STATE UNIVERSITY OF NEW YORK AT ALBANY Department of Economics. Ph. D. Preliminary Examination: Macroeconomics Fall, Partial Answer Key

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1 STATE UNIVERSITY OF NEW YORK AT ALBANY Department of Economics Ph. D. Preliminary Examination: Macroeconomics Fall, 2008 Partial Answer Key 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.. If consumers receive utility from government goods, then government spending has no wealth effects. 2. A higher saving/income ratio causes faster long-run economic growth. 3. Risk averse investors will always prefer assets that have a known rate of return. 4. Real business cycle theory accurately describes the current macroeconomic situation: the oil price is up, auto sales are down, exports are booming, house prices are falling, and a credit crunch has whacked the banks. 5. Firms that have constant returns to scale technologies and take factor prices as given cannot make positive profits. 6. The high expected rate of return to investors explains the low personal saving rate during the 990 s stock boom and the 2000 s housing boom.

2 Section 2. (Suggested Time: 2 Hours, 5 minutes) Answer any 3 of the following 4 questions. 7. Proportional income taxes in simple dynamic model with a government. Consider the following economy: Time: Discrete; infinite horizon Demography: Continuum of mass of (representative) consumer/worker households, and a large number of profit-maximizing firms, owned jointly by the households. Preferences: The instantaneous household utility function is u(c) where c is household consumption and u(.) is strictly increasing and strictly concave. The discount factor is β (0, ). Technology: There is a constant returns to scale technology for which labor is the only input so that a firm that hires h units of labor produces zh units of output. Endowments: Each household has unit of time per period to allocate between work and leisure. Institutions: There is a government that has to meet an exogenous stream of expenditures, {g t }. 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 τ t w t h t,whereτ 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. (a) Households solve X max β t u(c t ) {c t,h t,s t } t=0 t=0 s.t. c t = ( τ 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 rate on bonds. Clearly, h t =, for all t (Alternatively, one can simply note that households do not value leisure, so that h t =for all t, anddroph t from the household s problem.) The intertemporal optimality condition for saving is u 0 (c t )=β ( + r t+ ) u 0 (c t+ ). Firms solve n o max h f t zh f t w t h f t, 2

3 and it follows that for any firm that hires a strictly positive (but finite) amount of labor, w t = z, for all t. (b) The government s flow (period-by-period) budget constraint is g t +(+r t )b t = b t + τ t w t h t. (c) A competitive equilibrium is a sequence of prices, {w t,r t } a sequence of tax rates {τ t } andanallocation{c t,h t, h f t,b t,s t } such that: given prices and tax rates, the allocation solves the households problem and the firms 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 the consumer s budget constraint becomes c t =( τ t )z +(+r t ) b t b t, thegovernmentbudgetconstraintbecomes g t +(+r t )b t = b t + τ t z, and we can rewrite the intertemporal optimality condition as +r t+ = u0 (z g t ) βu 0 (z g t+ ). (d) In this model, Ricardian equivalence holds. The timing of τ t does not affect the equilibrium allocation, which is summarized by c t = z g t for all t. (e) If the utility function was replaced by u(c t, h t ),whereu(.,.) is strictly increasing in both arguments, the household would care about leisure and the proportional tax would distort the labor-leisure 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 first-order condition for the labor leisure choice will imply u (zh t g t, h t )z( τ t )=u 2 (zh t g t, h t ). Only in exceptional cases (e.g., u(.,.) is separable and logarithmic in both arguments) would the solution to this equation not depend on τ t. 3

