Foundations Of Modern Macroeconomics 1st Edition Ben J. Heijdra, Frederick Van Der Ploeg - Solutions

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(b) Suppose n>^>r. Show that opening to trade makes everyone in the economy
Will it ever run fiscal deficits or surpluses? Can you discern any rule by which the government should set taxes?(a) Suppose that the world interest rate equals the autarky rate, r^. Show that a small permanent rise in the world interest rate that occurs on date benefits not only the date old, but
Assume the gov- ernment maximizes the lifetime utility of the representative individual, but (exoge- nously) must spend G, < Y, in resources per period on projects that yield no benefits. Will the government be indifferent as to the path of taxes that finance its expendi- tures?
Explain this result intuitively. [Hint: Analyze a representative vintage's generational account.] Tax smoothing and deficits la Barro (1979). Consider a small, open representative- consumer economy with an infinite-horizon (on the lines of Chapter 2). The subjec- tive time preference factor
Debt and deficits in the Weil (1989a) model. Suppose the public debt follows an arbi- trary (non-Ponzi) path in Weil's overlapping generations model. Show that aggregate per capita consumption satisfies - {+r) + (+) (+) = (1-8) n(ds+1-ds) 6. & + (117) (y's -B)). +(1+r)b, + s=t so that current and
4. 5. Public debt and the intertemporal terms of trade. In the global-equilibrium model of section 3.6, can the initial young and future generations ever benefit when the government introduces a steady state public debt financed entirely by taxes on the young?
Consider an overlapping-generations economy (as in section 3.5) that is open to trade. Now, however, the world interest rate r lies below the population growth rate n, which is positive
(d) Suppose that the young can borrow and their endowment grows according to Y+1 =(1+g)y. However yo=0 and y remain constant over time. How does the date saving rate depend on g? Do the same exercise assuming it is the endowment of the middle aged that grows at rate g while those of the young and
How does your answer change when the young can borrow against future earnings?
(c) Suppose e rises. What is the effect on the saving rate?
What is the aggregate saving rate out of total output Y,?
(b) Let the growth rate of total output be g > 0, where y+1 = (1 + g)y).
1. 2. Saving and growth in a three generation model. Consider a pure endowment economy facing a given world interest rate r = 0. Residents' lifetimes last three periods, and on any date three distinct generations of equal size (normalized to 1) coexist. The young cannot borrow at all, and must save
(f) In the model of the text, we showed that marginal q equals average q. Is that true in this formulation?
(e) Now suppose that the system is initially in a steady state corresponding the pro- ductivity level A, but the firm learns (by surprise) on date that A will rise perma- nently to A' at some future date T. (Perhaps the firm learns that it will be able to use a new invention that aids productivity
(d) Using your graph, show what happens when there is an unanticipated permanent rise in A to A'. Show the new steady state and the transition to the new steady state.
(c) Assume that the productivity parameter is constant at A, and draw the phase diagram of the system with q on the vertical axis and K on the horizontal axis. Find the steady state. Does it depend on adjustment costs?
(b) Show that the first-order conditions imply the following system in q and K: - 91 == K+1 K = - X 1 91+1 - 9 = q - A++ F (K +91-1).
(a) Differentiate the firm's objective function with respect to I, and K, to find the first-order conditions characterizing efficient investment.
A simplified q model. Suppose that a firm facing a market interest rate 1+r has a production function given by Y, A, F(K,), where A is a productivity parameter and we treat L as fixed. The firm's objective function is to maximize the present discounted value of profits. However, the firm faces
Early critics of the intertemporal approach to the current account (e.g., models such as those in this chapter) argued that a major empirical flaw of the approach is its inability to yield countercyclical current accounts. Are they right?
Unstable debt-output ratios and transversality. In the unstable case of the model of a small country's debt-output ratio (appendix 2A), why is the transversality condition not violated as B/Y ?business cycle and the current account. In the simplest static Keynesian models of undergraduate
An alternative form of the Campbell (1987) saving test. Show that eq. (43) in the text holds if and only if the variable CA+1 AZt+1(1+r)CA, is statistically uncorrelated with date (or earlier) variables. Derivation of eq. (43). Equation (42) in the text implies that CA, = Z - E, Z, where Z is net
6. Because p>0. consumption innovations now are more variable than output innova- tions when individuals desire smooth consumption. (d) Compute the current account response to output innovations in this case. Verify the claim at the end of section 2.3.3.
