Department of Eonomis University of Maryland Eonomis 35 Intermediate Maroeonomi Analysis Midterm Exam Suggested Solutions Professor Sanjay Chugh Fall 009 NAME: Eah problem s total number of points is shown below. Your solutions should onsist of some appropriate ombination of mathematial analysis, graphial analysis, logial analysis, and eonomi intuition, but in no ase do solutions need to be exeptionally long. Your solutions should get straight to the point solutions with irrelevant disussions and derivations will be penalized. You are to answer all questions in the spaes provided. You may use one page (double-sided) of notes. You may not use a alulator. TOTAL PART / 50 TOTAL PART / 50 TOTAL / 00
Problem : Core Inflation and Non-Core Inflation in the Two-Period Eonomy (30 points). Two distint measures of inflation alled ore inflation and non-ore inflation generally attrat attention by poliy-makers and the media. The ore inflation rate is the rate of growth of pries of so-alled ore goods (suh as food, lothing, and shelter), while the non-ore inflation rate is the rate of growth of pries of so-alled non-ore goods (generally energy items). Consider our usual two-period eonomy (with no government), in whih the representative onsumer has no ontrol over his nominal inome. Rather than there being only one type of good the onsumer purhases eah period, however, suppose that eah period there are two types of goods: ore good and non-ore goods. The lifetime utility funtion of the representative onsumer is NON NON NON NON (,,, ) ln( ) ln( ) ln( ) ln( ) u = + + +, where ln stands for the natural logarithm, stands for onsumption of ore goods in period NON, stands for onsumption of non-ore goods in period, and similarly for and NON. The representative onsumer begins period one with zero assets (i.e., A 0 = 0). The period-byperiod budget onstraints of the representative onsumer are thus P + P + A = Y NON NON P + P + A = Y + ( + i) A NON NON where NON P denotes the nominal prie of ore goods in period, P denotes the nominal NON prie of non-ore goods in period, and similarly for P and P. As usual, Y and Y denote nominal inome in periods and, respetively, and i is the nominal interest rate. Finally, we an onstrut as usual the representative onsumer s nominal lifetime budget onstraint, whih here is: NON NON NON NON P P Y P + P + + = Y+ + i + i + i This nominal LBC has the same interpretation as always: the PDV of all lifetime nominal onsumption (whih here takes into aount both onsumption of ore and non-ore goods) is equal to the PDV of all lifetime nominal inome. That is, in a lifetime sense, all onsumption spending equals all inome regardless of how many goods there are to purhase. (OVER)
Problem ontinued a. (8 points) Using an appropriate nominal Lagrange formulation (you are free to hoose either a lifetime formulation or a sequential formulation), derive four optimality onditions: one between period- ore onsumption and period- non-ore onsumption; one between period- ore onsumption and period- non-ore onsumption; one between period- ore onsumption and period- ore onsumption; and one between period- non-ore onsumption and period- non-ore onsumption. These optimality onditions should be based on the utility funtion given above. Show all important steps. Solution: Taking the lifetime approah, the Lagangian is The FOCs are NON NON ( ) + ( ) + ( ) + ( ) ln ln ln ln Y P P + λ Y + P P + i + i + i NON NON NON NON NON NON λp = 0 λp = 0 NON λp = 0 + i NON λp = 0 + i Solving out for λ appropriately and then rearranging, we have that the four optimality onditions are, in order,
Problem a ontinued (if you need more spae) NON P = NON P NON P = NON P P ( + i) = P NON NON P ( + i) = NON NON P b. (6 points) In eah of the following three diagrams, appropriately label the slope of the budget line in terms of variables defined above. Briefly desribe how you determined the relevant slopes (you may refer to your work in part a if needed). NONCOR NONCOR COR COR COR COR Solution: Based on the omputed optimality onditions in part b, it is easy to onlude that the NON slope in the left-most diagram is P / P ; the slope in the middle diagram is NON P / P ; and the slope in the right-most diagram is P ( + i) / P. This follows from the fat that in eah of the optimality onditions in part b, the right-hand-side represents the (absolute value of the) slope of the relevant budget line. Alternatively, you ould have take the NON NON LBC onstruted in part a and suessively rearranged it to isolate,, and, whih would give you (in eah line, the fous is just on the relevant slope term): NON P = + (other stuff) NON P NON P = + (other stuff) NON P P ( + i) = + (other stuff) P 3
Problem ontinued Suppose that the Fed hanges the nominal interest rate, and suppose that this hange in poliy NON NON does not at all affet P, P, P, or P, nor does it affet Y or Y.. (3 points) At the resulting new optimal hoie, will this hange in monetary poliy affet onsumers MRS between period- ore and period- non-ore onsumption? If so, briefly explain why/how; if not, briefly explain why not. (You may refer to the diagrams in part b if needed.) Solution: Simply examining the analysis in parts a and b (and with the stated assumption that the monetary poliy ation has no effet on pries), it is lear that the slopes of the budget lines in the left-most and middle diagrams above will be ompletely unaffeted. Thus, at onsumers optimal hoies, their MRS along the period- ore/period- non-ore margin (i.e., the left-most diagram above) will be unaffeted. d. (3 points) At the resulting new optimal hoie, will this hange in monetary poliy affet onsumers MRS between period- ore and period- ore onsumption? If so, briefly explain why/how; if not, briefly explain why not. (You may refer to the diagrams in part b if needed.) Solution: In ontrast to the analysis above, now the fous is on the right-most diagram above. The right-most diagram desribes onsumption deisions that span time periods. Hene, the relevant prie aross time periods involves interest rates (reall our general interpretation of interest rates as the prie of time ). Thus, when the Fed hanges interest rates (note that in this problem, hanges in nominal interest rates and hanges in real interest rates are one and the same beause you are told to assume that inflation never hanges), the slope of the budget line in the right-most diagram will indeed be affeted. Hene, at onsumers optimal hoies, their MRS along the period- ore/period- ore margin will be affeted. 4
Problem : Quasi-Linear Utility (0 points). In the stati onsumption-leisure model, suppose the representative onsumer has the following utility funtion over onsumption and leisure, ul (,) = ln() + Al, where, as usual, denotes onsumption and l denotes leisure. In this utility funtion, ln( ) is the natural log funtion, and A is a number (a onstant) smaller than one that governs how muh utility the individual obtains from a given amount of leisure. Suppose the budget onstraint the individual faes is simply = ( t) w n, where t is the labor tax rate, w is the real hourly wage rate, and n is the number of hours the individual works. (Notie that this budget onstraint is expressed in real terms, rather than in nominal terms.) a. (3 points) Does this utility funtion display diminishing marginal utility in onsumption? Briefly explain. Solution: Marginal utility with respet to onsumption is the slope of the utility funtion when viewed as a funtion of onsumption alone. The given funtion is logarithmi in onsumption, and the natural log funtion, is stritly inreasing and stritly onave, meaning the slope with respet to onsumption is always dereasing and asymptotes to zero. Hene, this funtion does display diminishing marginal utility in onsumption. b. (3 points) Does this utility funtion display diminishing marginal utility in leisure? Briefly explain. Solution: Just as above, marginal utility with respet to leisure is the slope of the utility funtion when viewed as a funtion of leisure alone. The given funtion is linear in leisure, hene its slope with respet to leisure is onstant. Thus, this funtion does not display diminishing marginal utility in leisure.. (4 points) Assume (as usual) the representative onsumer maximizes utility. For the given utility funtion, plot this representative onsumer s labor supply funtion, explaining the logi behind your plotted funtion. Also, how