STATE UNIVERSITY OF NEW YORK AT ALBANY Department of Economics

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1 STATE UNIVERSITY OF NEW YORK AT ALBANY Department of Economics Ph. D. Comprehensive Examination: Macroeconomics Fall, 202 Answer Key to Section 2 Questions Section. (Suggested Time: 45 Minutes) For 3 of the following 6 statements, state whether the statement is true, false, or uncertain, and give a complete and convincing explanation of your answer. Note: Such explanations typically appeal to speci c macroeconomic models.. A tax on consumption is equivalent to a tax on labor income. 2. An increase in government spending without a commensurate increase in taxes will increase national income. 3. In the past few months, the Federal Reserve has dramatically increased the money supply, but output, employment and prices have all increased very slowly. Such an outcome is evidence in favor of the quantity theory. 4. An increase in government spending without a commensurate increase in taxes will increase welfare. 5. A permanent increase in productivity should cause permanent increases in both consumption and employment. 6. The lowering of a country s sovereign debt rating is equivalent to an increase in everyone s income tax rate and having the revenues thrown into the ocean.

2 Section 2. (Suggested Time: 2 Hours, 5 minutes) Answer any 3 of the following 4 questions. 7. The preferences of the representative household are X E 0 t=0 t [ln (C t + G t ) L t ] ; while the production function is Y t = L t ; 0 < < : (PRF) Government expenditures, G t, are driven by uctuation in taxes: G t = ( S t )W t L t ; (GBC) with S t = `t, the after-tax rate on labor income, following an AR() process around the log of its steady state value: bs t ln (S t =S) = bs t + " t ; 0 < ; (TS) where f" t g is a zero-mean i.i.d. process. (a) We set about nding the competitive allocation. Recall that the representative producer solves max t = Lt L t0 W t L t ; so that the rst order condition for pro t maximization is ( ) L t = ( ) Y t L t = W t ; (PM) and pro ts are t = Y t : The household s ow budget constraint is C t + B t = ( + r t ) B t + S t W t L t + t : (FBC) Recalling that the household takes government spending as given, the rst order conditions for utility maximization are S t W t = ; (LL) C t + G t = ( + r t ) E t : (EE) C t + G t C t+ + G t+ 2

3 (b) Combining the consumer s, rm s and government s budget constraints yields the capital accumulation equation: B t = ( + r t ) B t + S t W t L t + t C t = ( + r t ) B t + W t L t ( S t )W t L t + t C t = ( + r t ) B t + Y t G t C t : With no capital and identical consumers, the equilibrium quantity of bonds is zero, and the resource constraint is Y t = G t + C t : (RC) Combining equations (PM), (LL) and (RC) yields the labor-leisure condition: S t ( ) Y t = L t C t + G t ) L t = S t : (LL 0 ) Inserting this result into equation (PRF) yields Y t = S t S t : (PRF 0 ) Using this result, and the resource constraint, we can rewrite the Euler equation as + r t = S E t S : (EE 0 ) t (c) It follows from equation (PRF 0 ) that higher labor income taxes (lower values of S t ) decrease output. By reducing the after-tax wage, the substitution e ect of higher taxes is less work. In equilibrium, there is essentially no income e ect, because labor taxes are returned to the consumer in the form of government goods. Because the government goods are perfectly substitutable for private consumption, they are equivalent to lump-sum tax refunds. The only income e ect is that in equilibrium the consumer works less and thus has less total consumption. This allows the substitution e ect to dominate. (d) Let lower-case letters with carats b denote deviations of logged variables around their steady state values. It follows from equation (LL0) that L t = S t ) exp b`t = L t = S t = S L ss = exp (bs t ) : t+ Logging both sides yields b`t = bs t : (LL 00 ) 3

4 Proceeding similarly, it follows from equation (PRF 0 ) that exp (by t ) = exp (( )bs t ) ; The log of average labor productivity is ) by t = bs t ; > 0: (PRF 00 ) capl t = by t b`t = bs t : (e) Note that average labor productivity can be written as capl t = by t: With =( ) > 0, it immediately follows that increases in by t are accompanied by decreases in apl c t, so that labor productivity is countercyclical. Fluctuations in output are driven solely by uctuations in the tax rate. With decreasing returns to labor, however, a increase in labor that cause output to rise will cause average productivity, Y =L, to fall. This feature of the model is inconsistent with the data, where labor productivity is pro-cyclical. One way to improve the models performance would be to introduce labor hoarding. When labor hoarding occurs, rms use their workers more intensively during economic expansions, causing the standard productivity measures, with do not adjust for intensity, to increase. 8. Proportional income tax in simple dynamic model with a government and speci c utility functions Consider the following economy: Time: Discrete; in nite horizon Demography: Continuum of mass of (representative) consumer/worker households, and a large number of pro t maximizing rms, owned jointly by the households. Preferences: the instantaneous household utility function is c =2 where c is household consumption. The discount factor is 2 (0; ): Technology: There is a constant returns to scale technology for which labor is the only input so that a rm that hires h units of labor produces zh units of output. Endowments: Each household has unit of time per period to allocate how ever they like between work and leisure. Institutions: There is a government that has to meet an exogenous stream of expenditures, fg t g : Government spending is thrown into the ocean. The government can levy taxes and issue bonds in order to meet its expenditure requirement. Taxes are restricted to being proportional to labor income so that in period t; the tax revenue from a household which provides labor services h t is then t w t h t where t is the period t tax rate and w t is the wage rate. Every period there are markets for labor, government bonds and consumption goods. 4

