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ECOM009 Macroeconomics
YOU ARE NOT PERMITTED TO START READING THE CONTENTS OF THIS QUESTION PAPER UNTIL INSTRUCTED TO DO SO BY AN INVIGILATOR. Answer TWO question from Section A and TWO questions from Section B. CALCULATORS ARE PERMITTED IN THIS EXAMINATION. PLEASE STATE ON YOUR ANSWER BOOK THE NAME AND TYPE OF MA- CHINE USED. COMPLETE ALL ROUGH WORKINGS IN THE ANSWER BOOK AND CROSS THROUGH ANY WORK THAT IS NOT TO BE ASSESSED. IMPORTANT NOTE: THE ACADEMIC REGULATIONS STATE THAT POSSESSION OF UNAUTHORISED MATERIAL AT ANY TIME WHEN A STUDENT IS UNDER EXAMINATION CONDITIONS IS AN ASSESS- MENT OFFENCE AND CAN LEAD TO EXPULSION FROM QMUL. PLEASE CHECK NOW TO ENSURE YOU DO NOT HAVE ANY NOTES, MOBILE PHONES OR UNAUTHORISED ELECTRONIC DEVICES ON YOUR PERSON. IF YOU HAVE ANY THEN PLEASE RAISE YOUR HANDANDGIVE THEMTOAN INVIGILATOR IMMEDIATELY. PLEASE BE AWARE THAT IF YOU ARE FOUND TO HAVE HIDDEN UNAUTHO- RISED MATERIAL ELSEWHERE, INCLUDING TOILETS AND CLOAK- ROOMS IT WILL BE TREATED AS BEING FOUND IN YOUR POS- SESSION. UNAUTHORISED MATERIAL FOUND ON YOUR MOBILE PHONE OR OTHER ELECTRONIC DEVICE WILL BE CONSIDERED THE SAME AS BEING IN POSSESSION OF PAPER NOTES. MOBILE PHONES CAUSING A DISRUPTION IS ALSO AN ASSESSMENT OF- FENCE. EXAM PAPERS CANNOT BE REMOVED FROM THE EXAM ROOM.
1 Section A 1. [30 marks] Assume consumers have infinite lifetimes and maximize the utility function Ut = Et ∞∑ s=t −βs (c¯− ct+s) 2 2 , where ct+s is consumption at time t+ s, β > 0 is the subjective discount factor and Et is the expectation operator conditional on all information available to consumers at time t. The coefficient c¯ is a positive parameter large enough for the marginal utility of consumption to be always positive over the relevant range. Consumers can freely borrow and lend at a constant riskless rate r such that β (1 + r) = 1. Labour income yt follows the stochastic process yt = yt−1 + λyt−2 + t, with λ > 0 and εt a white-noise process distributed normally with zero mean. (a) This is standard and results in the Euler equation ct = Et(ct+1). (b) Guess the linear function ct = α0 + α1at + α2yt + α3yt−1. Replacing in the Euler equation and taking expectations yields α1at + α2yt + α3yt−1 = α1at+1 + α2(yt + λyt−1) + α3yt, which can be rearranged as at+1 − at = α−11 [−α3yt + (α3 − α2λ)yt−1] (1) Replacing for ct in the dynamic budget identity yields at+1 − at = rat + yt − (α0 + α1at + α2yt + α3yt−1). (2) Equating coefficients on the same variables on the right hand sides of both (1) and (16) one obtains the consumption function ct = rat + r r(1 + r)− λ [(1 + r)yt + λyt−1]. (3) The consumption response to an innovation is ct − ct−1 = ct − Et−1ct = r(1 + r) r(1 + r)− λεt. As λ > 0, income is more persistent than a random walk and consumption responds more than one-to-one to an income innovation. Extra points for students who mention Deaton’s paradox. 