(b) Suppose output follows a nonstationary stochastic process like eq. (38) in the text, - Y+1 Yp(Y-Y-1) ++1 where 0 t, (E+1-E)Y = (1+ p + ... + p - (1+1)) ( +1 = 1-ps-r 1-p (c) Conclude that for the permanent-income consumption equation (32), the con- sumption innovation on date +1 is C++1 - C =
4. Details on Deaton's paradox. Consider the linear-quadratic stochastic consumption model, in which (1+r) = 1 and G = 1 = 0 on all dates. (a) Use the current account identity together with consumption equation (32) in the text to show that the change in consumption is the present value of changes
(a) In any given period, an individual lives on to the next period with probability
3. Uncertain lifetimes and infinite horizons. One way to motivate an infinite individual planning horizon is to assume that lives, while finite in length, have an uncertain terminal date. In this case, the terminal period + 7 of section 2.1.1 becomes a random variable. Let E,U, denote the
(c) Have you now proved that current account sustainability requires only that coun- tries pay an arbitrarily small constant fraction of interest owed each period, rolling over the remaining debt and interest? [Hint: Consider the case of an endowment econ- omy with G=I=0 and constant output Y. How
(b) Show directly that the intertemporal budget constraint is satisfied for any > 0. [Hint: Show why TB=- (1+r)^ &r B = = (1 + r) B..] s=1 Note that even if & is very small, so that trade balance surpluses are very small, debt repayments grow over time as B, grows more negative.
(a) Using the current account identity and the definition of the trade balance, show that under this policy, net foreign assets follow the equation B+1 = [1 + (1 )r] B.
1. Current account sustainability and the intertemporal budget constraint. Suppose that a country has negative net foreign assets and adopts a policy of running a trade balance surplus sufficient to repay a constant small fraction of the interest due each period. It rolls over the remaining
(b) Explain why $ \zeta^* - 1 > 0 $ at the world interest rate associated with the optimal tax.
(a) Show that the optimal ad valorem borrowing tax [such that $ (1 + \tau)(1 + r^*) $ is the gross interest rate domestic residents face when the world rate is $ 1 + r^* $] is given by $ \tau = \frac{1}{(\zeta^* - 1)} $, where $ \zeta^* $ is the elasticity with respect to $ 1 + r $ of
(c) By differentiating this consumption function $ C_1 $ (including differentiation of $ W_1 $) with respect to $ R $, show that$$ \frac{dC_1}{dR} = -\frac{C_1}{1 + R} + \frac{Y_2}{1 + R} + \frac{y}{1 + R} \left[ -1 - \log(\beta/R) \right]. $$
(b) What is the optimal level of $ C_1 $, given $ W_1, R, $ and $ \beta $?
(a) Solve for $ C_2 $ as a function of $ C_1 $, $ R $, and $ \beta $ using the consumer’s intertemporal Euler equation.
7. Interest rates and saving with exponential period utility. A country's representative individual has the exponential period utility function$$ u(C) = -\gamma \exp(-C/\gamma) $$($ \gamma > 0 $) and maximizes $ U_1 = u(C_1) + \beta u(C_2) $ subject to$$ C_1 + R C_2 = Y_1 + R Y_2 = W_1 $$[where
(b) Do the same exercise for a rise in $ A^*_2 $ (Foreign's date 2 productivity), still assuming Home has a positive current account on date 1. Is the effect on Home's current account simply a mirror image of the answer to part a?
(a) Suppose date 2 Home investment productivity, $ A_2 $, rises slightly. If Home has a date 1 surplus on current account and Foreign a date 1 deficit before the productivity rise, how does the change affect date 1 current accounts?
6. Future productivity shocks when current accounts are initially unbalanced. Let Home have the production function $ Y = A F^*(K) $, and Foreign the function $ Y^* = A^* F^*(K^*) $, on each of two dates. Let the corresponding lifetime utility functions of residents be \( U_1 = u(C_1) + \beta
5. Endowment shifts and world interest rates. In the two country endowment model of borrowing and lending, show algebraically that a rise in $ Y_1 $ or $ Y_1^* $ lowers $ r $, whereas a rise in $ Y_2 $ or $ Y_2^* $ raises $ r $.
(d) Does zero intertemporal substitutability necessarily imply a literally constant consumption path, as in parts a-c, under all possible preference assumptions? [Hint: Suppose lifetime utility is$$ U_1 = \frac{1}{1 - 1/\sigma} \left( C_1^{-1/\sigma} + \beta/\sigma C_2^{-1/\sigma} \right). $$Show
(c) Calculate that$$ \frac{dC_1}{dr} = \frac{Y_1 - Y_2}{(2 + r)^2}. $$What is your interpretation?
