Problem Set # 2 Dynamic Part - Math Camp Consumption with Labor Supply Consider the problem of a household that hao choose both consumption and labor supply. The household s problem is: V 0 max c t;l t X t u (c t ; l t ) t0 s:t: : A 0 given A t+ ( + r) (A t + w t l t + I t c t ) (and a no ponzi game condition) where l t ihe household s hours worked at t, w t ihe wage rate, I t iransfer income, and preferences are given by: B u (c t ; l t ) log c t l t where B > 0 and > are parameters. The household takes w and I as exogenous variables. You may also assume that ( + r). (a) De ne the state variables and the control variables. Set up the problem as a dynamic programing problem, write down the Bellman Equation. Derive the rst order conditions and the envelope condition. (b) Show that > insurehat the marginal disutility of work is increasing in hours worked. (c) Derive the following three conditions for optimal behavior by the household:
A static rst-order condition relating today s labor supply to today s wage and today s consumption An Euler Equation relating today s consumption and tomorrows consumption An Euler Equation relating today s labor supply and wages with tomorrow s labor supply and wage.. Solution (a) At ihe only endogenous state variable. Ct and Lt are the control variables. The Bellman Equation is: V t (A t ) max u (C t ; L t ) + V t+ (A t+ ) s:t:a t+ ( + r)(a t + w t L t + I t C t ) (b) disutility of labor is: DU (B)L t and the marginal disutility of labor is: MDU BL t, nally, the derivative of the marginal disutility of labor with respect to labor is dmdudl t ( )BL 2 t which is positive i >. (c) The FOC s with respect to the control variables C and L are: u C (C t ; L t ) ( + r)v 0 (A t+ ) u L (C t ; L t ) w t ( + r)v 0 (A t+ ) Combining these two FOC we have: u C (C t ; L t ) u L(C t ; L t ) w t Interpretation: the LHS ihe marginal bene t of working an extra hour today, assuming that the wage is spent on consumption; the RHS ihe marginal utility cost of working an extra hour today. At an optimum these must be equqlized. Plugging the functional form for utility we can rewrite thies Euler Equation as: L t wt BC t () 2
Envelope Condition: Since ( + r) : Using the functional form: Combining and 2: V 0 (A t ) ( + r)v 0 (A t+ ) V 0 (A t ) u C (C t ; L t ) V 0 (A t+ ) u C (C t+ ; L t+ ) u C (C t ; L t ) ( + r)u C (C t+ ; L t+ ) u C (C t ; L t ) u C (C t+ ; L t+ ) C t+ C t C t C t+ (2) L t+ wt L t w t+ Intuition: the LHS ihe marginal utility cost of earning a dollar today. The RHS ihe discounted marginal utility cost of earning a dollar in present discounted value tomorrow. At an optimum these should be equal. 2 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 maximizehe following discounted utility function: TX t [u (c t ) + d(l t )] t0 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: c t f(x t ; ) 3
where x r is expenditure on some group of goods and services and representime 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: a t+ ( + r)(a t + w t h t x t ) where a t is nancial wealth at the start of period t; wt is a wage rate, which householdake 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 a T + 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 time satis es l t h t (a) Write down the Bellman Equation for this problem. Derive the rst order conditions, the envelope condition and an Euler Equation. (b) Show that (x t ) is an increasing function of w t ; that is, as wages rise, people substitute expenditures for time in the production of consumption. Now assume that u(c) log c; that f(x t ; ) x t s t ; that d(l t ) l t and that ( + r). (c) Show that expenditure will constant over the life cycle. (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. 2. Solution (a) State Variable: a t Control Variables: x t ; and h t Bellman Equation: V t (a t ) max u (f (x t ; ))+d ( h t )+V t+ (( + r) (a t + w t h t x t )) 4
The FOC are: Envelope: x t : u 0 (c t ) ( + r) V 0 t+ (a t+ ) : f st u 0 (c t ) d 0 ( h t ) h t : d 0 ( h t ) ( + r) w t V 0 t+ (a t+ ) V 0 t (a t ) ( + r) V 0 t+ (a t+ ) u 0 (c t ) (b) Combining FOC with Envelope: u 0 (c t ) ( + r) Vt+ 0 (a t+ ) d0 ( h t ) f u 0 (c t ) w t w t f st w t Thus, if w t rises f must rise too. Given concavity assumptions on f(x; s) this implies xt must rise. Intuitively, if w t risehe opportunity cost of time spent in home production instead of working for wages rises. Therefore, it is optimal to substitute expenditures for time in the production of consumption. (c) Under the assumptions ( + r) and u(f(x; s)) log x t +( ) log, the Euler Equation becomes: x t x t+ Thus, expenditures are constant over time. (d) From (b) we had f w t! x t w t and under the assumptions f x t So, holding x t xed, higher w implies lower s. From FOC f st u 0 (c t ) d 0 ( h t )! ( ) l t, thus holding x xed, higher w rises s which reduces l t. Since l t h t this impliehat h t increases. 5