程序代写代做代考 THE UNIVERSITY OF MANCHESTER MACROECONOMIC ANALYSIS Semester 1 2020/21

THE UNIVERSITY OF MANCHESTER MACROECONOMIC ANALYSIS Semester 1 2020/21
INSTRUCTIONS SPECIFIC TO THIS EXAM:
• Answer ONE question from Section A and TWO questions from Section B. Each section is worth 50 points.
• Please submit typed responses. Hand-drawn diagrams are acceptable. Alterna- tively, you may write all your answers by hand and scan them into PDF format. Either way, you must include the cover page provided on Blackboard.
• Ensure that your answers are oriented correctly. Marks will be deducted if your answers are rotated 90 degrees, upside down, etc.
• Ensure that your answers are legible and the scanned image is clear.
• Students are not permitted to discuss their answers with other students before
submission.
• Candidates are expected to demonstrate to the examiners a competent knowledge
of all computations.
• Candidates are also advised that the examiners attach considerable importance to
the clarity with which answers are expressed.
⃝c The University of Manchester, 2020/21 Page 1 of 12
ECON60111

ECON60111
SECTION A Answer ONE question
1. Growth, saving and real interest rate
Thomas Piketty, in Capital in the Twenty-First Century, argues that a fall in the growth rate of the economy is likely to lead to an increase in the difference between the real interest rate and the growth rate. This problem asks you to investigate this issue in the context of the Ramsey-Cass-Koopmans (RCK) model. Specifically, consider a RCK economy that is on its balanced growth path, and suppose there is a permanent fall in g.
(a) How, if at all, does this affect the k ̇ = 0 and c ̇ = 0 curves? Illustrate the change in these curves in a diagram and give economic intuition. (10 points)
See Figure 1.
The equation describing the dynamics of the capital stock per unit of effective labor is
k ̇(t) = f(k(t)) − c(t) − (n + g)k(t).
For a given k, the level of c that implies k ̇ = 0 is given by c = f(k)−(n+g)k. Thus a fall in g makes the level of c consistent with k ̇ = 0 higher for a given k. That is, the k ̇ = 0 curve shifts up. Intuitively, a lower g makes break- even investment lower at any given k and thus allows for more resources to be devoted to consumption and still maintain a given k. Since (n + g)k falls proportionately more at higher levels of k, the k ̇ = 0 curve shifts up more at higher levels of k.
The equation describing the dynamics of consumption per unit of effective
labor is given by
c ̇(t) = f′(k(t)) − ρ − θg c(t) θ
Thus, the condition required for c ̇ = 0 is given by f′(k) = ρ + θg. After the fall in g, f′(k) must be lower for c ̇ = 0. Since f′′(k) < 0, this means that the k needed for c ̇ = 0 therefore rises. Thus the c ̇ = 0 curve shifts to the right. For intuition, keep in mind that c is consumption per effective worker, where c = C/A and C is consumption per worker. Therefore, c ̇/c = C ̇ /C − A ̇/A and the Euler equation can be re-written as C ̇ /C = (f (k) − ρ)/θ. Thus, c ̇/c = 0 when C ̇/C = A ̇/A = g. Thus, when g decreases, C ̇/C must be lower in the new steady state. Households optimally choose to lower the growth rate of C (consumption per worker) when the real interest rate is lower (which happens when k increases). 2 of 12 (b) At the time of the change, does c rise, fall, or stay the same, or is it not possible to tell? What about in the long run? Explain. (10 points) (c) At the time of the change, does r − g rise, fall, or stay the same, or is it not possible to tell? What about in the long run? Explain. (10 points) ECON60111 Consumption per effective unit of labor, c, can jump at the time of the shock. For the economy to reach the new balanced growth path, c must jump at the instant of the change so that the economy is on the new saddle path. However, we cannot tell whether the new saddle path passes above or below the original point E. Thus we cannot tell whether c jumps up or down or remain the same at the instant that g falls. Thereafter, c and k rise gradually to their new balanced-growth-path values; these are higher than their values on the original balanced growth path. At the time of the change in g the value of k, the stock of capital per unit of effective labor, is given by the history of the economy, and it cannot change discontinuously. It remains equal to the k∗ on the old balanced growth path. Therefore, on impact, k does not change. Thus, f′(k) ≡ r does not change. So the immediate impact of the decrease in g is that (r − g) increases. Inthelongrun,r=ρ+θgandso(r−g)=ρ+(θ−1)g. Thus ∂(r − g) = (θ − 1) ∂g Thus, the long-run impact of a decrease in g on r − g depends on θ. Specifi- cally, if θ < 1, then ∂(r−g)/∂g < 0, and a decrease in g would cause (r−g) to rise. In the case of logarithmic utility, with θ = 1, the change in g would have no long-run impact on (r − g). Finally, if θ > 1, then ∂(r − g)/∂g > 0 and so a decrease in g would cause (r − g) to fall. Intuitively, a high value of θ means that the individual has a strong desire to smooth consumption over time. Since consumption grows at rate g on the balanced growth path, a higher value of g would require more compensation (i.e., an even higher r) to tolerate this higher consumption growth. Similarly, a decline in the growth rate g means a decline in the required compensation.