would a derease in the tax rate t affet the optimal amount of labor supply (i.e., inrease it, derease it, or leave it unhanged)? Carefully explain your logi/derivation. (Note: Be sure to base your analysis here on the utility funtion that is given above.) Solution: The onsumption-leisure optimality ondition (whih an be derived using a Lagrangian, whih is omitted here beause the general derivation proeeds exatly as we ve seen several times) is ul A = = ( t) w, u / from whih we get that A = ( t) w at the onsumer s optimal hoie. Substituting the given budget onstraint into this (i.e., substituting for ) we have A ( t) w n = ( t) w. Caneling terms and solving for n, we find n =, A 5
Problem ontinued (if you need more spae) whih shows that labor supply here is independent of taxes, hene hanges in the tax rate annot affet the quantity of labor. The labor supply funtion, plotted with the wage (pre- or after-tax, it doesn t make a differene) on the vertial axis and n on the horizontal axis, is a vertial line at the numerial value / A. This (perfetly inelasti) labor supply is learly unaffeted by hanges in taxes; indeed, it is ompletely unaffeted by hanges in hanges in the pre-tax real wage w as well. Further disussion (whih was not required): The reason why labor (equivalently, leisure) here doesn t depend at all on the (pre- or after-tax) wage is that there is no diminishing marginal utility in leisure (i.e., utility is linear with respet to leisure, as we saw above). When a multidimensional utility funtion is linear in one argument and has diminishing marginal utility in its other argument(s), it is said to be quasi-linear. Quasi-linear utility funtions give rise to demand funtions for the linear objet that are ompletely insensitive (inelasti) to prie here, the demand for leisure (the flip side of whih is the supply of labor) is ompletely insensitive (inelasti) to the wage. 6
Problem 3: Hyperboli Impatiene and Stok Pries (8 points). In this problem you will study a slight extension of the infinite-period eonomy from Chapter 8. Speifially, suppose the representative onsumer has a lifetime utility funtion given by u ( ) + γβu ( ) + γβ u ( ) + γβ u ( ) +..., 3 t t+ t+ t+ 3 in whih, as usual, u(.) is the onsumer s utility funtion in any period and β is a number between zero and one that measures the normal degree of onsumer impatiene. The number γ (the Greek letter gamma, whih is the new feature of the analysis here) is also a number between zero and one, and it measures an additional degree of onsumer impatiene, but one that ONLY applies between period t and period t+. This latter aspet is refleted in the fat that the fator γ is NOT suessively raised to higher and higher powers as the summation grows. The rest of the framework is exatly as studied in Chapter 8: at is the representative onsumer s holdings of stok at the beginning of period t, the nominal prie of eah unit of stok during period t is S t, and the nominal dividend payment (per unit of stok) during period t is D t. Finally, the representative onsumer s onsumption during period t is t and the nominal prie of onsumption during period t is P t. As usual, analogous notation desribes all these variables in periods t+, t+, et. The Lagrangian for the representative onsumer s utility-maximization problem (starting from the perspetive of the beginning of period t) is 3 u ( t) + γβu ( t+ ) + γβ u ( t+ ) + γβ u ( t+ 3) +... + λ t Yt + ( St + Dt ) at P t t Sta t + γβλ Y + ( S + D ) a P S a t+ t+ t+ t+ t t+ t+ t+ t+ γβ λ t+ Yt+ ( St+ Dt+ ) at+ Pt+ t+ St+ at+ 3 γβ λ t+ 3 Yt+ 3 ( St+ 3 Dt+ 3) at+ Pt+ t+ St+ a t+ + + + + + + 3 3 3 3 +... NOTE CAREFULLY WHERE THE ADDITIONAL IMPATIENCE FACTOR γ APPEARS IN THE LAGRANGIAN. (OVER) The idea here, whih goes under the name hyperboli impatiene, is that in the very short run (i.e., between period t and period t+), individuals degree of impatiene may be different from their degree of impatiene in the slightly longer short run (i.e., between period t+ and period t+, say). Hyperboli impatiene is a phenomenon that routinely reurs in laboratory experiments in experimental eonomis and psyhology, and has many farreahing eonomi, finanial, poliy, and soietal impliations. 7
