ECN 505 – Mathematical Economics
Problem Set 4
Spring 2022
Áron Tóbiás
Department of Economics
Syracuse University
Instructions
This problem set is due Wednesday, April 13, at 3:45 P.M. in class. Submissions handed
in late or by email will not be accepted.
Please submit a legible paper copy of your write-up and staple multiple pages together.
Please indicate your name at the top of the first page of your submission.
You are allowed to work together with your classmates, but each of you is expected
to submit a complete and independent write-up.
Please justify all your answers. Merely stating the final result without any explanation is not sufficient for full credit, even if your answer is correct.
Problem 1
Consider the following function:
f (x1 , x2 , x3 ) = x21 x42 x53 .
(a) Compute the gradient vector of the function f .
(b) Compute the Hessian matrix of the function f .
Problem 2
Suppose that the growth of the national economic output of a country Y depends on
how much capital K is available in the economy and also on the progress of technological innovation g according to the following aggregate production function:
Y(g, K) = 0.5gK.
Suppose furthermore that the evolution of technological growth and that of the aggregate capital stock over time t are given as
√
g(t) = 1 + t,
K(t) = 10e0.02t .
The chain rule tells us that the total derivative dY /dt of national output with respect to
time is:
dY
∂Y dg ∂Y dK
=
+
.
dt
∂g dt
∂K dt
1
(?)
ECN 505 – Mathematical Economics
Problem Set 4
Spring 2022
Áron Tóbiás
Department of Economics
Syracuse University
(a) Provide an economic interpretation of the formula (?).
(b) Use the chain-rule formula (?) to compute the total derivative dY /dt of national
output with respect to time given the functional forms Y(g, K), g(t), and K(t) specified above. Make sure to express the formula you compute for dY /dt solely as a
function of t.
Problem 3
Consider the following production function:
h
i− 1 / 2
2
−2
y = f (x1 , x2 ) = 0.4x−
+
0.6x
.
1
2
(a) Use the implicit-function theorem to find the marginal rate of technical substitution
dx2 /dx1 corresponding to this production function.
(b) Show that the isoquants corresponding to this production function are convex.
[Hint: Differentiate the formula you have computed for dx2 /dx1 to find an expression for the second-order derivative d2 x2 /dx21 .]
2