(b) Derive the consumption function for this case,$$ C_1 = \left( \frac{1 + r}{2 + r} \right) Y_1 + \left( \frac{1}{2 + r} \right) Y_2. $$Show that $ C_2 = C_1 $ using this consumption function and the current account identity.
(a) Show that the Euler equation (25) approaches $ C_2 = C_1 $, so that a flat consumption path is chosen irrespective of the market interest rate $ r $.
4. Problem on $ \sigma = 0 $. The individual has an isoelastic period utility function and exogenous endowments. This exercise considers the limit as $ \sigma \to 0 $.
(d) Using $$ I_2 = -K_2 $$ and the results of parts a-c, explain why
(c) Derive Home's date 1 consumption function, and show it can be written as$$ C_1(r) = \frac{1}{1 + \beta} \left( Y_1 - I_1 + \frac{Y_2 - I_2}{1 + r} \right) $$
(b) Show that Home's II schedule can be written as$$ I_1(r) = K_2 - K_1 = \left( \frac{\alpha A_2}{r} \right)^{\frac{1}{\alpha}} - K_1 $$
(a) Investment is determined so that the marginal product of capital equals r. Show that this equality implies Intertemporal Trade and the Current Account Balance$$ K_2 = \left( \frac{\alpha A_2}{r} \right)^{\frac{1}{\alpha}} $$
Adding investment to the last exercise. Assume date 2 Home output is a strictly concave function of the capital stock in place multiplied by a productivity parameter,$$ Y_2 = A_2 K_2^{\alpha} $$($\alpha < 1$), with a parallel production function in Foreign. (Date 1 outputs are exogenous because
(f) How does an increase in Foreign's rate of output growth affect Home's welfare? Observe that a rise in the ratio Y/Y raises the equilibrium world interest rate. Then show that the derivative of U with respect to ris dU dr =====
(e) Confirm that the country with an autarky interest rate below r will run a current account surplus on date I while the one with an autarky rate above r will run a deficit.
(d) Check that it lies between the autarky rates r^ and ^*
(c) Compute the equilibrium world interest rate. -Y2.
(b) Show that Home saving is S(r) Y C(r) = Y - 1+ B (1+B)(1+r)
(a) Home receives perishable endowments Y and Y2 in the two periods. Show that the Home date 1 consumption function is a function of r, C(r)= 1 1+B Y2 +r (This equation shows a general property of logarithmic preferences that we will use many times in this book: expenditure shares on the available
ci Logarithmic case of the two-country endowment model. Consider the pure endow- ment model, in which equilibrium holds when S + S = 0. Home's utility function is U = log C+ Blog C. (42) Foreign has an analogous log utility function, with its consumption levels and time- preference factor
(c) Explain why the answer in b implies that a country benefits from a rise in the world interest rate if and only if its terms of intertemporal trade improve. (d) Let W = Y + Y2/(1+r) (that is, W is lifetime wealth in units of date 1 con- sumption). Show that a small percentage gross interest rate
(b) Use the Euler condition together with the (differentiated) budget constraint to compute the total derivative du (C1, C2) au (C1, C2) dr ac (Y - C).
1. Welfare and the terms of trade. Let the representative individual in a small open economy maximize U(C1, C2) subject to C + C/(1+r) Y+ Y2/(1+r), where Y and Y2 are fixed. (a) Show that the intertemporal Euler equation takes the form aU (C1, C2)/aC = (1+r)au(C1, C2)/aC2.
