Christiano411, Winter 2005FINAL EXAMAnswer three of the following equally-weighted four questions. If a ques-tion seems ambiguous, state why, sharpen it up and answer the revised ques-tion. You have 1 hour and 55 minutes. Good luck!3. Answer for question 3.(a) i. Sequence of markets equilibrium. At each date, t, the house-hold maximizes discounted utility from then on:1Xj=tjtu(cj);subject to a sequence of budget constraints:cj+ ipj rjkpj+ wjn  Tj; j  t;where wjand rjare market prices beyond the control of thehousehold. The household uses its entire endowment of timefor labor eort, n; because it does not value leisure. The rmschoose ntand kp;tsuch that prots are maximized, whereprots are dened as follows:kg;tn(1)tkpt wtnt rtkt:A sequence of markets equilibrium is a set of prices and quan-tities, frt; wt; t  0g; fyt; ct; n; ipt; igt; t  0g and taxes, fTt; t 0g such that given taxes and prices, the quantities solve the householdproblem. given the prices, the quantities solve the rm problem. given the quantities and a value of s, the government bud-get constraint is satised.1 the resource constraint is satised.(b) the rst order condition for the household isuc;t= uc;t+1[rt+1+ 1  p];and the rm sets fkp;t+1= rt+1; where fkp;t+1is the marginalproduct of private capital. Combining these, and taking functionalforms into account:ct+1ct= [ nkg;t+1kp;t+1!(1)+ 1  p]:Let gcdenote the gross growth rate of consumption in a balancedgrowth path. Then,(gc)= [(ns)(1)+ 1  p]:Suppose gccorresponds to some given positive net growth rate,ie., gc> 1: Then,s =1n(1"gc+ p 1#)11:The number in square brackets is positive, so that s is well dened.Thus the Euler equation is consistent with constant consumptiongrowth in steady state. To fully answer the question, we needto establish (i) that the other equations - the household budgetequation and the resource constraint - are also satised with aconstant consumption growth rate and (ii) that the other quantityvariables display positive growth too. Let ggand gpdenote thegross growth rates of government and private capital, respectively.Then, the government's policy for choosing kg;timplies:gg= gp= g;say. Note that output can be writtenk(1)gtkptn(1)= kgt(kpt=kgt)n(1)= kgtsn(1):2Divide the resource constraint by kgt:ctkgt+ gt+1 (1  g) + gt+1 (1  p) = sn(1):So, in a constant growth steady state (i.e., gt+1= g; constant)the consumption to public capital ratio is a constant, equal to thefollowing:sn(1)+ (1  g) + (1  p) + 2g:But, the consumption to public capital ratio being constant im-plies:gc= g:The household budget constraint is trivially satised, since it isequivalent with the resource constraint given the rst order condi-tions of rms, linear homogeneity of the production function withrespect to rms' choice variables, and the government budget con-straint.(c) T he planner's problem is: choose ct; kg;t+1; kp;t+1; t  0 to maxi-mize discounted utility. After substituting out consumption usingthe resource constraint, the problem becomes:maxfkg;t+1;kp;t+1g1Xt=0tu[k(1)gtn(1)kpt+ (1  g)kg;t+ (1  p)kp;tkp;t+1 kg;t+1];subject to the object in square brackets (consumption) being non-negative at all dates, and to kg;t; kp;t 0: The planner's rst orderconditions are:uc;t= uc;t+1[fkp;t+1+ 1  p]uc;t= uc;t+1[fkg;t+1+ 1  g];for t = 0; 1; 2; :::: With the functional forms:ct+1ct= [kg;t+1 nkp;t+1!(1)+ 1  p]ct+1ct= [(kg;t+1)1n(1)(kp;t+1)+ 1  g]:3Substituting out consumption using the resource constraint, thesetwo equations represent a vector dierence equation in k; k0; k00,where k = [kgkp]0: There are many solutions to this equation thatare consistent with the given initial condition, k0= [kg;0kp;0]: Onecan construct the whole family of solutions by indexing them byk1: dierent values of k1give rise, by iterating on the euler equa-tion, to dierent sequences of capital. Not all are optimal. Onlythe one solution that also satises the transversality condition isoptimal. Thus, satisfying the Euler equation is not sucient foran optimum.(d) Setting  = 1   and equating the planner's two rst order con-ditions, we get:[ nkg;t+1kp;t+1!(1)+ 1  p]= [(1  )n(1) kp;t+1kg;t+1!+ 1  g];which requires thatkp;t+1kg;t+1be a particular constant for t = 0; 1; ::::Call this constant s : By setting s = s the government cannot dobetter, since this achieves the planner's optimum.4. suppose that the production function, f; is strictly concave and utilityis strictly concave and positive. Then, using the rst order and enve lopeconditions,"f0(k) 1#[k  g(k)]  0:Then,k > k ) f0(k) 1< 0 ) g(k)  kk < k ) f0(k) 1> 0 ) g(k)  k:Here, strict concavity of f has been used. The last weak inequalities arein fact strict because, by property (iv), k > 0; k 6= k implies g(k) 6= k:4Finally,k < k ) g(k) < k k > k ) g(k) > k by property