5 (a) Write down and solve the problems faced by the representative household and the representative rm. Households solve: (Students might spot immediately that households do not value leisure and so h t = for all t in which case as long as they mention it, they can drop h t from the household s problem.) max fc t;h t;s tg t=0 X t=0 t c =2 t s.t. c t = ( t )w t h t + ( + r t ) s t s t where s t is the saving (in bonds) by a household in period t and r t is the interest on bonds Clearly, h t = for all t and there is an intertemporal optimality condition Firms solve c =2 t = ( + r t+ ) c =2 t+ n max zh f h f t t w t h f t So for any rm that hires a strictly positive (but nite) amount of labor, w t = z for all t o (b) Write down the government s (period by period) budget constraint. g t + ( + r t )b t = b t + t w t h t (c) De ne and a characterize a competitive equilibrium. A competitive equilibrium is a sequence of prices, fw t ; r t g a sequence of tax rates f t g and an allocation fc t ; h t; h f t ; b t ; s t g such that, given prices and tax rates, the allocation solves the households problem and the rms problem; the government budget constraint holds and the markets for labor, consumption goods and bonds all clear. The market clearing conditions are: h f t = h t = b t = s t z = g t + c t Using these we get c t = ( t )z + ( + r t ) b t b t 5

6 and the government budget constraint becomes g t + ( + r t )b t = b t + t z Characterization of equilibrium: + r t+ = z z =2 gt+ g t (d) Does Ricardian equivalence hold? Explain Yes, the timing t does not a ect the equilibrium allocation which is summarized by c t = z g t for all t: (e) How would your answer to part (d) change if the utility function was replaced by 2c =2 t ( h t ) =2 Explain your answer. If households care about leisure then the proportional tax will distort the laborleisure time allocation. Goods market clearing will be zh t = g t + c t and h t will depend on the timing of taxes. The equilibrium wage will still be z for all t but the rst-order condition for the labor leisure choice will imply So (zh t g t ) =2 ( h t ) =2 z( t ) = (zh t g t ) =2 ( h t ) =2 ( h t )z( t ) = zh t g t and c t = zh t g t for all t but h t is a function of t. 9. (Adapted from a question by Rody Manuelli.) We are considering a Lucas tree model where dividends follow a two-state growth process. The preferences of the representative consumer are E 0 X t=0 t c t! ; 0 < < ; > 0: Output is produced by an in nite-lived tree: each period, the tree produces d t units of non-storable output. The growth rate of dividends follows a stationary process, with G t d t =d t following a two-state Markov chain. In particular, G t can take on two values:, and >. The conditional probabilities are so that zero growth is permanent. f (; ) = Pr (G t+ = j G t = ) = ; f (; ) = Pr (G t+ = j G t = ) = ; f (; ) = Pr (G t+ = j G t = ) = 0; f (; ) = Pr (G t+ = j G t = ) = : 6

7 (a) Writing the consumer s problem as a Lagrangean, we get ( V (x t ; d t ; G t ) = min f t0g max fc t0; s t+ ; z(); z()g + t x t c t p t s t+ X + X G t+ 2f;g The FOC for an interior solution are: c t = t ; c t q(g t+ ; G t )z(g t+ ) G t+ 2f;g V ([p t+ + d t+ ] s t+ + z(g t+ ); d t+ ; G t+ ) f(g + ; G t ) t p t = E [t + t+ [p t+ + d t+ ] t q(g t+ ; G t ) = Since (following Benveniste-Scheinkman), [t + t+ f(g + ; G t ); G t+ 2 f; g: ) : the Euler equations t = t = c t ; q(g t+ ; G t ) = c(g t+; d t+ ) c(g t ; d t ) f(g + ; G t ); G t+ 2 f; g; (EE) p t c t = E t c t+ [p t+ + d t+ ] : (EE2) (b) Given the random variables d 0 and G 0, the conditional density f (G t+ ; G t ), and the initial endowments s 0 = and z = z = 0, a recursive rational expectations equilibrium consists of pricing functions p (d; G) and q (G 0 ; G), a value function V (x; d; G), and decision functions c (x; d; G), s (x; d; G), and z (x; d; G 0 ; G) such that:. Given the pricing functions p (d; G) and q (G 0 ; G), the value and policy functions V (x; d; G), c (x; d; G), s (x; d; G), and z (x; d; G 0 ; G) solve the consumer s problem. 2. Markets clear: for x = p (d; G) + d, c (x; d; G) = d, s (x; d; G) =, and z (x; d; ; G) = z (x; d; ; G) = 0. (c) Noting that d t+ = G t+ d t, it follows from equation (EE) that equilibrium contingent claims prices obey q(g t+ ; G t ) = (G t+d t ) f(g d + ; G t ); t = Gt+f(G + ; G t ) G t+ 2 f; g; (EE 0 ) It follows from our speci cation of the stochastic process for G t that: 7