Along the p = 0 line there is full employment (y = y). It is upward sloping because an increase in the domestic price level reduces output via the real balance effect. To restore full employment, the nominal exchange rate must depreciate. For points to the right of the p = 0 line, output is below
Figure 11.16. Phase diagram for the Dornbusch model arrows in Figure 11.16. More formally we can derive the same result by noting that I 1.74) implies:(a e EMYEYQ which shows that the interest parity condition introduces an unstable element into the economy in the sense that exchange rate movements
Along the e = 0 line the domestic interest rate equals the foreign interest rate (r = r*).It is downward sloping in view of our assumption (made above) that EmyEm < 1.For points above the e = 0 line the nominal (and the real) exchange rate is too high, output is too high, and the domestic rate of
By substituting (11.72)-(11.73) and (T5.5) into (T5.3) and (T5.4), we obtain the dynamic representation of the model:The only sign that is ambiguous in the Jacobian matrix on the right-hand side of (11.74) is the one for ae/ap. This is because an increase in the domestic price level has an
The Foundation of Modern Macroeconomics by monetary policy, but can be affected by fiscal policy. But we are really interested in the short-run dynamics implied by the model. To study this, we first reduce the model to two differential equations in e and p. For given values of the nominal exchange
Up to this point we have always assumed that r = r* under perfect capital mobility,- hich would be correct if investors never expect the exchange rate to change. Whilst this may be reasonable under a (tenable) fixed exchange rate regime, it is a somewhat inconsistent assumption to make about
11.3.1 The Dornbu!Up to this point we which would be correct this may be reasonable what inconsistent as, of freely flexible excha nerally will) fluctut:I.in the exchange rate. T duce the assumption oi ex dectations; see Chap perfect capital mobility E., nations (T5.1) an('_ a small open economy
vof the investment:Ee yield gap F. --- ( 1 ± r ) — (1 + r*)—I = (1 + r) — (1 + r*) (1 +AEe Eo Eo= (1 + r) — (1 + r* + AEe+ r*AEe) --- r (r* +AEe E0 (11.69) Eo Eo ) 'where the cross-term r* AEe 1E0 can be ignored because it is of second-order magnitude. Equation (11.69) can be written in
Up to this point we have been somewhat inconsistent in our discussion of the economy operating under flexible exchange rates. The nature of this inconsistency can be gleaned by looking at the uncovered interest parity condition. Consider a domestic investor who has f 100 to invest either at home,
spending:so that:(1 + e + - + r)ogN gc - gN = > 0,[1 + 0 + (C) 2 ] (1 + 0 + - + 02[1 + 0 + (r) 2 i -OgN - + r)C0A[1 + 0 + (01(1 + 0 + 2)- + 02 From (11.65) we can conclude that gN is larger, and gN is smaller, in the asymmetric than under the symmetric case (for which = > 0)—compare
The opposite holds if there is real wage rigidity in both countriesc = < 0), as is illustrated in Figure 11.15. Fiscal policy constitutes a beggar-thy-neighbour policy and uncoordinated actions by national governments lead to spending levels that are too high. The coordinated policy solution
,comotive policy. In the absence of coordination, however, individual countries do not take into account that their own fiscal spending also aids the other country. They therefore both underestimate the benefit of their own spending and consequently choose spending levels that are too low. In the
Chapter 11: The Open Economy wage rigidity in both countries “. = = 1), the cooperative solution involves the higher spending levels in the two countries. This is illustrated in Figure 11.14, where point C designates the cooperative solution. The intuition behind this result is obvious. With
,ye behaviour to the ones ons (11.58)-(11.59)), it is Ynlicitly takes into account v the terms premultiplied lying (11.62)-(11.63) for g'eels under coordination(11.64)rative and non-cooperative 1.64). If there is nominal(11.62)gN =+ e - op+ 0)2 - + +19 +g* + e -= (1 + 0)2 - -* 1 + + 0 [ (
The relative size of government spending in the cooperative and non-cooperative scenario's can be judged by comparing (11.60) and (11.64). If there is nominal wage rigidity in both cc the higher spending it where point C designatL:is obvious. With nomina locomotive policy. In the not take into
Figure 11.15. International coordination of fiscal policy under real wage rigidity in both countries under a coordinated fiscal policy is:min LG± 1(g — y)2 + i (g* rg —tg*,g)+ tlg2 + (g*) 2 22 ag* = (g + — + (g* + —y) + og* = o.By comparing these first-order conditions under cooperative
Figure 11.14. International coordination of fiscal policy under nominal wage rigidity in both countries 293 Chapter 11: The Open Economy lomy and nominal yaw--lulating domestic ou!nt target, p > 0) without'tiive rise to large deficit)me cost function, LG:(11.55 domestic economy, given e foreign
c. i.e. y features in bot'•ending level independently, country into account. In chooses its spending level sample, the policy maker g0 gN 11y T/10 +
The non-cooperative Nash equilibrium is defined as that equilibrium in which each country's spending plan is optimal given the other country's spending plan. Since the reaction functions designate such conditionally optimal spending plans, the non-cooperative Nash equilibrium is obtained by finding
Similarly, the foreign country has a reaction function (RR*) which relates its optimal(non-coordinated) level of government spending to its full employment target and the spending level of the domestic country:= (g* + *g — + 9g* = 0 g* = RR*. (11.59) ag* 1 + 0
Suppose that both governments choose their own spending level independently, i.e. without taking the possible repercussions for the other country into account. In this case, fiscal policy is uncoordinated and each country chooses its spending level conditional upon the other country's spending
Assume that the domestic government is interested in stimulating domestic output(to get as close as possible to some given full employment target, y > 0) without, however, creating a large government sector (which could give rise to large deficits).We assume that the domestic policy maker minimizes
0 < < 1 if there is real wage rigidity in the domestic economy and nominal wage rigidity in the foreign economy (see section 2.3).