(d) Letting s = [f(k∗)−c∗]/f(k∗) denote the fraction of output that is saved on the balanced growth path, find an expression for ∂s/∂g. Can you tell whether this expression is positive or negative? Explain. (10 points)
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(e) For the case where the production function is Cobb-Douglas, f(k) = kα, rewrite your answer to part (d) in terms of ρ, n, g, θ, and α. (Hint: use the fact that
f′(k∗) = ρ + θg.)
(10 points)
ECON60111
On a balanced growth path, the fraction of output that is saved and invested is given by [f(k∗) − c∗]/f(k∗). Since k is constant, or k ̇ = 0 on a balanced growth path, then from k ̇ = f(k)−c−(n+g)k, we can write f(k∗)−c∗ = (n+g)k∗. Using this, we can re-write the fraction of output that is saved on a balanced growth path as
s = [(n + g)k∗]/f(k∗)
Differentiating both sides of this equation with respect to g yields ∂s = f(k∗)[(n + g)(∂k∗/∂g) + k∗] − (n + g)k∗f′(k∗)(∂k∗/∂g)
∂g
which simplifies to
∂s ∂g =
[f(k∗)]2
(n + g)[f(k∗) − k∗f′(k∗)](∂k∗/∂g) + f(k∗)k∗ [f(k∗)]2
(A-1)
Since k∗ is defined by f′(k∗) = ρ + θg, implicitly differentiating both sides of this expression with respect to g gives us f′′(k∗)(∂k∗/∂g) = θ. Solving for
∂k∗/∂g yields
Substituting this back into Equation (A-1) yields
∂s = (n + g)[f(k∗) − k∗f′(k∗)]θ + f(k∗)k∗f′′(k∗) ∂g [f(k∗)]2f′′(k∗)
∂k∗ θ
∂g =f′′(k∗)<0. (A-2) The first term in the numerator is positive whereas the second is negative. Thus, the sign of ∂s/∂g is ambiguous. We cannot tell whether the fall in g raises or lowers the saving rate on the new balanced growth path. When the production function is Cobb-Douglas, f(k) = kα, f′(k) = αkα−1 and f′′(k) = α(α − 1)kα−2. Substituting these facts into Equation (A-2) yields ∂s = (n + g) [(k∗)α − k∗α(k∗)α−1] θ + (k∗)αk∗α(α − 1)(k∗)α−2 ∂g (k∗)α(k∗)αα(α − 1)(k∗)α−2 4 of 12 ECON60111 This simplifies to ∂s = (n + g)(k∗)α(1 − α)θ − (1 − α)(k∗)αα(k∗)α−1 ∂g [−(1 − α)(k∗)α(α(k∗)α−1)(α(k∗)α−1)/α] Dividing the top and bottom of the right-hand side of this expression by (1 − α)(k∗)α, and using the fact that α(k∗)α−1 = f′(k∗) = ρ + θg on the balanced growth path, we get: ∂s =−α[(n+g)θ−(ρ+θg)] ∂g (ρ + θg)2 Thus, finally, we have ∂s =−α(nθ−ρ) =α(ρ−nθ) ∂g (ρ+θg)2 (ρ+θg)2 Figure 1: Ramsey Question 2. Optimal Monetary Policy Consider the following variant of the canonical New Keynesian model: y =E[y ]−1(i −E[π ])+uIS (1) t tt+1 θt tt+1 t πt = βEt[πt+1] + κ(yt − ytn) (2) where (uIS,yn) are stochastic shocks and yn is the natural level of output. Suppose ttt central bank sets the nominal interest rate it according to it = rtn + φπEt[πt+1] (3) where rtn is the economy’s natural rate of interest. Let y ̃t = yt − ytn. 5 of 12 (a) Show that Equations (1), (2) and (3) can be written, in matrix form, as  y ̃   E y ̃   π   E π   1  κ (1 − φ )/θ  π t =A tt+1 ,A= ttt+1 π (4) Carefully explain each step of your derivation. (10 points) (b) What does the system of equations, given by Equation (4), simplify to when φπ = 1? What are the eigenvalues of the matrix A in this case? (10 points) β + κ(1 − φ )/θ  ECON60111 First, note that the natural rate of interest is given by rn=θ􏰀E[yn ]−yn+uIS􏰁 t tt+1tt Then, substituting into Equation (3), we get: i=θ􏰀E[yn ]−yn+uIS􏰁+φE[π ] t tt+1 t t πtt+1 Substitute this expression in the New Keynesian IS curve, Equation (1): y =E[y t t t+1 ]−E[yn ]+yn −uIS −φπE[π ]+1E[π ]+uIS t t+1 t t θ t t+1 θ t t+1 −yn ]+1−φπE[π ] t (A-3) y −yn =E[y t t t t+1 t+1 θ t t+1 Usingthaty ̃ =y −yn andy ̃ =y −yn ,thiscanbewrittenas: t t t t+1 t+1 t+1 􏰂 1 − φπ 􏰃 y ̃t = Et[y ̃t+1] + θ Et[πt+1] Next, substitute Equation (A-3) into the New Keynesian Phillips curve, Equation (2): 􏰂 􏰂 1 − φπ 􏰃 􏰃 πt = βEt[πt+1] + κ Et[y ̃t+1] + θ Et[πt+1] 􏰂 κ(1−φπ)􏰃 πt = κEt[y ̃t+1] + β + θ Et[πt+1] (A-4) Equations (A-3) and (A-4), written in matrix form, is equivalent to Equa- tion (4). When φπ = 1, the matrix A simplifies to 10 A=κ β (A-5) The characteristic equation of A is defined by det(A−γI), where I is a 2×2 6 of 12 (c) Suppose we look for self-fulfilling movements in y ̃ and π of the form πt = λtZ, y ̃t =cλtZ,|λ|≤1. Whenφπ =1,forwhatvaluesofλandcdoessucha solution satisfy Equation (4)? Thus, what form do the self-fulfilling movements in inflation and output take? (20 points) ECON60111 identity matrix, and the solutions of the equation, γ, are the eigenvalues. Recall that the determinant of a 2 × 2 matrix, such as  B=ab cd is given by |B| = ad−bc. Therefore, the characteristic equation of A is given by 􏰑􏰑 􏰑􏰑1−γ 0 􏰑􏰑 􏰑 􏰑 = (1 − γ)(β − γ) = 0 􏰑􏰑 κ β−γ􏰑􏰑 The characteristic equation implies that the eigenvalues are γ = β and γ = 1. To analyze the self-fulfilling movements, we first make the observations that Et[y ̃t+1] = λy ̃t (A-6) Et[πt+1] = λπt (A-7) Substituting Equations (A-5), (A-6) and (A-7) into Equation (4) and per- forming the multiplication yields y ̃10λy ̃ λy ̃  t=×t=t (A-8)  π   κ β   λπ   κλy ̃ +βλπ  tttt From Equation (A-8), we obtain two equalities. The first is given by y ̃t = λy ̃t (A-9) There are two possibilities that satisfy Equation (A-9): y ̃t = 0 or λ = 1. The second equality that we obtain from Equation (A-8) is πt = κλy ̃t + βλπt (A-10) Consider the first possible solution for Equation (A-9). Substituting y ̃t = 0 into (A-10), we obtain or simply πt = βλπt λ = 1/β (A-11) 7 of 12 (d) Suppose φπ is slightly (that is, infinitesimally) greater than 1. Are both eigen- values inside the unit circle? Is it possible for there to be self-fulfilling equilibria? Why or why not? (10 points) ECON60111 Since β < 1, Equation (A-11) implies λ > 1, which contradictions the as- sumption of the problem. Therefore, we can eliminate the possibility that y ̃t =0andconcludeλ=1. Wecannowsubstituteλ=1,πt =λtZ,and y ̃t = cλtZ into Equation (A-10) to obtain
This implies that
Z = κcZ + βZ.
c = (1 − β)/κ
Thus, the self-fulfilling movements of π and y ̃ result in both variables re- maining constant at values Z and (1 − β)/κ]Z, respectively.
