Suppose we are given the constant returns-to-scale CES production

Q1. Suppose we are given the constant returns-to-scale CES production function q = [k + l]1/ where k
represents capital and l represents labor
a. Show that MPk = (q/k)1 and MPl = (q/l)1 .
b. Show that RTS = (k/l)1 ; use this to show that elasticity of substitution between labor and capital
= 1/(1 – ).
c. Determine the output elasticities for k and l; and show that their sum equals 1.
Note: Output elasticity measures the response of change in q to a change in any input.
Elasticity of output wrt k is eq,k = %q/%k = (q/k)*(k/q) or (q/k)*(k/q) or lnq/lnk
Similarly for elasticity of output wrt l, eq,l
d. Prove that q/l = (q/l) and hence that ln(q/l) = ln(q/l)

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Q2. Suppose the production of airframes is characterized by a CobbDouglas production function: Q =
LK. The marginal products for this production function are MPL = K and MPK = L. Suppose the price of
labor is $10 per unit and the price of capital is $1 per unit. Find the cost-minimizing combination of labor and capital if the manufacturer wants to produce 121,000 airframes.