Econ 510a (second half)Yale UniversityFall 2006Prof. Tony SmithHOMEWORK #5This homework assignment is due at 5PM on Friday, December 8 in Marnix Amand’s mailbox.1. (a) In the Mehra-Prescott model that we discussed in lecture on November 30, forwhat value of γ (the degree of risk aversion) does the model generate an average(annual) equity premium of 6%? (Set β = 0.99 and fix the other parameters atthe values that we used in lecture.) For this degree of risk aversion, what is theaverage (annual) risk-free rate? Is it close to the historical average of 1%?(b) In the Mehra-Prescott model, set γ = 1 (log utility) and β = 0.99, but supposethat the high-growth state lasts longer (on average) than the low-growth state:set φ11= 0.6 (but keep φ22= 0.43). Recalibrate the values of µ and δ so thatthe average (annual) growth rate of per capita consumption is 0.018 and thestandard deviation of the (annual) growth rate of per capita consumption is 0.036.Calculate the average equity premium and the average risk-free rate for this newparametrization.(c) In the Mehra-Prescott model with γ = 1, β = 0.99, and the parameters of theprocess for consumption growth set equal to their values from class (i.e., φ11=φ22= 0.43, µ = 0.018, and δ = 0.036), find the average rate of return on a two-period risk-free bond, i.e., a sure claim to one unit of the consumption good twoperiods from now.Let the average (net) rate of return on the two-period bond be given by r2.Similarly, let r1be the average (net) rate of return on a one-period risk-free b ond.The gross two-period return 1 + r2can be decomposed into the product of twosuccessive (gross) one-period returns 1 + ˜r2, where ˜r2= (1 + r2)1/2− 1. The termstructure of interest rates is upward-sloping if ˜r2> r1; otherwise, it is downward-sloping. Which way does the term structure slope if the Mehra-Prescott model iscalibrated as in part (a)?Solution1. Using the model and notations covered in class, the expected returns are:rf= π 1prL− 1 +1prH− 1!re= π φHH (1 + peH)λHpeH− 1!+ φLH (1 + peH)λHpeL− 1!!+(1 − π) φHL (1 + peL)λLpeH− 1!+ φLL (1 + peL)λLpeL− 1!!which, after further simplification, and using the expressions covered in class for theprices of risk free bonds and equity iin each state:−β(φLHλ1−γH+ φLLλ1−γL) = βφLHλ1−γHpeH+ (βφLLλ1−γL− 1)peL−β(φHLλ1−γL+ φHHλ1−γH) = (βφHHλ1−γH− 1)peH+ βφHLλ1−γLpeLThis, with Matlab, deliversγ = 17.304rf= 0.1468Intuition: γ is much to high, and so is the risk free rate (14.7 % !). Obviously, thequantitative predictions of the Mehra-Prescott model are way off target.2. First, we have to compute the new probability transition matrix:P = 0.6 0.570.4 0.43!which gives you the new unconditional probabilities (i.e. the invariant distribution):π = 0.590.41!And we have to recalibrate λ1and λ2such that:1 + 0.018 = π1λ1+ π2λ20.0362= π1λ21+ π2λ22− (π1λ1+ π2λ2)2Which delivers:λ1= 1.061λ2= 0.975which gives us:rf= 0.0270re− rf= 0.0013Intuition: Again, the quantitative results are way off target. The risk free rate isridiculously low (0.13%). The risk free rate is alittle too high.3. By definition, with i = H, L, we have:pf2,i= β2E ct+2ct−γ!which, after some straightforward algebra, gives:pf2,i= βXj=L,Hλ−γjφi,jprjthus:rf2= πL1pf2,L− 1+ πH1pf2,H− 1Matlab tells us:rf2= 0.055Using the calibration from section (a), Matlab tells us:¯r2= 0.16752. Consider the planning problem for a neoclassical growth model with logarithmic utility,full depreciation of the capital stock in one period, and a production function of theform y = zkα, where z is a random shock to productivity. The shock z is observedbefore making the current-period savings decision. Assume that the capital stock cantake on only two values: i.e., k is restricted to the set {¯k1,¯k2}. In addition, assume thatz takes on values in the set {¯z1, ¯z2} and that z follows a Markov chain w ith transitionprobabilities pij= P (z0= ¯zj|z = ¯zi).(a) Let ¯z1= 0.9, ¯z2= 1.1, p11= 0.95, and p22= 0.9. Find the invariant distributionassociated with the Markov chain for z. Use the invariant distribution to computethe long-run (or unconditional) expected value of z; that is, compute E(z) =π1¯z1+ π2¯z2, where π1and π2determine the invariant distribution.(b) Let β = 0.9, α = 0.36,¯k1= 0.95kss, and¯k2= 1.05kss, where kssis the steady-statecapital s tock in a version of this model without shocks and with no restrictions oncapital (i.e., kss= (αβ)1(1−α)). Let g(k, z) denote the planner’s optimal decisionrule. Prove that g(ki, zj) = kjfor all i and j.(c) The decision rule from part (b) and the law of motion for z jointly determinean invariant distribution over (k, z)-pairs. Find this distribution. (That is, findprobabilities πij= P (k = ki, z = zj) that “reproduce” themselves: if πijis theunconditional probability that the economy is in state (ki, zj) today, then it is alsothe unconditional probability that the economy is in this state tomorrow. For amore complete discussion of this concept, see pp. 78 and 79 in the lecture notesby Per Krusell.) Use your answer to compute the long-run (or unconditional)expected values of the capital stock and of output.(d) In Matlab, use the optimal decision rule, the law of motion for z, and a randomnumber generator to create a simulated time series {kt, yt}Tt=0, given an initial con-dition (k0, z0). Compute T−1PTt=1ktand T−1PTt=1ytfor a suitably large value ofT and confirm that these sample means are close to the corresponding populationmeans that you computed in part (c). (You may find useful the Matlab code byLjunqvist and Sargent for simulating a Markov chain that I have posted on thecourse web site.)Solution (a) Given the transition matrixP = 0.95 0.050.1 0.9!(1)we can calculate the stationary distribution π according to the formulaπ0= π0P⇒(π1= 0.95π1+ 0.1π2π2= 0.05π1+ 0.9π2The solution to this equation isπ1= 2π2(2)Imposing the condition that π1+ π2= 1, the solution is(π1=23π2=13(3)Correspondingly, the long run expected value isE(z) = π0z =23× 0.9 +13× 1.1 =2930(4)(b) Form the dynamic programming problem asvki, zi= maxk0∈{k1,k2}lnzikαi− k0+ βpi1vk0, z1+ pi2vk0, z2(5)Since (ki, zi) takes on only 4 values, we assume that the policy function takes thefollowing form.g(k1, z1) = k1g(k2, z1) = k1g(k1, z2) = k2g(k2, z2) =