4 8. The production function for this economy is given by Y t = G α t L α t, 0 <α<. (PRF) The preferences of the representative household are µ X E 0 t=0 βt γ C γ t ( L t ), 0 <β<, 0 <γ<. The capital accumulation equation is K t+ =(+r) K t + G α t L α t C t G t, (CA) with government spending following an AR() process around the log of its steady state value: bg t ln (G t /G ss )=φbg t + ε t, 0 φ<, (TS) where {ε t } is an exogenous i.i.d. process. (a) In recursive form, the social planner s problem is V (K t,g t )= max {C t,l t } γ C γ t ( L t )+βe t V ( + r) Kt + G α t Lt α C t G t,g t+. The first order conditions are µ V C γ (Kt+,G t+ ) t ( L t ) = βe t, (FOC) K t+ γ C γ t = βe t µ V (Kt+,G t+ ) K t+ Using Benveniste and Scheinkman s results, we find that µ V (K t,g t ) V (Kt+,G t+ ) = βe t ( + r). K t K t+ Inserting equation (FOC), this reduces to and (FOC) becomes V (K t,g t ) K t = C γ t ( L t )(+r), ( α) G α t L α t. (FOC2) C γ t ( L t )=β( + r) E t C γ t+ ( L t+ ) = E t C γ t+ ( L t+ ). (EE) Combining equations (FOC) and (FOC2) yields γ C γ t = C γ t ( L t )( α) G α t L α t L α t = ( γ)( α) G α t, (LL) L t C t The capital accumulation equation (CA) was derived above when formulating the social planner s problem. 4

5 (b) Log both sides of equation (LL): ln ( exp (ln L t )) + α ln (L t ) = ln (( α)( γ)) ln (C t )+α ln (G t ). Implicitly differentiating this expression yields αd ln L t exp (ln L t) exp (ln L t ) d ln L t = d ln C t + αd ln G t. Letting lower-case letters with carats b denote deviations of logged variables around their steady state values, this implies b t α + L ss ( bc t + αbg t ) L ss θ [αbg t bc t ], θ > 0, (LL 0 ) with the last line following from α, L ss (0, ). Proceeding similarly, it follows from equation (PRF) that (c) Implicitly differentiate equation (CA): by t = ( α) b t + αbg t [ + ( α) θ] αbg t ( α) θbc t, α( + λ)bg t λbc t, 0 <λ<θ. (PRF 0 ) exp (ln (K t+ )) = ( + r)exp(ln(k t )) + exp (ln (Y t )) exp (ln (C t )) exp (ln (G t )), exp (ln (K t+ )) d ln (K t+ ) = (+r)exp(ln(k t )) d ln (K t )+exp(ln(y t )) d ln (Y t ) exp (ln (C t )) d ln (C t ) exp (ln (G t )) d ln (G t ), so that Then insert (PRF 00 )toget K ss b kt+ ( + r) K ss b kt + Y ss by t C ss bc t G ss bc t, b kt+ ( + r) b k t + Y K by t C K bc t G K bg t = (+r) b k t +(ψ + ζ r) by t ψbc t ζbg t. b kt+ ( + r) b k t +(ψ + ζ r)[α ( + λ) bg t λbc t ] ψbc t ζbg t, so that the log-linear approximation for the capital accumulation equation is b kt+ = (+r) b k t + ω bg t ω 2 bc t, (CA 0 ) ω = (ψ + ζ r) α ( + λ) ζ, ω 2 = (ψ + ζ r) λ + ψ. Since (ψ + ζ) >r, λ>0 and ψ>0, it follows that ω 2 > 0. The sign of ω, however, is uncertain. 5

6 (d) It is straightforward to log-linearize the Euler equation, and show that: bc t = η b k t + µbg t, (Thisisgiveninthequestion.). Combining equations (LL 0 ), (PRF 0 ) and (CF), we get by t = α( + λ)bg t λ ³η b k t + µbg t = [α + λ (α µ)] bg t λη b k t, h b t = θ αbg t ³η b k t + µbg t i = θ (α µ) bg t θη b k t, capl t = by t b t = [α +(λ θ)(α µ)] bg t (λ θ) η b k t. 2. Given our assumption that µ<α, an unexpected government spending increase will increase both output and labor. However, with θ>λthe effect on the average product of labor is uncertain. When government spending goes up, two things occur. The first is that the spending itself crowds out private consumption, increasing labor supply and output. In this case, the diminishing marginal product of labor implies that apl c t will decrease. Graphically, this appears as a rightward shift of the labor supply curve. The second effect is that government spending acts as a productivity increase, raising output, labor and apl c t. Graphically, this appears as a rightward shift of the labor demand curve. Combining these effects means output and hours will rise, but the effect on the average product of labor is uncertain. Extrapolating from the uncertain initial effects means apl c t can be either pro- or counter-cyclical. Initial Responses to a Productive Government Spending Shock wages, w t S l 0 S l w 0 d l d l 0 l 0 l Labor hours, l t 6