Problem 3 ontinued a. (4 points) Compute the first-order onditions of the Lagrangian above with respet to both a t and a t +. (Note: There is no need to ompute first-order onditions with respet to any other variables.) Solution: The two FOCs are λ S + γβλ ( S + D ) = 0 t t t+ t+ t+ γβλ S + γβ λ ( S + D ) = 0 t+ t+ t+ t+ t+ b. (4 points) Using the first-order onditions you omputed in part a, onstrut two distint stok-priing equations, one for the prie of stok in period t, and one for the prie of stok in period t+. Your final expressions should be of the form S t =... and S t + =... (Note: It s fine if your expressions here ontain Lagrange multipliers in them.) Solution: Simply rearranging the two FOCs above and aneling the γ term (along with one β term) in the seond FOC, we have γβλ S = ( S + D ) t+ t t+ t+ λt βλ S = ( S + D ) t+ t+ t+ t+ λt + For the questions next, observe that the S t expression and the S t+ expression are subtly, but importantly, different here. They would be idential to eah other (other than the fat that the time subsripts are different, but that is as usual) if and only if γ =. If γ <, whih is the ase of hyperboli impatiene, then stok pries are determined in a somewhat different way in the very short run ompared to the longer short run or medium run. 8
Problem 3 ontinued For the remainder of this problem, suppose that it is known that D t+ = D t+, and that S t+ =S t+, and that λ t = λ t+ = λ t+.. (5 points). Does the above information neessarily imply that the eonomy is in a steadystate? Briefly and arefully explain why or why not; your response should make lear what the definition of a steady state is. (Note: To address this question, it s possible, though not neessary, that you may need to ompute other first-order onditions besides the ones you have already omputed above.) Solution: No, none of these statements neessarily imply that the eonomy is in a steady state, whih, reall, means that all real variables beome onstant and never again hange. There are two ways of observing that the above information does not imply the eonomy is in steady state. First, the above statements are all about nominal variables, and in a steady state it an be the ase that nominal variables ontinue flutuating over time, even though all real variables do not. Another way of arriving at the orret onlusion here is that the statements above only refer to periods t, t+, and t+. In a steady-state, (real) variables settle down to onstant values forever, not just for a few time periods. d. (5 points) Based on the above information and your stok-prie expressions from part b, an you onlude that the period-t stok prie (S t ) is higher than S t+, lower than S t+, equal to S t+, or is it impossible to determine? Briefly and arefully explain the eonomis (i.e., the eonomi reasoning, not simply the mathematis) of your finding. Solution: You are given that nominal stok pries, nominal dividends, and the Lagrange multiplier in period t+ and t+ are equal to eah other. Let s all these ommon values S, D, and λ (that is, S = St+ = St+ D = Dt+ = Dt+, and λ = λt+ = λt+ ). Inserting these ommon βλ values in the period-t+ stok prie equation, we have S = ( S+ D). Caneling terms, we λ have that the nominal stok prie in period t+ (and t+) is S = β S+ β D (whih we ould of β ourse solve for the stok prie as S = D if we needed to). β Now, using the ommon values of S, D, and the multiplier in the period-t stok prie equation gives us St = γβ( St+ + Dt+ ) = γβ( S + D) = γ( βs+ βd). Note that the final term in parentheses is nothing more than S, hene we have St = γ S. If γ <, then learly the stok-prie in period t is smaller than it is in period t+ (and period t+). The eonomis of this is due to the hyperboli impatiene whih makes onsumers more impatient to purhase onsumption in the very short run (period t) ompared to the longer short run. All else equal, this means that in the very short run, onsumers do not are to save as muh (due to the their extreme impatiene in the very short run), whih means their demand for saving --- i.e., their demand for stok is lower. Lower demand for stok means a lower prie of stok, all else equal. 9
