Question

Consider an individual with utility function c^αl^1−α, where c is consumption, l is leisure, and α ∈ (0, 1). The individual is endowed with R units of nonlabor income and T units of time. The individual earns wage w for each unit of time worked. The price of a unit of consumption is p.
(a) What is the budget constraint for this individual?
(b) What is the price of leisure?
(c) Set up the appropriate Lagrangian for this agent’s problem.
(d) Find the first order conditions.
(e) Solve the demand functions for l and c in terms of exogenously given parameters.
(f) What did this procedure assume about the solution to our problem?
(g) Find the indirect utility function.
(h) Find the utility when the agent chooses not to work at all.
(i) Using the last two parts, set up an equation that describes the reservation wage.
(j) Using the slope of the indifference curve, find the reservation wage.
(k) Perform comparative statics on the reservation wage for T, α, and R
(l) sketch the budget set and the indifference curves.

Answer 1

U(c,l) = c^al^1-a

His total consumption c is constrained by total income i.e. both labor and nonlabor income.

Total labour = T-l. Hence total labor income = w(T-l)

Non Labor income = R

Hence the budget constraint can be written as:

p*c = (T-l)w + R

p*c = (wT + R) - wl

p*c + w*l = wT + R

b)

One hour of leisure implies one hour of labor foregone and thus one hour of real wage income w/p. Hence the price of leisure is w/p.

c)

Max U(c,l) = c^al^1-a

subject to:

p*c + w*l = wT + R

Setting up Lagrange

L = c^al^1-a + (wT + R - p*c - w*l)

d)

First order conditions can be found by differentiating L with respect to c, l and

e)

Setting the First order conditions equal to zero we get,

Dividing both we get

((1-a)*c)/al = w/p

c = (w*a*l)/p(1-a) ............1

Put this in third lagrangian equation we get,

wT + R - w*a*l/(1-a) - w*l = 0

l = ((1-a)(wT+R))/w

Put this value of l in equation 1 we get

c = a(wT+R)