08 Jun Question University of Toronto Mississauga
Question
University of Toronto Mississauga
Department of Economics
ECO 325H: Advanced Economic Theory − Macro
Prof. Kambourov
Problem Set 1
Tuesday, September 22, 2015
1. Which of these production functions generally display constant returns to scale in
labor and capital (i.e. if K and L double, does Y double)? For those that do not
display constant returns to scale for all parameter values, find the parameter values
that make them satisfy constant returns to scale.
(a) ln Y = A+α ln K + β ln L (Cobb-Douglas)
(b) Y = A min(K, L) (Leontieff)
(c) ln Y = A + B ln(α0 K ρ + α1 Lρ ) (CES)
(d) ln Y =
A + α0 ln K + α1 ln L
+ 2α4 ln K ln L + α5 (ln L)2 )
1
+ 2 (α3 (ln K)2
(Translog)
(e) Y = A + α0 K + α1 L (Linear)
2. Let F (K, L) = G(K, L) + H(K, L).
(a) Prove that F is homogeneous of degree one (i.e., constant returns to scale) if
both G and H are.
(b) Give a counter-example to show that the fact that F is homogeneous of degree
one does not always imply that G and H are homogeneous of degree one.
(c) Let H(K, L) = α0 L. Prove that F is homogeneous of degree one if and only if
G is also homogeneous of degree one.
3. Romer, Chapter 1, Problem 1.1. Use the fact that the growth rate of a variable
equals the time derivative of its log to show:
(a) The growth rate of the product of two variables equals the sum of their growth
˙
˙
˙
Z(t)
X(t)
rates. That is, if Z(t) = X(t)Y (t) then Z(t) = X(t) + Y (t) .
Y (t)
(b) The growth rate of the ratio of two variables equals the difference of their
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