What is the hange in national savings in period one? Provide intuition for the result you otain d Now suppose again that t 8 and also that redit onstraints on the onsumer, of the type desried in Question, are in plae, with lenders stipulating that onsumers annot e in det at the end of period one (ie, the redit onstraint again takes the form a ) Will this redit onstraint affet onsumers optimal deisions? Explain why or why not Is this redit onstraint welfare-enhaning, welfare-diminishing, or welfare-neutral? e Now with the redit onstraint desried aove in plae, onsider again the tax ut of unit in the first period, with no hange in government purhases (That is, t falls from 8 units to 7 units) What is the hange in national savings in period one that arises due to the tax ut? Provide eonomi intuition for the result you otain 4
a Using the government s LBC, find that t Using the same proedure as in question a aove (speifially, starting with the ondition that u (, ) ( + r) u (, )) and with the given funtions, we get that at the optimal hoie, ( + r) Plugging this into the LBC of the y g eonomy and solving for yields y g + + r, from whih it immediately follows that, whih then implies that With government purhases unhanged, a hange in the timing of lump-sum taxes leads to no hange in onsumption and hene no hange in national savings This is the Riardian Equivalene proposition onsumers inrease their private savings after the tax ut in antiipation of the tax inrease that must our in period two d Examine the period-one udget onstraint of the onsumer: + a y t (rememer, the onsumer has zero initial assets here) This expression, along with the value of you found in part aove an e used to determine that a 9 Thus, onsumers optimally (ie, under no redit onstraints) want to e detors at the end of period one With the imposition of the redit onstraints, onsumers an no longer do so, and will hoose a eause that is the losest they an get to their unrestrited hoie while also satisfying the redit onstraint The period one udget onstraint, with a, yields y t The redit onstraint diminishes welfare eause onsumers are eing fored to hoose a onsumption alloation different from the one they would otherwise hoose graphially, they are on a lower indifferene urve than the one that maximizes utility sujet to the LBC of the eonomy e With t 7, the redit onstraint is still inding, and y t Thus, nat eause s y g, national savings falls y exatly the amount y whih onsumption rises, whih is one This ours eause Riardian Equivalene fails if apital ontrols/orrowing onstraints are inding (another reason, eyond distortionary taxes as desried in the Leture Notes, why Riardian Equivalene fails) The reason here is that onsumers were not at their unrestrited optimal hoie to egin with they wanted to onsume more in period one than they were restrited to Thus, any relaxation of their period one udget onstraint (ie, in the form of lower taxes in period one) indues them to inrease their onsumption, dragging down national savings 3 Marginal Propensity to Consume for Various Utility Funtions An old (Keynesian) idea in maroeonomis is the marginal propensity to onsume, areviated MPC Briefly, the MPC is the fration of urrent-period inome that is onsumed in the urrent period For example, if inome in period one is and onsumption in period one is 8, the MPC 9 Consider the standard two-period onsumption-savings model in whih the representative onsumer has no ontrol over his real laor inome in periods and, denoted y y and 5
y, respetively As usual, denote y r the real interest rate etween period and period, and assume the individual egins his life with zero initial assets ( A ) Make the additional assumption that in present-disounted-value terms, his real inome in eah of y the two periods is the same that is, y Using the LBC and the intertemporal + r optimality ondition, derive for eah of the following utility funtions the period- MPC that is, derive what fration of period- real inome the onsumer devotes to period- onsumption (In other words, derive the oeffiient MPC in the expression MPC y+ onst ) Note that not all of these utility funtions satisfy the property that utility is stritly onave in oth its arguments ut this is irrelevant for the exerise here (Hint: Set up the Lagrangian in order to solve) +, with u and u a u (, ) u (, ) ln( ) + (No, this is not a typo ), with u and u u (, ), with u and u ( ), where is a onstant suh that < < (This type of utility funtion is alled the Co-Douglas utility funtion) General Comment: In all of the following, the important point that omes out of the solution of the Lagrangean is that at the optimal hoie, the intertemporal MRS (ie, the ratio u/ u ) equals + r, a derivation that y now you should e familiar with (indeed, y now you should e familiar with at least this result without having to set up and solve the Lagrangean) Of ourse, you need to ompute the marginal utility funtions aove, the marginal utility funtions for eah given utility funtion are presented, whih you needed to ompute yourself u So we have the onsumption-savings optimality ondition + r along with the LBC u y here, whih with our assumption of y + r, is + y The general solution + r proedure here is use the given funtional forms with the optimality ondition and the LBC to generate the MPC Solving the LBC for, y We an then use + r the optimality ondition to solve for in terms of, sustitute into the LBC, and generate the appropriate relation etween and y +, with u and u a u (, ) 6
Solution: The optimality ondition here states that + r, from whih we get that ( + r) Inserting this into the LBC and solving for we get y + r Thus, MPC + r u (, ) ln( ) + (No, this is not a typo ), with u and u Solution: The optimality ondition here states that + r, whih oviously is independent of Optimal period-one onsumption is thus oviously + r, independent of y Hene the period-one MPC for this utility funtion is zero (ie, period-one onsumption does not depend on period-one inome period-one onsumption is said to e autonomous here u (, ), with u and u ( ), where is a onstant suh that < < (This type of utility funtion is alled the Co-Douglas utility funtion) Solution: The optimality ondition here states that + r, from whih we get that ( + r) Inserting this into the LBC and solving for we get y Thus, MPC 7