7 (e) Attached are the background derivations not needed for the answers above behind the shortcuts provided in the exam. We begin by logging both sides of Equation (EE): γ ln C t +ln( exp (ln L t )) = ln E t C γ t+ ( L t+ ) E t ( γ ln C t+ +ln( exp (ln L t+ ))), with the second line using the approximation ln (E t (X t )) E t (ln (X t )). Implicitly differentiating, we get γdln C t + exp (ln L µ t) exp (ln L t ) d ln L t E t γdln C t+ + exp (ln L t+) exp (ln L t+ ) d ln L t+, or γbc t + Inserting equation (LL 0 ), we get γbc t + L µ ss t b E t γbc t+ + L ss t+ b. L ss L ss L ss L ss θ [αbg t bc t ] E t µ γbc t+ + L ss θ [αbg t+ bc t+ ]. L ss Note that ω 4 L ss θ = L ss α + L ss (0, ). L ss L ss L ss Using this notation, we have which can be written as (γ ω 4 ) bc t + ω 4 αbg t E t ((γ ω 4 ) bc t+ + ω 4 αbg t+ ), bc t + ω 3 bg t E t (bc t+ + ω 3 bg t+ ), (EE 0 ) ω 3 = ω 4 α = α γ ω 4 γ/ω 4 Next, let s derive µ and η. Combining equations (CF) and equation (CA 0 ) yields b kt+ =(+r ω 2 η) b k t +(ω ω 2 µ) bg t. (CA 00 ) Inserting equation (CF) into the right-hand-side of equation (EE 0 ) yields E t (bc t+ + ω 3 bg t+ )=η b k t + µbg t + ω 3 bg t, Inserting equation (CF) into the left-hand-side of equation (EE 0 ) yields ³ E t (bc t+ + ω 3 bg t+ )=E t η b k t+ + µbg t+ + ω 3 bg t+. Inserting (CA 00 ) into this expression and recalling equation (TS) yields E t (bc t+ + ω 3 bg t+ ) = E t ³η h( + r ω 2 η) b i k t +(ω ω 2 µ) bg t + µbg t+ + ω 3 bg t+, = η ( + r ω 2 η) b k t +(η[ω ω 2 µ]+φ [µ + ω 3 ]) bg t. 7

8 IftheexpressionsforE t (bc t+ ) aretobeequal, η ( + r ω 2 η) = η, η[ω ω 2 µ]+φ [µ + ω 3 ] = µ + ω 3. The first of these two equations has the non-zero solution +r ω 2 η = η = r, ω 2 while the second equation implies that ηω ( φ) ω 3 = µ ( φ + ηω 2 ) = µ ( φ + r), so that µ = = = ηω φ + r ( φ) ω 3 φ + r rω ω 2 ( φ + r) ( φ) ω 3 φ + r rω ω 2 ( φ + r) α γ/ω 4 µ φ. φ + r Note that ω ω 2 = = < (ψ + ζ r) α ( + λ) ζ (ψ + ζ r) λ + ψ (ψ + ζ r) αλ + αψ ( α) ζ αr (ψ + ζ r) λ + ψ (ψ + ζ r) αλ + αψ (ψ + ζ r) λ + ψ = α. Then with φ< we have µ + ( φ) ω 3 φ + r = rω ω 2 ( φ + r) < r φ + r α<α, and for positive values of ω 3,wehaveµ<α. In general, however, the sign and size of µ are indeterminate, reflecting the multiple effects of government spending that were discussed above. 8

