# 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 to

include a consumption-leisure choice (as we had in LN1). In particular, we now assume that

household utility is given by

u (c1 ) + v (l1 ) + ? [u (c2 ) + v (l2 )] ,

where lt is household leisure in period t. Here, u has the same properties as described in

LN3 (i.e., strictly increasing, strictly concave, and limc?0 u0 (c) = ?). The function v here

is new, and gives the household’s utility of leisure in each period. We will assume v has

similar properties to u: v 0 (l) > 0 when l < 1, v 00 (l) < 0, and liml?0 v 0 (l) = ?. We will also

make the additional assumption that v 0 (1) = 0. As in LN1, we assume that the household

has a total endowment of one unit of time, so that 0 ? lt ? 1, with labour supply equal to

Nts = 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 the

simple production function yt = zt (nt + k)

exogenous level of capital in each period. Notice that f here is not (strictly) concave, and in

particular 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 the

fact 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 we

can no longer treat yt as pure endowment income. Further, the distinction between

labour 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 then

combine them into a single lifetime budget constraint (LBC). Interpret the LBC in your

own words.

(b) (5 marks) Using the assumption that v 0 (1) = 0, argue that the NNC on labour (i.e., the

constraint that lt ? 1) will never bind, and can therefore be ignored. That is, argue that

no 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 required

FOC’s, and combine them to eliminate all Lagrange multipliers. Regardless of your

method, 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 to

c2 . Interpret these three conditions in your own words.

1 Econ 4021B – Winter 2017

Dana Galizia, Carleton University (d) (10 marks) Write down the firm’s labour-demand problem for period t and solve it (the

problem is the same in both periods, so you don’t have to do this twice). The condition

you get should be a bit unusual, in that it won’t depend on nt . Explain what would

happen 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 the

labour-demand condition you got holds, what is the firm’s period-t profit (i.e., ?t ) equal

to?

(e) (6 marks) Noting that, as in LN3, it is not possible in this economy for goods to be

stored from period one to period two, write down the equilibrium conditions for the

goods market and the labour market in period t (they are the same in both periods, so

you don’t have to do this twice), and determine what this implies for the equilibrium

level of savings, s.

(f) (6 marks) Substitue the equilibrium conditions and the results from the firm’s problem

into the household’s optimality conditions (i.e., into the two static and one intertemporal

conditions from part (c)) to eliminate ct , nt , wt , and ?t , t = 1, 2, leaving a system of

three equations in three endogenous variables: l1 , l2 , and r (or, if you prefer, ? instead

of r).

(g) (6 marks) Suppose z1 = z2 = z¯ for some z¯. Using your answer from (f), argue that, in

equilibrium, we must have c1 = c2 and l1 = l2 , and solve for the equilibrium interest rate

in this case.

(h) (12 marks) Suppose z2 increases by a small amount (with no change in z1 ). For each of

l1 , l2 , c1 , c2 , and r, determine whether it will increase, decrease, remain unchanged, or

whether 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 response

to 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 modify

our model to allow y1 to potentially increase in response to an increase in z2 , and see whether

this “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, so

that yt = zt f (nt , µt kt ). We also assume that the amount of depreciation from period 1 to 2

is increasing in the amount of capital used in period 1. In particular, the law of motion for

capital is now k2 = [1 ? ?(µ1 )]k1 + i, where the function ?(µ) is strictly increasing (? 0 (µ) > 0)

2 Econ 4021B – Winter 2017

Dana Galizia, Carleton University and strictly convex (? 00 (µ) > 0). Capital utilization µt will be an additional choice variable

for the firm, with the restriction that this choice must satisfy 0 ? µt ? 1. In all other respects

the model will be the same as in LN6 (including that households inelastically supply one unit

of labour each period).

(a) (4 marks). Write down the goods market and labour market equilibrium conditions for

each period. Use the labour market equilibrium conditions to eliminate n1 and n2 in the

goods 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 respect

to 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) any

similarities or differences.

(c) (4 marks) As in LN6, the share price in period 2 is given simply by p2 = d2 , and the

manager disburses all profits as dividends, i.e., d2 = ?2 . Write down the manager’s

objective function for period 2 (i.e., taking k2 as given). Without taking any first-order

conditions, 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 , and

d1 + i = ?1 . Assuming µ2 = 1, and that the manager wishes to maximize p1 , write down

his maximization problem (i.e., the objective function and any constraints) in terms of

n1 , n2 , i, k2 , and µ1 . Set up the Lagrangian for the firm, and obtain the FOC’s (assume

that the inequality constraints on µ1 never bind). Substitute the Lagrange multiplier out

of 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 equations

in the three endogenous variables c1 , k2 , and µ1 . (HINT: two of these conditions will

look similar to equations (32) and (33) from LN6.)

(f) (6 marks) Using your results from part (e), argue that, in equilibrium, µ1 will not change

in response to a change in z2 (i.e., that dµ1 /dz2 = 0). Has the comovement problem

been “solved” by adding variable capital utilization to the model? 3

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Originally posted 2019-06-04 14:31:03.