1. Suppose we are given the constant returns-to-scale CES production function q = [ k ρ + l ρ]1/ρ. a. Show that MPk = ( q / k )1–ρ and MPl = ( q / l )1–ρ. b. Show that RTS = ( k / l )1–ρ; use this to...

1. Suppose we are given the constant returns-to-scale
CES
production function
q
= [k
ρ +
l
ρ]1/ρ.

a. Show that
MPk
= (q/k)1–ρ and
MPl
= (q/l)1–ρ.

b. Show that
RTS
= (k
/l)1–ρ; use this to show that σ = 1/(1 – ρ).

c. Determine the output elasticities for
k
and
l; and show that their sum equals 1.

d. Prove that
q l
= aq l
B σ and hence that ln a
q l
b = σ lnq
l
b .
Note:
The latter equality is useful in empirical work because we may approximate ∂q
/∂l
by the competitively determined wage rate. Hence
s
can be estimated from a regression of ln(q/l) on ln
w.