Let’s compute the eigenvalues for the general form of matrix A. First, note
that the characteristic equation of A is defined by det(A − γI), where I is a
2×2 identity matrix, and the solutions of the equation, γ, are the eigenvalues.
Therefore, the characteristic equation of A is given by 􏰑􏰑
􏰑􏰑 1 − γ (1 − φπ)/θ 􏰑􏰑 = (1−γ)(β+κ(1−φ )/θ−γ)−κ(1−φ )/θ = 0 􏰑􏰑 􏰑􏰑 π π
􏰑 κ β+κ(1−φπ)/θ−γ 􏰑
Defining α ≡ κ(1 − φπ)/θ, we can re-write the characteristic equation as
(1 − γ)(β + α − γ) − α = 0 β+α−γ−βγ−αγ+γ2 −α=0 γ2 − (1 + β + α)γ + β = 0
Using the quadratic formula, we can obtain the two possible eigenvalues:
1+β+α−􏰟(1+β+α)2 −4β 2 1+β+α+􏰟(1+β+α)2 −4β 2
where naturally λ1 < λ2. Note that when, φπ = 1, α = 0, and these eigenvalues simplify to λ1 = β and λ1 = 1 (since (1+β)2 −4β = 1+β2 −2β = (1−β)2). Therefore, for a small increase in φπ, λ1 will still equal be close to β, and thus will still be inside the unit circle (i.e., less than 1). However, when λ1 = λ2 = 8 of 12 ECON60111 φπ increases above 1, α will become negative. Therefore, λ2 will decrease slightly, and thus λ2 will be less than 1 (and thus inside the unit circle). To see this formally, let’s evaluate when λ2 < 1: 1+β+α+􏰟(1+β+α)2 −4β 2 􏰟(1 + β + α)2 − 4β < 1 − β − α (1 + β + α)2 − 4β < (1 − β − α)2 1 + β2 + α2 + 2β + 2α + 2βα − 4β < 1 + β2 + α2 − 2β − 2α + 2βα 4α < 0 Since α = κ(1 − φπ)/θ, then this condition will be satisfied when φπ > 1. Therefore, when φπ is slightly greater than 1, both eigenvalues will be inside the unit circle.
In this case, it will not be possible for there to be self-fulfilling equilibria, according to the Blanchard and Kahn conditions. The central bank threatens to respond, sufficiently strong, to an increase in expected inflation, so as to rule out self-fulfilling equilibria.
<1 SECTION B Answer TWO questions 1. Define the golden-rule level of capital in the Solow model. Define the modified golden- rule level of capital in the Ramsey model. Explain the difference between the two and how they relate to each other. (25 points) Key points: • The golden-rule level of capital is the one that maximizes consumption of households in the long-run. • The modified golden-rule level of capital is the long-run level of capital that maximizes households’ overall welfare. • The modified golden-rule level of capital is lower than the golden-rule level of capital. • The golden rule takes into account only the long run, while the modified golden rule takes into account the transition as well – i.e., that it is costly to accumulate capital, as households have to save and forgo consumption. 2. How are assets priced in the Consumption Capital Asset Pricing Model (Consumption 9 of 12 CAPM)? Derive the risk premium for assets in this model and give economic intuition. Use this model to explain the equity premium puzzle. (25 points) ECON60111 In the Consumption Capital Asset Pricing Model, the expected-return premium of an asset is proportional to the covariance of its return with consumption. Assume now that the individual can invest into many different risky assets, each with stochastic gross return Ri = 1 + ri . Suppose that the household also t+1 t+1 has access to a risk-free asset with gross return R ̄ = 1 + r ̄ . t+1 t+1 )􏰊 An Euler equation holds for each asset: Risky asset i: u′(C ) = 1 E 􏰉Ri u′(C (B-1) (B-2) t 1+ρt t+1 t+1 Risk-free asset: u′(Ct) = R ̄t+1 Et [u′(Ct+1)] 1+ρ Using E[XY ] = E[X]E[Y ] + Cov(X, Y ), combine (B-1) and (B-2) and re-write: E 􏰉Ri 􏰊E [u′(C t t+1 t t+1 E 􏰉Ri u′(C )􏰊=R ̄ E [u′(C )] t t+1 t+1 t+1 t t+1 )]+Cov 􏰀Ri ,u′(C )􏰁=R ̄ E [u′(C )] t t+1 t+1 t+1 t t+1 􏰉 􏰊 Cov 􏰀Ri ,u′(C )􏰁 ERi−R ̄=−tt+1 t+1 t t+1 t+1 Et [u′(Ct+1)] That is, the individual is not directly concerned with Var (Ri ), but rather how t t+1 world when marginal utility is higher (i.e., consumption is lower). One of the most important implications of this analysis of assets’ expected returns concerns the case where the risky asset is a broad portfolio of stocks. To see this assume that individuals have