The symmetric two-country model of the world economy that was developed in the previous subsections can be used to study the issue of international policy coordination.Since we do not wish to carry on with the rather extensive notational burden introduced in the development of this model, the
Figure 11.13. Monetary policy with real wage rigidity in Europe and nominal wage rigidity in the United States This explains why foreign output rises. Similarly, the real exchange rate appreciation causes European producer wages to fall, thus also enabling an increase in output there.11.2.4
A monetary expansion in Europe has no real effects (see above), but expansionary US monetary policy (a rise in m*) constitutes a locomotive policy for Europe. This has been illustrated in Figure 11.13. The increase in the US money stock shifts GME -,-down and LM*(AS'„',) to the left. The European
A US fiscal expansion (a rise in g*) shifts both GMER and GME'k. In terms of Figure 11.12, the new equilibrium is at e2. The rate of interest is higher, there is a real depreciation in Europe, but output falls because real producer wages in Europe rise.Output and employment in the US rise, so that
The European fiscal impulse constitutes a locomotive policy since it ends up simultaneously stimulating US output and employment.
The Foundation of Modern Macroeconomics Figure 11.12. Fiscal policy with real wage rigidity in Europe and nominal wage rigidity in the United States exchange rate of Europe appreciates and the new equilibrium is at el. Both y and y*increase, though the latter increases by more than the former (see
and GME1,'„ with the former experiencing the larger shift (as ri < 1). The real 289 GM ER (gi, go*)GMER (go,g1 *)GM ER (go, go*)GME,; (go, g1 *)GMV; (g1 , go*)GME,; (go, go*)Europe United States LM*(AS)y =7 *y
Figure 11.11. Fiscal policy with real wage rigidity in both countries wage rigidity well describes the European countries. Letting Europe denote the home country and the US the foreign country (and ignoring the rest of the world for the time being), the model describing this configuration is
In an influential paper, Branson and Rotemberg (1980) argue on the basis of empirical evidence, that nominal wage rigidity characterizes the US economy whilst real r*0 Yi *Yo=- Yo *Yi Y,Y *Figure 11.1'(11.53) both counit wage rigidity well describe country and the US the fc time being), the model
Equation (11.53) provides a clear statement of the beggar-thy-neighbour property of fiscal policy when both countries experience real wage rigidity.Not surprisingly, monetary policy has no real effects under real wage rigidity. As none of the equilibrium conditions is affected, the interest rate,
In sharp contrast to our conclusion in the previous section, fiscal policy constitutes a beggar-thy-neighbour policy under real wage rigidity. This can be illustrated with the aid of Figure 11.11. The increase in government spending in the domestic country(g) raises the interest rate and causes a
The key thing to note is that own and foreign fiscal policy affect have the same output effects in both countries.Monetary policy in the domestic country, on the other hand, does not benefit but harm the foreign country. This is illustrated with the aid of Figure 11.10. The increase in the domestic
For future reference we derive the expressions for the output multipliers. First, we use (GMEN) and (GMEN) to derive the effect of domestic and foreign fiscal policy on the world interest rate:dr*=dr* (1 + 77)EyG(1 + (DNENwEmY) >dg dg* 2 [EyR(1 + (i,,TENwemy) + WNENWEMR](11.50)Next, we use
Fiscal policy in the domestic country (represented by a rise in g) shifts up both GMEN and GMEN but, provided the own effect of government spending dominates(so that ri < 1), the former shifts by more than the latter (i.e. ar*/ag is largest for GMEN). The new equilibrium is at el , the domestic
The curves LM(ASN) and LM*(AS7v ) are drawn in the left-hand panel of Figure 11.9, and coincide in the initial equilibrium due to the symmetry assumption. The goods market equilibrium schedule under nominal wage rigidity, GME N, is obtained by substituting LM(ASN) into the IS curve and solving for

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Foundations Of Modern Macroeconomics 1st Edition Ben J. Heijdra, Frederick Van Der Ploeg - Solutions

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