Dynamic Problem Set 2 Solutions Jonathan Kreamer July 22, 2011

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Consumption with Labor Supply

Consider the problem of a household that has to choose both consumption and labor supply. The household’s problem is: V0 = max ct ,lt

∞ X

β t u(ct , lt )

t=0

s.t. : A0 given At+1 = (1 + r)(At + wt lt + It − ct ) plus a no-ponzi game condition, where lt is the household’s hours worked at t, wt is the wage rate, It is transfer income, and preferences are given by:   B γ l u(ct , lt ) = log ct − γ t where B > 0 and γ > 1 are parameters. The household takes w and I as exogenous variables. You may also assume that β(1 + r) = 1. (a) Define the state variables and the control variables. Set up the problem as a dynamic programming problem, write down the Bellman Equation. Derive the first order conditions and the envelope condition. State variables are At . Control variables are ct , lt , and At+1 . The Bellman equation is v(At ) = max {u(ct , lt ) + βv ((1 + r)(At + wt lt + It − ct ))} ct ,lt

This setup uses substitution to incorporate the constraint. Alternatively, you can put a Lagrangian in the Bellman Equation. First-order conditions are uc (t) = β(1 + r)v 0 (At+1 )

(1)

−ul (t) = βwt (1 + r)v 0 (At+1 )

(2)

v 0 (At ) = β(1 + r)v 0 (At+1 )

(3)

The Envelope condition is

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(b) Show that γ > 1 insures that the marginal disutility of work is increasing in hours worked.   γ−1 γ The utility function is u(ct , lt ) = log ct − B , meaning γ lt . The marginal utility of work is ul (t) = −Blt that the marginal disutility of work is −ul (t) = Bltγ−1 . To determine the change in marginal utility we take the second derivative, which is ull (t) = − (γ − 1) Bltγ−2 , or if we want the rate of change in marginal disutility, −ull (t) = (γ − 1) Bltγ−2 . If we have γ − 1 > 0, or γ > 1, then we have −ull > 0, meaning that the marginal disutility of labor is increasing with labor. (c) Derive the standard three conditions: labor-leisure, consumption Euler equation, and labor Euler equation. The labor-leisure condition derives from (1) and (2). Together these yields −

ul (t) = wt uc (t)

In other words, the ratio of the marginal utilities of the two goods is equal to their relative price. Now . Of observe that from (1) and (3), we have v 0 (At ) = uc (t) and from (2) and (3) we have v 0 (At ) = − uwl (t) t course, these also imply v 0 (At ) = uc (t) and v 0 (At ) = − uwl (t) respectively. Substituting these expressions back t into (3), we get the Euler equations uc (t) = β(1 + r)uc (t + 1)  ul (t) = β(1 + r)

wt wt+1

 ul (t + 1)

It’s important to note that these three equations are not only standard and should become second nature, but they have a very intuitive interpretation. In any maximization problem with an interior solution, we equalize the marginal values of all uses of each resource at our disposal. Here we have one budget constraint, and we must allocate our endowment of time plus investment earnings between leisure and consumption. The relative price of leisure and consumption in each period is wt . p1 1 The basic equation of economic optimization is Up11 = Up22 , or more commonly U U2 = p2 . Here the relative price of leisure in terms of consumption is w: we have to give up w units of consumption to buy one unit of leisure. Therefore we have − uucl (t) (t) = wt . We can also substitute between consumption/leisure today and consumption/leisure tomorrow. The price of a unit of consumption today is 1+r units of consumption tomor(t) row, so the standard equation becomes βuucc(t+1) = 1 + r, which is just a rearrangement of the consumption Euler equation. Finally, the price of an hour of leisure today is ul (t) βul (t+1)

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=

wt (1+r) wt+1 ,

wt (1+r) wt+1

units of leisure tomorrow, so we have

which is just the leisure Euler equation.

Consumption Versus Expenditure over the life cycle

Suppose there is an agent who lives for T periods in an environment with no uncertainty, and who maximizes the following discounted utility function T X

β t [u(ct ) + d(lt )]

t=0

where c is consumption and l is leisure. Assume that both d and u are increasing and concave. Assume that consumption is related to expenditure and time inputs as follows ct = f (xt , st )

