Question

how do you find the demand functions for leisure and consumption and the labor supply function when all that is given is utility function and price of a good?

Answer 1

We suppose the Labor Supply is denoted by L, Consumption is denoted by C and leisure is denoted by l. The wage rate is given as W.

Now, an worker has a total time endowment of 24 hours a day and he has to decide how much time to spend on leisure(l) and how much time to work(L). Hence we can write,

L+l=24........(1)

Now, the price of a good is given as P(say). Now, if the worker consumes C units of that good, then total expense is P.C.

Now, he will buy this good from his labor income only i.e. W.L.

Hence, His budget constraint will be when

Expenditure=Labor Income

or, P.C=W.L

or, P.C=W.(24-l) {From equation 1)

or, P.C+W.l = W.24..........BL

This is the budget line between Consumption and Leisure.

Nowz according to the question, the utility function of the worker is also given. Now, the worker will gain utility by two things. One is leisure and another is consumption. Hence, the utility function will look like U(C,l).

Hence, we have in our hand,

The Utiltiy of the Worker: U(C,l)

Budget Constraint: P.C+W.l=24.W

Now, the worker will maximize his utility subject to the budget constraint i.e.

Max U(C,l) subject to P.C+W.l=24.W

We can also set the Lagrange's Equation.

J = U(C,l)+a(24W - P.C - W.l), where a>0.

The First Order Conditions

dJ/dC= Uc(C,l)-a.P=0........(2)

dJ/dl=Ul(C,l)-a.W=0...........(3)

Here, solving from the First Order Conditions, we get the optimal value of Leisure i.e. l*(P,W) and Optimal Consumption i.e. C*(P,W)

Also, from equation (1) we get

L*=24-l* i.e. the labor supply function L*(P,W).

Hence, in this way we will get the labor supply funcrion L*(W,P), laisure l*(W,P) and Consumption C*(W,P).

Hope the explanation is clear to you my friend.