Assume that preferencse are given by u(c,L) = 2*ln(c) +ln(L). The wage rate is $15 and the person has 120 hours of time available. Profits are 0. Taxes are 0. Write down two equations that describe the constraints faced by this consumer. Find an equation for the Marginal Rate of Substitution. Find the equilibrium level of consumption and leisure.
For convenience, let c= $15 of consumption. (Dont worry, you would get the same answer if c=$1 of consumption)
So, your equations are:
Max(U)=2*ln(c)+ln(L)
subject to:
c+L=120
To get one L,the person must give up one c. So the price of L in terms of C is 1. Soooooo, we want the marginal rate of substitution in utility to be equal to ratio of prices of L and c
So, next take the partial derivitives MU(c) and MU(L). These are 2/c and 1/L.
The marginal rate of substitution is MU(L)/MU(c) = (1/L)/(2/c) = c/2L
To maximize c/2L = 1 or c=2L, subject to c+L=120.
c=80, L=40