Problem 3 ontinued Now also suppose that the utility funtion in every period is u() = ln, and also that the real interest rate is zero in every period. e. (5 points) Based on the utility funtion given, the fat that r = 0, and the basi setup of the problem desribed above, onstrut two marginal rates of substitution (MRS): the MRS between period-t onsumption and period-t+ onsumption, and the MRS between period-t+ onsumption and period-t+ onsumption. Solution: This only requires examining the lifetime utility funtion (the first line of the Lagrangian above). By definition, the MRS between period t onsumption and t+ onsumption u'( t) t+ is =, and the MRS between period t+ onsumption and t+ onsumption is γβu'( t+ ) γβt γβu'( t+ ) u'( t+ ) t+ = =. Note that the form of the two MRS funtions is different: the γβ u'( t+ ) βu'( t+ ) βt+ hyperboli impatiene affets the former MRS, but not the latter MRS. f. (5 points Harder) Based on the two MRS funtions you omputed in part e and on the fat that r = 0 in every period, determine whih of the following two onsumption growth rates OR t+ t+ t t+ is larger. That is, is the onsumption growth rate between period t and period t+ (the fration on the left) expeted to be larger than, smaller than, or equal to the onsumption growth rate between period t+ and period t+ (the fration on the right), or is it impossible to determine? Carefully explain your logi, and briefly explain the eonomis (i.e., the eonomi reasoning, not simply the mathematis) of your finding. Solution: The basi onsumption-savings optimality ondition states that the MRS between two onseutive time periods is equated to (+r). You are told here that r = 0 always. Based on the two MRS funtions onstruted above, then, it follows immediately that the onsumption growth rate between period t and t+ is smaller than the onsumption growth rate between period t+ and period t+. This follows beause γ <. The eonomis is similar to above: hyperboli impatiene makes onsumers onsume muh more in the very short run (i.e., period t), whih means that the growth rate of onsumption between period t (already a very high onsumption period) and t+ will be low, ompared to the similar omparison one period later. 0
Problem 4: Government Debt Ceilings ( points). Just like we extended our two-period analysis of onsumer behavior to an infinite number of periods, we an extend our two-period analysis of fisal poliy to an infinite number of periods. The government s budget onstraints (expressed in real terms) for the years 009 and 00 are g + b = t + ( + r) b g + b = t + ( + r) b 009 009 009 008 00 00 00 009 and analogous onditions desribe the government s budget onstraints in the years 0, 0, 03, et. The notation is as in Chapter 7: g denotes real government spending during a given time period, t denotes real tax revenue during a given time period (all taxes are assumed to be lump-sum here), r denotes the real interest rate, and b denotes the government s asset position (b 008 is the government s asset position at the end of the year 008, b 009 is the government s asset position at the end of 009, and so on). At the end of 008, the government s asset position was roughly a debt of $0 trillion (that is, b 008 = -$0 trillion). The urrent fisal poliy plans/projetions all for: g 009 = $4 trillion, t 009 = $ trillion, g 00 = $3 trillion, and t 00 = $ trillion. Finally, given how low interest rates are right now and how low they are projeted to remain for the next few years, suppose that the real interest rate is always zero (i.e., r = 0 always). a. (3 points) Assuming the projetions above prove orret, what will be the numerial value of the federal government s asset position at the end of 009? Briefly explain/justify. Solution: Using the given numerial values and using the 009 government budget onstraint given above, it is straightforward to alulate b 009 = -$ trillion. b. (3 points) Assuming the projetions above prove orret, what will be the numerial value of the federal government s asset position at the end of 00? Briefly explain/justify. Solution: Using the given numerial values, the value for b 009 found in part a, and using the 00 government budget onstraint given above, it is straightforward to alulate b 00 = -$3 trillion.
