Question

Example: Consider a person with this utility function: U = C – N^2.5 Subject to: C = w(1-t)N , With the wage w and the tax rate t exogenous.

a. Find the optimal labor input (N) solely as a function of exogenous parameters.

b. What’s the elasticity of labor supply for this person? Answer with a precise number.

c. If the tax rate is 50% and the wage is 10, exactly how many units of labor will this person supply? Answer with a precise number.

Answer 1

A).

Consider the given problem here the maximization problem is given by.

=> maximize “U=C-N^2.5” subject to “C=W(1-t)*N”.

So, here the slope of the budget line is given by, “dC/dN = W(1-t)”. Now, the MUc=1” and “MUn = (-2.5)*N^1.5”, => the MRS=MUn/MUc = (-2.5)*N^1.5/1”.

=> MRS = (-2.5)*N^1.5, => the absolute slope of the utility function and the budget line must be same at the equilibrium.

=> (2.5)*N^1.5 = W(1-t), => N^1.5 = W(1-t)/2.5, => N = [W(1-t)/2.5]^2/3.

SO, here the optimum labor supply is given by, “N*= [W(1-t)/2.5]^2/3”.

B).

So, here the labor supply is given by.

=> N = [W(1-t)/2.5]^2/3, => dN/dW = (2/3)*[W(1-t)/2.5]^(2/3-1)*[(1-t)/2.5].

=> the elasticity is given by, => e = (dN/dW)*(W/N).

=> e = (2/3)*[W(1-t)/2.5]^(2/3-1)*[(1-t)/2.5]*(W/N).

=> e = (2/3)*[W(1-t)/2.5]^(2/3-1)*[W(1-t)/2.5]*(1/[W(1-t)/2.5]^2/3).

=> e = (2/3)*[W(1-t)/2.5]^(-1)*[W(1-t)/2.5] = (2/3), => e = 2/3 = constant.

C).

Now, if “t=50%” and “W=10”, then the labor supply is given by.

=> N = [W(1-t)/2.5]^2/3, => N = [10*(1-0.5)/2.5]^2/3 = [2]^2/3 = 1.58 hour.

=> N = 1.58 hours.