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ECON 411.3 (01): Monetary Theory T2: 2020/21 Assignment 2 due on February 25 th Question 1: Money in a Production Economy The evolution of the per-capita capital stock in an OLG production economy is given by kt+1 = σn f(kt ), where f(kt ) = kαt . Assume that n = 2, α = 13 and σ = 12 . The depreciation rate of capital is δ = 13 . a) Find the steady-state value k of the capital stock for the economy without money. b) What is the return R on capital? c) Is the economy dynamically efficient? Now the government introduces money. The gross rate of money supply growth z is chosen such v that the return on money in a stationary equilibrium is t+1 = nz = 2 34 . vt d) What will the return on capital, R, be after money has been introduced? e) What is the new steady-state capital stock? f) Given that the young cohort’s saving is still the same share σ = 12 of per-capita output as before, what are the real balances vM held by the young cohort in per-capita terms? N g) How much seignorage does the government earn (again, in per-capita terms of the young cohort)? h) Bonus (15%): We represented the non-monetary equilibrium of this economy graphically. Can you determine the monetary equilibrium graphically? Where do real balances appear in your figure? Question 2: Asset Pricing Consider the consumption problem of an individual that lives for two periods. She has two available means of saving. She can either buy a totally riskless bond for a price of one per unit that pays a gross real return R next period. Or she can buy tomato plants at a price of pT (relative to the consumption good) each. Tomato plants yield a risky (i.e. stochastic) return of Q in the following period. The individual’s expected utility is given by u(cy ) + βE [u(co )]. Her period budget constraints are cy = y y − aB − pT aT and co = y o + RaB + QaT , where aB and aT are, respectively, the number of bonds and tomato plants bought. a) Solve the saving problem. Show that the first-order conditions imply that u′ (cy ) = RβE [u′ (co )] and u′ (cy ) = p1T βE [Qu′ (co )]. Now assume that the period utility function is u(c) = ln c and that the gross return on the riskless bond is R = 1. In the second period, the weather can either be good or bad. Each outcome is equally likely. If the weather is good, consumption is high, co = 1 , and the return on tomato plants is good, Q = 1 . If the weather is bad, consumption is lower, co = 0.5 and the tomato plants carry no fruit, i.e. Q = 0 . E[Qu′ (co )] b) Use your results from part a) to show that pT = RE[u′ (co )] . c) Compute the expected values based on the information provided and show that pT = 13 . Q d) The expected return on tomato plants can be written as E[ pT ]. Calculate this value. What is the risk premium associated with growing tomatoes? -1-
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