Problem 4 ontinued Under urrent federal law, the U.S. government s debt annot be larger than $ trillion at any point in time. This limit is known as the debt eiling.. (3 points) Based on your answer in part a above, does the debt eiling pose a problem for the government s fisal poliy plans during the ourse of the year 009? If it poses a problem, briefly desribe the problem; if it poses no problem, briefly desribe why it poses no problem. Solution: No, the debt eiling poses no problem for the fisal poliy plans for the year 009. This is beause the t and g plans all for a debt at the end of 009 of $ trillion, whih does not exeed the eiling. d. (3 points) Based on your answer in part b above, does the debt eiling pose a problem for the government s fisal poliy plans during the ourse of the year 00? If it poses a problem, briefly desribe the problem; if it poses no problem, briefly desribe why it poses no problem. Solution: Yes, the debt eiling poses a problem for the fisal poliy plans for the year 00. This is beause the t and g plans all for a debt at the end of 00 of $3 trillion, whih violates the eiling. Suppose that the Obama administration only beomes aware of the $ trillion debt eiling at the very end of 009 to be preise, suppose the administration only beomes aware of it on Deember 3, 009, when all of the year s spending and tax olletions have ended. Furthermore, suppose Congress does not alter the debt eiling at all. e. (3 points) Will the government be fored (note the emphasis here) to hange t 00 ompared to the projetion of t 00 = $ trillion? If not, explain why not. If so, explain in whih diretion (up or down)? Solution: Beause all that matters for the end-of-year debt position is the fisal flow during the year 00 (i.e., what matters is the differene g 00 t 00 ), no, the government does not have to inrease taxes in 00 in order to stay within the debt eiling. The government ould instead ahieve the entire adjustment required to stay within the debt eiling by utting government spending in 00, and leave taxes in 00 unhanged.
Problem 4 ontinued While we did not formally study the idea of a government utility funtion, in omplete analogy with onsumer theory, we an imagine that the government has a utility funtion for its own spending. Suppose the government s lifetime utility funtion, starting from the perspetive of the very beginning of the year 009, an be desribed by the funtion ug ( 009, g00, g0, g 0,...), and this utility funtion satisfies all the usual properties we have been studying (i.e., it is stritly inreasing in eah argument, with diminishing marginal utility in eah argument). f. (7 points Harder) The following diagram (on the next page) fouses on the two-year time span 009-00 and plots the government s budget onstraint over the two-year time span along with the government s hoies of g 009 and g 00 as desribed in parts a and b. The diagram below depits these hoies of g as optimal hoies. Note that this budget line is NOT a LIFETIME budget onstraint beause the government is NOT assumed to ease operations at the end of 00. Suppose that the debt eiling law never hanges. Furthermore, beause of politial reasons, it sometimes seems muh easier to hange government spending than to hange taxes. Let s make this idea blak-and-white by now supposing that taxes an never hange. One the Obama administration beomes aware of the debt eiling on Deember 3, 009, illustrate in the diagram below any and all effets that must happen to omply with the debt eiling. If there are no effets to illustrate, explain why there are none. If there are effets to illustrate, be lear to illustrate all of them (it is up to you determine what and how many effets there are), and briefly explain your illustration. (Note: examples of effets to illustrate may be things suh as the budget line pivots outward, et.) Ideally, the government s utility funtion is benevolent in the sense that it should reflet the needs and desires of its itizens, but orruption et. an sometimes distort government utility funtions. But let s leave aside suh issues here and think of the government as benevolent. 3
Problem 4g ontinued g 00 t 00 b 00 Note: This value of b 00 plotted here is under the fisal plans desribed at the very beginning of the problem. t 009 + (+r)b 008 Optimal government spending hoies under urrent fisal plans slope = -(+r) g 009 Solution: You are told that the administration only beomes aware of the debt eiling at the end of 009, at whih point it is too late to hange either g 009 or t 009. Furthermore, you are told that t 00 annot hange (due to politial reasons, say). The only way for the debt eiling law to not be violated in 00, then, is for the government to lower g 00. Thus, the restrited hoie of government spending must then lie diretly vertially below the original optimal hoie shown in the diagram (diretly vertially below beause, again, g 009 is something that an no longer be hanged one Deember 3, 009 arrives). Passing through this new point will be a budget line that is parallel to the original one, but on whih the vertial interept is lower, by exatly the amount by whih g 00 falls. (Indeed, the fall in g 00 is exatly mirrored by an equal rise in b 00 see the 00 budget onstraint above.) END OF EXAM 4