2 (c) The saving function is st = − λ r(1 + r)− λ [yt + ryt−1]. Again, saving responds negatively to an income innovation as income is more persistent than a random walk. 2. The question is rather standard. Only part c) requires some extra thought. (a) max It,Kt V0 = ∫ ∞ 0 [ Kαt − It − I2t 2 ] e−rtdt (4) s.t. K0 given (5) K˙t = It − δKt (6) The necessary conditions for an optimum are ∂L0 ∂It = − [1 + It] + qt = 0 (7) ∂L0 ∂Kt = αKα−1t − δqt + q˙t − rqt = 0 (8) lim t→∞ qtKte −rt = 0 (9) This can be reduced to a system of two differential equations K˙t = (qt − 1) (10) q˙t = (r + δ)qt − αKα−1t (11) (b) Saddle path convergence to steady state from the right. (c) The K˙ locus is expected to shift up in the (Kt, qt) space at time t1. The steady state capital stock after t1 is going to be higher. At time t0 the economy jumps onto the unique path which crosses the stable arm of the new saddle path at time t1. qt jumps up at t0, keeps rising until t1 and falls from t1 onwards.The capital stock increases from t0 until the new steady state is reached.. 3. (a) The individual optimization problem is max c1t ,c2t+1 log c1t + 1 1 + ρ log c2t+1 (12) s.t. a1t = Wt − τ − c1t (13) c2t+1 = τ(1 + n) + a1t (1 + rt+1) . (14) where we have already imposed solvency. The Euler equation can be written as c2t+1 = 1 + rt+1 1 + ρ c1t. (15) 3 Replacing in the IBC and solving for c1t yields c1t = 1 + ρ 2 + ρ [ Wt − τ + τ(1 + n) 1 + rt+1 ] . (16) The associated individual saving function is s1t = Wt − τ − c1t = 1 2 + ρ ( Wt − τ − τ(1 + n)(1 + ρ) 1 + rt+1 ) (17) (b) Given the Cobb-Douglas technology it is Wt = (1− α) k˜αt , with k˜t = Kt/(Lt), and rt+1 = αk˜ α−1 t+1 − δ. Hence individual saving is a function of the current stock of capital s1t = 1 2 + ρ ( (1− α) k˜αt − τ − τ(1 + n)(1 + ρ) 1 + αk˜α−1t+1 − δ ) . (18) The stock of capital at t+ 1 equals the total saving of the young at time t or Kt+1 = Lt 1 2 + ρ ( (1− α) k˜αt − τ − τ(1 + n)(1 + ρ) 1 + αk˜α−1t+1 − δ ) (19) or in per capita terms k˜t+1 = 1 (2 + ρ)(1 + n) ( (1− α) k˜αt − τ − τ(1 + n)(1 + ρ) 1 + αk˜α−1t+1 − δ ) . (20) Rearranging yields, k˜t+1 + τ(1 + n)(1 + ρ) 1 + αk˜α−1t+1 − δ = 1 (2 + ρ)(1 + n) ( (1− α) k˜αt − τ ) . (21) Equation (21) implies that, for any level of k˜t, kt+1 is lower and, the function has a lower slope, than if τ = 0. Dynamics is standard. (c) In steady state it has to be k˜t+1 = k˜t. If τ > 0 the solution to (20) is a lower steady state level of k˜∗ for the stable steady state equilibrium B (see Figure 1). A pay as you go social security system depresses capital formation. Intuition: private saving by the young generation falls and the government uses the contribution not to invest in physical capital but to pay the pensions of the currently old. Hence aggregate saving, and investment, fall at given kt. If the marginal product of capital at the decentralized equilibrium with τ = 0 is below the Golden Rule marginal product of capital αk˜α−1GR = δ + n, the economy is dynamically inefficient. There exist τ > 0 which yields a Pareto improvement relative to the decentralized equilibrium with τ = 0. 4 kt kt+1 B B′ A′ A k∗k∗ ′ Figure 1: Steady state with pay-as-you-go pension system. Section B 4. Usual stuff, driven by concavity of preferences and uneveness of income over time or states of nature. 5. Marginal cost of adjustment is non-zero in the convex adjustment cost theory. The two coincide in steady state with zero depreciation rate as there is no adjustment and therefore the marginal cost of adjustment is zero. 6. The theory predicts that risk premia are driven by the covariance between the future marginal utility of consumption and asset returns. Empirically the theory, in its incarnation taught to students, fares pretty badly. Students should discuss the equity premium and risk-free rate puzzles. 7. Standard aggregation in the life cycle model.