9 9. We are considering an economy populated by N Democrats and N Republicans. The preferences of both consumers are µ X E 0 t=0 βt c σ t, 0 <β<, σ > 0. σ Output is produced by an infinite-lived tree: each period, the tree produces d units of non-storable output. However, there is political uncertainty. When Democrats are in charge of the government, they force each Republican to transfer m units of output to Democrats. When Republicans are in charge, the situation reverses. The indicator e t {D t,r t }, which denotes which party is in charge, follows a stationary, symmetric two-state Markov chain with stationary transition density f (e 0,e): f (e, e) = Pr(e t+ = D e t = D) =π = f (R, R), f (D, R) = Pr(e t+ = D e t = R) = π = f (R, D). (a) Let q (e 0,e) denote the price of the one-step-ahead contingent claim that pays off when e t+ = e 0. Writing the Republican s problem as a Lagrangean, we get V (x t,e t )= min λ t 0 max c t 0, s t+,z(e 0 ) + β X e t+ {D,R} c σ t σ + λ t f (e t+,e t ) V TheFOCforaninteriorsolutionare: c σ t = λ t, λ t p t = β X e t+ {D,R} x t c t p t s t+ X q (e 0,e t ) z (e 0 ) e 0 {D,R} ÃÃ z (et+ )+[p t+ (e t+ )+d] s t+ + m 2 I et+ =D f (e t+,e t ) V [t +] x t+ [p t+ (e t+ )+d], λ t q (e 0,e t ) = β V (x t+ (e 0 ),e 0 ) x t+ f (e 0,e t ), e 0 {D, R}. Since (following Benveniste-Scheinkman), the Euler equations are V [t] x t = λ t,!!, e t+, p t c σ t = βe t ct+ (e t+ ) σ [p t+ (e t+ )+d], (EE) µ σ q (e 0 ct,e t ) = β f (e 0,e c t+ (e 0 t ), e 0 {D, R}. (EE2) ) 9

10 (b) We consider first an arrangement where Democrats and Republicans refuse to trade with each other: Democrats will trade only with Democrats, etc.. Evaluating equation (EE2) shows that the equilibrium pricing kernel for Democrats is µ σ d + m q (D, D) = β π = βπ, d + m µ σ d + m q (R, D) = β ( π), d m µ σ d m q (D, R) = β ( π), d + m µ σ d m q (R, R) = β π = βπ. d + m The kernel for Republicans is the mirror opposite: q (D, D) = βπ, µ σ d m q (R, D) = β ( π), d + m µ σ d + m q (D, R) = β ( π), d m q (R, R) = βπ. 2. It follows from arbitrage arguments that if an asset pays w (e) units of consumption goods when e t+ = e, itspriceis Z p w t = w (e t+ ) q (e t+,e) de t+. When the asset is a risk-free discount bond, w (e) =. Imposing the pricing kernel, it follows that the price of a risk-free bond, R (e t ),isgivenby R (e) = X q (e t+,e t ). e t+ {D,R} For Republicans we have µ σ d + m R (R) = q (D, R)+q (R, R) =β ( π)+βπ, d m µ σ d m R (D) = q (D, D)+q (R, D) =βπ + β ( π). d + m Again, everything reverses for Democrats: µ σ d m R (R) = q (D, R)+q (R, R) =β ( π)+βπ, d + m µ σ d + m R (D) = q (D, D)+q (R, D) =βπ + β ( π). d m 0

