Substitue the equilibrium conditions and the results from the firm’s problem into the household’s optimality conditions

Due: Tuesday, April 4, 2017 (in class)1. (60 marks total) In this question, we modify our two-period endowment economy model toinclude a consumption-leisure choice (as we had in LN1). In particular, we now assume thathousehold utility is given byu (c1 ) + v (l1 ) + ? [u (c2 ) + v (l2 )] ,where lt is household leisure in period t. Here, u has the same properties as described inLN3 (i.e., strictly increasing, strictly concave, and limc?0 u0 (c) = ?). The function v hereis new, and gives the household’s utility of leisure in each period. We will assume v hassimilar properties to u: v 0 (l) > 0 when l < 1, v 00 (l) < 0, and liml?0 v 0 (l) = ?. We will alsomake the additional assumption that v 0 (1) = 0. As in LN1, we assume that the householdhas a total endowment of one unit of time, so that 0 ? lt ? 1, with labour supply equal toNts = 1 ? lt . On the production side, we will assume that output is produced according the¯ (i.e., we have f (n, k) = n + k), where k¯ is thesimple production function yt = zt (nt + k)exogenous level of capital in each period. Notice that f here is not (strictly) concave, and inparticular fnn = fkk = fnk = 0. This will not turn out to be too big of a problem in this case.In all other respects, the model is the same as the one we encountered in LN3, including thefact that households save s from the first period to the second at interest rate r. As usual,we will let ? ? 1/(1 + r).(a) (5 marks) Unlike in LN3, household income is now endogenous, and as a result wecan no longer treat yt as pure endowment income. Further, the distinction betweenlabour income and dividend income will now matter (as it did in LN1). In view of this,write down the household’s budget constraints for period one and period two, and thencombine them into a single lifetime budget constraint (LBC). Interpret the LBC in yourown words.(b) (5 marks) Using the assumption that v 0 (1) = 0, argue that the NNC on labour (i.e., theconstraint that lt ? 1) will never bind, and can therefore be ignored. That is, argue thatno household would ever choose lt = 1.(c) (10 marks) Set up the Lagrangian for the household using either the sequence-of-budgetconstraints approach or the single LBC approach (it’s up to you). Obtain all the requiredFOC’s, and combine them to eliminate all Lagrange multipliers. Regardless of yourmethod, you should be able to express the result as two “static” optimality conditions(one for each t = 1, 2) relating ct to lt , and one “intertemporal” condition relating c1 toc2 . Interpret these three conditions in your own words.1 Econ 4021B – Winter 2017Dana Galizia, Carleton University (d) (10 marks) Write down the firm’s labour-demand problem for period t and solve it (theproblem is the same in both periods, so you don’t have to do this twice). The conditionyou get should be a bit unusual, in that it won’t depend on nt . Explain what wouldhappen to the quantity of labour the firm demands if this condition weren’t satisfied,and argue that this can’t happen in an equilibrium (HINT: you will have to use part (b)in your argument). Draw the firm’s labour demand curve. Finally, assuming that thelabour-demand condition you got holds, what is the firm’s period-t profit (i.e., ?t ) equalto?(e) (6 marks) Noting that, as in LN3, it is not possible in this economy for goods to bestored from period one to period two, write down the equilibrium conditions for thegoods market and the labour market in period t (they are the same in both periods, soyou don’t have to do this twice), and determine what this implies for the equilibriumlevel of savings, s.(f) (6 marks) Substitue the equilibrium conditions and the results from the firm’s probleminto the household’s optimality conditions (i.e., into the two static and one intertemporalconditions from part (c)) to eliminate ct , nt , wt , and ?t , t = 1, 2, leaving a system ofthree equations in three endogenous variables: l1 , l2 , and r (or, if you prefer, ? insteadof r).(g) (6 marks) Suppose z1 = z2 = z¯ for some z¯. Using your answer from (f), argue that, inequilibrium, we must have c1 = c2 and l1 = l2 , and solve for the equilibrium interest ratein this case.(h) (12 marks) Suppose z2 increases by a small amount (with no change in z1 ). For each ofl1 , l2 , c1 , c2 , and r, determine whether it will increase, decrease, remain unchanged, orwhether the direction of the change is ambiguous. Interpret your results.2. (40 marks total) In the two-period model with investment of LN6, we found that, in responseto an anticipated increase in productivity, c1 and i always moved in opposite directions (the“comovement problem”). The intuition for this was that, since z2 has no effect on y1 in equi¯ which doesn’t depend on z1 ), and since c1 + i = y1 ,librium (in equilibrium, y1 = z1 f (1, k),the only way for i to increase is if c1 decreases, and vice versa. In this question, we modifyour model to allow y1 to potentially increase in response to an increase in z2 , and see whetherthis “fixes” the comovement problem.Specifically, we modify our model of LN6 by introducing variable capital utilization. In particular, let µt denote the fraction of capital that is actually used by the firm in period t, sothat yt = zt f (nt , µt kt ). We also assume that the amount of depreciation from period 1 to 2is increasing in the amount of capital used in period 1. In particular, the law of motion forcapital is now k2 = [1 ? ?(µ1 )]k1 + i, where the function ?(µ) is strictly increasing (? 0 (µ) > 0)2 Econ 4021B – Winter 2017Dana Galizia, Carleton University and strictly convex (? 00 (µ) > 0). Capital utilization µt will be an additional choice variablefor the firm, with the restriction that this choice must satisfy 0 ? µt ? 1. In all other respectsthe model will be the same as in LN6 (including that households inelastically supply one unitof labour each period).(a) (4 marks). Write down the goods market and labour market equilibrium conditions foreach period. Use the labour market equilibrium conditions to eliminate n1 and n2 in thegoods market conditions, leaving two equations (one for each period).(b) (6 marks) Totally differentiate the expression you got in part (a) for period 1 with respectto z2 . Based on the result, is it possible for c1 and i to move in the same direction?Compare your answer to the case we encountered in LN6, and explain (in words) anysimilarities or differences.(c) (4 marks) As in LN6, the share price in period 2 is given simply by p2 = d2 , and themanager disburses all profits as dividends, i.e., d2 = ?2 . Write down the manager’sobjective function for period 2 (i.e., taking k2 as given). Without taking any first-orderconditions, argue that the manager will always set µ2 = 1.(d) (12 marks) As in LN6, the share price in period 1 is given by p1 = d1 + ?d2 , andd1 + i = ?1 . Assuming µ2 = 1, and that the manager wishes to maximize p1 , write downhis maximization problem (i.e., the objective function and any constraints) in terms ofn1 , n2 , i, k2 , and µ1 . Set up the Lagrangian for the firm, and obtain the FOC’s (assumethat the inequality constraints on µ1 never bind). Substitute the Lagrange multiplier outof these five FOC’s to obtain four optimality conditions for the firm(e) (8 marks) As in LN6, the household’s optimality condition is ?u0 (c2 )/u0 (c1 ) = 1/(1 + r).Combine this with the law of motion for capital, the equilibrium conditions from part (a),and the firm’s optimality conditions from (b) and (c) to get a system of three equationsin the three endogenous variables c1 , k2 , and µ1 . (HINT: two of these conditions willlook similar to equations (32) and (33) from LN6.)(f) (6 marks) Using your results from part (e), argue that, in equilibrium, µ1 will not changein response to a change in z2 (i.e., that dµ1 /dz2 = 0). Has the comovement problembeen “solved” by adding variable capital utilization to the model? 3

 
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