8 . When the current state is, we have 2. When the current state is, we have q(; ) = 0 = 0; q(; ) = = : q(; ) = = ; q(; ) = ( ) = ( ): (d) It follows from arbitrage arguments that if an asset pays w (G 0 ) units of consumption goods when G t+ = G 0, its price is p w t = X w (G t+ ) q (G t+ ; G t ) : G t+ 2f;g When the asset is a risk-free discount bond, w (G 0 ) =. Imposing equation (EE 0 ), it follows that the price of a risk-free bond, R (G t ), is given by R (G t ) = X Gt+f(G + ; G t ) = E t (Gt+): G t+ 2f;g Explicit solutions for the two values of G t are R () = E t Gt+ Gt = = ; R () = E t G Gt = t+ = + ( ) : (e) Let q (G 0 ; G t ) =f (G 0 ; G t ) denote the probability-adjusted contingent claim price. It follows from equation (EE 0 ) that q (G 0 ; G t ) =f (G 0 ; G t ) = (G 0 ). With > and > 0, it follows that q (; ) f (; ) = < R () = + ( ) < q (; ) f (; ) = This rank ordering re ects the diminishing marginal utility of consumption. A unit of output delivered when G t+ = is a unit delivered when total consumption is highest and the marginal utility of consumption is lowest. Adjusting for conditional probabilities, this unit of consumption is worth less than a unit of consumption delivered when G t+ =, when the marginal utility of consumption is high. The risk-free bond, being a weighted average, has an intermediate value. 0. Diamond Coconut Economy with uniform idiosyncratic preferences Time: Discrete, in nite horizon Geography: A trading island and a production island. Demography: A mass of of ex ante identical individuals with in nite lives. 8

9 Preferences: The common discount rate is r; consumption of own produce yields 0 utils, consumption of someone whose good you like yields u utils. The share of individuals whose goods I like is : Whether an individual likes my good or not is independent of whether I like hers. (So she likes my good with probability too.) Productive Technology: On the production island individuals come across a tree with a coconut with probability each period. The cost of obtaining the coconut is c which is uniformly distributed over (0; c] where c > u (so some trees will get rejected). Matching Technology: On the trading island people with coconuts meet each with probability ; a constant. Navigation: Travel between islands is instantaneous. Endowments: Everyone has a boat and starts o with one of their own coconuts (a) Write down the asset value (Bellman) equations for this economy. De ne the terms you introduce. The regular asset value equations are V T = 2 (u + V P ) + ( 2 )V T + r V P = + r fe [max fv T c; V P g] + ( )V P g where V T is the value to being on the trading island and V P is the value to being on the production island The ow value equations are rv T = 2 (u + V P V T ) rv P = E [max fv T c V P ; 0g] (b) De ne a search equilibrium A search equilibrium is a pattern of trade such that given everyone else conforms to that patterns no individual will wish to deviate from it. (c) Solve for an implicit equation that speci es the reservation tree height, c in terms of the parameters of the model. Let c = V T V P : Then rewrite V P as rv P = now subtract rv P from rv T to get Z c 0 [V T c V P ] dc r (V T V P ) = 2 u 2 (V T V P ) or by de nition of c ; so Z c 0 [V T c V P ] dc Z c (r + 2 )c = 2 u [c c] dc () 0 (r + 2 )c 2 u + c2 2 9 = 0

10 (d) How does c change with respect to (i.e. obtain the sign of the comparative static)? De ne (c; ) (r + 2 )c 2 u + c2 2 so that (c ; ) = 0: Then Then dc c=c = 2(c u) < 0 c=c (sign comes from equation = (r + 2 ) + c > 0: c=c So dc d > 0 Increasing rapidly improves my chances of getting to trade my good and return to the production island. Investment (in inventory) has a higher return and I am therefore more ready to invest. (e) Draw a diagram showing the population ows between the islands. Write down the steady-state equations and solve for the population on the trading island as a function of c. The diagram is the same as in the class notes for constant except that should be replaced with 2 and F (c ) replaced with c : Steady-state equations are: 2 n T = c n P n T + n P = where n T is proportion of the population on the trading island and n P is the proportion of the population on the production island. Thus n T = c 2 + c 0