11 (c) If a financial intermediary enters the economy and enables financial markets to be complete, consumers will take out contingent claims that eliminate all idiosyncratic political risk. Because there is no aggregate risk, the end result is that both types of consumers will consume the same amount, d, every period.. In equilibrium, with c t = d, equation (EE2) simplifies to q (D, D) = βπ = q (R, R), q (R, D) = β ( π) =q (D, R), for both types of consumers. 2. The equilibrium pricing function for bonds, R (E t ), becomes R (D) =q (D, D)+q (R, D) =β. 3. In equilibrium, with c t = d, equation (EE) simplifies to p t = βe t ([p t+ (e t+ )+d]) Since 0 <β<, it makes sense to solve this equation forward: so that E t βl p t = βet (d), p t = βl βd + b t = βd X t=0 βt + b t, with the bubble term b t obeying We now require that E t (b t+ )=β b t. lim E t β J p t+j = 0, t. (TVC) J Equation (TVC) will be satisfied only if b t =0and the price of a stock is p t = β β d. 4. With no aggregate risk, there is no uncertainty about the return on stocks, and the equity premium is zero.

12 0. Search with on-the-job wage changes. (Adapted from Rogerson, Shimer & Wright.) Time: Discrete, infinite horizon. Demography: Single worker who lives for ever. Preferences: The worker is risk-neutral (i.e. u(x) =x) and discounts the future at the rate r. Endowments: When unemployed: The worker receives b units of the consumption good per period. Also, with probability α she gets to sample a wage from the continuous distribution F with support (0, w] where w > b. When employed: The worker receives her current wage but the wage can change. There is no lay-off as such but instead with probability λ anewwageisdrawnfrom F. The worker can quit if she considers the new wage to be too low. Otherwise, she remains employed at the new wage and is again subject to the same probability of a new draw. (Note: the worker does not have the option of remaining in the jobatheroldwage.) (a) The flow asset value equations for V u (the value to being unemployed) and V e (w) (the value to being employed at the wage w) are rv u = b + αe w [max{v e (w) V u, 0}] rv e (w) = w + λe w 0 [max{v e (w 0 ) V e (w),v u V e (w)}] = w + λe w 0 [max{v e (w 0 ) V u, 0}]+λ(V u V e (w)). (b) The reservation wage w foranunemployedworkerconsideringanewjoboffer solves V e (w )=V u. Asanemployedworkerdoesnothavetheoptionofholding onto his old wage, his choice is always between continued employment at his current wage and unemployment. In short, continued employment at the wage w is the same as starting a new job at the wage w. This means that the reservation wage w foranemployedworkerconsideringwhethertoquitalsosolves V e (w )=V u, so that the reservation wage for unemployed workers is the same as thethresholdwagebelowwhichemployedworkerswillquit. (c) From the equations above and the definition of mathematical expectation we have Subtracting yields rv u = b + α rv e (w) = w + λ Z w w (V e (w) V u ) df (w), Z w r (V e (w) V u )=w b +(λ α) so that w (V e (w 0 ) V u ) df (w 0 )+λ(v u V e (w)). Z w w (V e (w 0 ) V u ) df (w 0 )+λ(v u V e (w)), Z w (r + λ)(v e (w) V u )=w b +(λ α) (V e (w 0 ) V u ) df (w 0 ). w () 2

13 Moreover, since V e (w )=V u evaluating () at w = w implies Z w 0=w b +(λ α) (V e (w 0 ) V u ) df (w 0 ). (2) w Inserting (2) into () to eliminate the integral term implies (r + λ)(v e (w) V u )=w w. Substituting this expressing back into (2) and rearranging, we get w = b + α λ Z w (w 0 w ) df (w 0 ). r + λ w (d) The preceding answer shows that if λ = α, w = b. If there is no opportunity cost (a smaller flow of wage offers) associated with accepting a job then I accept any offer that is better than my current income (b). (e) If there is a large number of such workers with mass normalized to, the steadystate flow diagram for movements between employment and unemployment is given by λf(w*) u - u α [ F( w*) ] Letting u denote the fraction of workers that are unemployed, we see that in a steady state α[ F (w )]u = λf (w )( u), so that u = λf (w ) α[ F (w )] + λf (w ). 3