Production and labor supply.  Consider an economy in which there is a worker with preferences for leisure l and a consumption good c, represented by the utility function U (c, l) = u (c) + v (l).  The consumption good is produced by a competitive firm that uses labor n and has access to technology y = f (n), where y is the quantity produced of the consumption good. The firm is owned by the representative agent and hires labor for a wage w in a competitive labor market. The representative agent has N¯ hours available, which she allocates to labor and leisure activities such that l + n = N¯ .  The problem of the representative agent is to choose how much to consume c and how many hours to work ns  in order to maximize her utility, subject to her budget and time constraints, that is:

n(a)s(x)u (c)+v (N¯ _ ns)

s●t●   c ≤ wns+ π

where π are the profits made by the firm (owned by the agent) and all relative prices are expressed in terms of the consumption good (i.e. the price of c is 1). The problem of the firm is to choose how much labor to hire nf  in order to maximize its profits, taking the wage w as given:

max f ╱nf 、_ wnf

An equilibrium in this economy is allocations ,c, ns , nf 、and a wage w such that: (i) given w, the allocations c and ns solve the worker’s problem, (ii) given w, the allocation nf  solves the firm’s problem, (iii) the market for the consumption good clears c = y, and (iv) the market for labor clears ns = nf .

The first-order conditions of the Lagrangian associated with the worker’s problem are:

u\ ( )c _ λ = 0

_v\ (N¯ _ ns)+λ w = 0

where λ is the Lagrange multiplier for the budget constraint.  Rearranging the above equations and assuming that the budget constraint binds, we get an equation for ns:

v\ (N¯ _ ns) = u\ (wns+π)w                                                           (1)

as a function of the wage w and the firm’s profits π .

The first-order condition of the firm’s problem is:

f \ ╱nf 、_ w = 0

which implies that wages equal the marginal product of labor w = f \ ╱nf 、and profits are π = f ╱nf 、_ f \ ╱nf 、nf . Plugging these into equation (1) and using the market clearing condition for labor nf  = ns = ne we get an equation for the equilibrium labor supply ne:

v\ (N¯ _ ne) = u\ (f (ne)) f \ (ne)                                                   (2)

Assume that f (n) = An1   α_ , u (c) =1(c)_(1_)σ(σ) , and v (l) = ψ1(l)   Then, their derivatives are f\ (n) =

1.  (10 points) Show that equation (2) can be rewritten as:

ne = N¯ _ ┌  (ne)α+σ (1_α)┐ ε(1)

(3)