constant-relative-risk-aversion. The Euler equation for asset i is the gross return Ri covaries with marginal utility, U′(C t+1 ). It suggests the risk-premium will positive when the covariance of returns with marginal utility is negative. Investors will need to be compensated for taking on the risk of investing in this asset, because this asset’s returns are expected to be lower in states of the Let gc = C t+1 t+1 C−θ= 1 E􏰉􏰀1+ri 􏰁C−θ􏰊. t1+ρt t+1t+1 /C − 1. Then, dropping time subscripts, t E 􏰉􏰀1 + ri􏰁 (1 + gc)−θ􏰊 = 1 + ρ Taking a first-order Taylor approximation around r = g = 0, etc., we get that the difference between the expected returns on two assets, i and j, satisfies E􏰉ri􏰊−E􏰉rj􏰊=θCov􏰀ri −rj,gc􏰁. t+1 10 of 12 3. Define the concept of Ricardian equivalence. Show how Ricardian equivalence holds in the Ramsey-Cass-Koopmans model and give economic intuition. Discuss reasons Ricardian equivalence might not hold in practice. (25 points) ECON60111 It is difficult to reconcile observed returns on stocks and bonds with the above equation. For the United States during the period 1890–1979, the difference between the average return on the stock market and the return on short-term government debt—the equity premium—is about 6 percentage points. The coef- ficient of relative risk aversion needed to account for the equity premium is the solution is close to θ = 25. This is an extraordinary level of risk aversion (other evidence suggests that risk aversion is much lower than this.). This is known as equity-premium puzzle. Ricardian equivalence is the proposition that the timing of taxes does not have any effect on household consumption decisions. Given the pattern of government spending, a cut in taxes today does not affect consumption, because households internalize that taxes must increase in the future. In the Ramsey-Cass-Koopmans model, the household’s budget constraint is 􏰗 ∞ −R(t) 􏰗 ∞ −R(t) e C(t)dt ≤ K(0) + D(0) + e [W (t) − T (t)] dt 00 Here C(t) is consumption at t, W(t) is labor income, T(t) is taxes, K(0) and D(0) are the quantities of capital and government bonds at time 0, respectively. Re-write the household’s budget constraint as: 􏰗 ∞ −R(t) e 􏰗 ∞ −R(t) 􏰗 −R(t) e W(t)dt − C(t)dt ≤ K(0) + D(0) + 000 T(t)dt G(t)dt Using the government’s budget constraint, −R(t) 􏰗 ∞ −R(t) 􏰗 ∞ e C(t)dt ≤ K(0) + 􏰗 ∞ −R(t)G(t)dt e 􏰗 ∞ 0 T (t)dt = D(0) + we can re-write the household’s budget constraint as e 000 Therefore, given the path of government spending, taxes have no effect on the household’s budget constraint and thus have no effect on household consumption. Therefore, it is government purchases which affect consumption. One reason Ricardian equivalence may not hold is that there is turnover in the population. When new individuals are entering the economy, some of the future e t=0 −R(t) 􏰗 −R(t) W(t)dt − 11 of 12 ECON60111 tax burden associated with a government bond issue is borne by individuals who are not alive at the when the bond issued. However, intergenerational links can cause a series of individuals with finite lifetimes to behave as if they are a single household with an infinite horizon. Furthermore, lifetimes are long enough that perhaps Ricardian equivalence is a good approximation. Another reason Ricardian equivalence may not hold is liquidity constraints. When the government issues a bond to a household to be repaid by higher taxes on the household at a later date, it is in effect borrowing on the household’s behalf. If the household already had the option of borrowing at the same interest rate as the government, the policy has no effect on its opportunities, and thus no effect on consumption. However, if the household faces a higher interest rate for borrowing than the government, and that household would borrow at the government’s interest rate if it were possible, it will respond by raising its consumption. Overall, the Ricardian equivalence result rests on the permanent-income hypoth- esis, which fails in quantitatively important ways. 12 of 12 END OF EXAMINATION