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where xt is expenditure on some group of goods and services and st represents time spent in home production. Assume that f is increasing and concave in both of its inputs. Assume that the household can borrow or lend at a risk-free rate r. The household’s dynamic budget constraint is: at+1 = (1 + r)(at + wt ht − xt ) where at is financial wealth at the start of period t; wt is a wage rate, which each households take as given; and ht is hours spent working, which households choose freely each period. The wage will in general be time varying; for instance, wages might rise early in the life cycle then fall as individuals age. However, there is no uncertainty, so assume that at time zero households know the entire path of wt from periods 0 to T . Assume that households have to satisfy the solvency constraint aT +1 = 0. Households have an endowment of one unit of time each period, which they spend on leisure, working for wages, and in home production. Thus leisure satisfies lt = 1 − ht − st (a) Write down the Bellman Equation for this problem. Derive the first order conditions, the envelope condition and an Euler Equation. The Bellman equation is vt (at ) = max u (f (xt , st )) + d (1 − ht − st ) + βvt+1 ((1 + r)(at + wt ht − xt )) xt ,st ,ht

First-order conditions are 0 u0 (ct )fx (t) = β(1 + r)vt+1 (at+1 )

(4)

u0 (ct )fs (t) = d0 (lt )

(5)

0 β(1 + r)wt vt+1 (at+1 ) = d0 (lt )

(6)

The Envelope condition is 0 vt0 (at ) = β(1 + r)vt+1 (at+1 )

Substituting (6) into (4) yields u0 (ct )fx (t) =

d0 (lt ) wt

Combined with (5), we get fx (t) =

fs (t) wt

From (4) and the envelope condition, we get v 0 (at ) = u0 (ct )fx (t). Substituting this (plus the version for at+1 ) into the envelope condition yields the Euler equation u0 (ct )fx (t) = β(1 + r)u0 (ct+1 )fx (t + 1) Once again, we can check that these equations make sense, and simultaneously gain intuition for them, by putting them in the familiar form. We have the substitution between time and expenditures on consumption

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d0 (lt ) 0 u (ct )fx (t)

= wt

Our second first-order condition above says that we should be indifferent between spending our time on home production, and working and then purchasing goods for the production of consumption. That is, the marginal productivity from devoting time to home production should equal the wage times the marginal productivity from devoting goods to home production wt fx (t) = fs (t) (b) Show that xstt is an increasing function of wt ; that is, as wages rise, people substitute expenditures from time in the production of consumption. Intuitively, an increase in wt will increase the agent’s wealth. This wealth effect should lead to an increase in consumption and leisure (since both are presumably normal goods). An increase in leisure means a decrease in ht + st , while an increase in consumption means an increase in f (xt , st ). Since wt is higher, there is also a substitution effect towards ht . The net effect will likely be an increase in xt relative to st . Mathematically, consider the equation wt = ffxs (t) (t) . Therefore an increase in wt must lead to an increase in ffxs (t) (t) . We have that f (xt , st ) is increasing and concave in each of its inputs. This is likely to mean an increase in xstt .   1−γ 1 and that β(1 + r) = Now assume that u(c) = log c; that f (xt , st ) = xθt s1−θ ; that d(l ) = t t 1−γ lt 1. (c) Show that expenditure will be constant over the life cycle. Take the Euler equation u0 (ct )fx (t) = β(1 + r)u0 (ct+1 )fx (t + 1). Observe that fx (t) = θ uc (t) =

1 ct .



st xt

1−θ

and

Substituting this into the Euler equation, along with β(1 + r) = 1, we obtain 1 ct



st xt

1−θ =

1



ct+1

st+1 xt+1

1−θ



st+1 xt+1

Now we substitute in ct = xθt s1−θ , which yields t 1 θ xt s1−θ t



st xt

1−θ

1 = θ 1−θ xt+1 st+1

1−θ

Simplifying this, we obtain xt = xt+1 (d) Show that, conditional on this constant level of expenditure X, hours worked are an increasing function of wages over the life cycle, while leisure time, hours spent in home production and consumption are decreasing functions of wages. Our equations are wt u0 (ct )fx (t) = d0 (lt ) wt fx (t) = fs (t)

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at+1 = (1 + r)at + wt ht − xt − ct Substituting functional forms into the first two and replacing xt = X, we obtain wt θ θ X s1−θ t  wt θ

X st



X st

θ−1

= lt−γ

θ−1

 = (1 − θ)

 wt θ

xt st

X st

θ

θ−1 = fs (t)

at+1 = (1 + r)at + wt ht − X − ct Simplifying the second equation, we get  st =

1−θ θ



X wt

which implies that st is decreasing in wt . Substituting this into the first equation and simplifying, we obtain  1 X γ lt = wt θ which shows that leisure is decreasing in wt . Since both lt and st are decreasing in wt , we have that ht = 1 − lt − st is increasing in wt . In particular, we have  ht = 1 −

X wt θ

 γ1

 −

1−θ θ



X wt

Finally, consumption is increasing in s, and since X is constant and s is decreasing in w, it follows that consumption is decreasing in w.

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