MAE-Seminar and Homework 2

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Mathematics Applied to Economics
2nd Seminar and Homework
Partial Derivatives. Applications to Economics
(1) Find f ′x(1, 2) =
∂f
∂x
(1, 2) and f ′y(1, 2) =
∂f
∂y
(1, 2) if f(x, y) = 2x4 + 3x2y + 4y3.
(2) Find f ′x(1, 1, 0) =
∂f
∂x
(1, 1, 0), f ′y(1, 1, 0) =
∂f
∂y
(1, 1, 0) and f ′z(1, 1, 0) =
∂f
∂z
(1, 1, 0) if f(x, y, z) =
2x2y − xy2z3 + 2x4z − y2 + 4.
(3) Compute all the first order partial derivatives of the following functions:
(a) f(x, y, z) = 2x2 + 3x4y + y2 − 2y2z + x2z2 + z3
(b) f(x, y, z) =
x3 + y3
xy
(c) f(x, y) =
√
x2 + y2
(4) Compute all the first and second order partial derivatives of the following functions:
(a) f(x, y) = y lnx
(b) f(x, y, z) = exy
2z3
(c) f(x, y) = 3x3 − 3x2y2
(d) f(x, y, z) = xey + y2ez
(5) Compute the indicated partial derivatives:
(a) f(x, y) = exy;
∂2f
∂x2
=?
(b) f(p, q) = ln
p + q
pq
;
∂2f
∂q∂p
=?
(6) The production for a certain country is described by the function Q(L,K) = 4L1/4K3/4.
(a) Compute Q′L =
∂Q
∂L
, Q′K =
∂Q
∂K
.
(b) What is the marginal productivity of labor (capital) when L = 625 and K = 10000? In this
case should the government encourage capital or labor investment in order to increase the
total output of that country?
(c) Write down the 1st order approximation of function Q around the point (625, 10000)
(d) Use the the previous formula to estimate Q(624, 10002).
(e) Show that the output will be double if both the capital investment and the size of the labor
force are doubled.
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(7) The production for a certain country is described by the function Q(L,K) = 324L1/3K2/3.
(a) Compute Q′L =
∂Q
∂L
, Q′K =
∂Q
∂K
.
(b) What is the marginal productivity of labor (capital) when L = 65 and K = 727? In this
case should the government encourage capital or labor investment in order to increase the
total output of that country?
(c) Write down the 1st order approximation of function Q around the point (625, 1000)
(d) Use the the previous formula to estimate Q(65, 727).
(e) Show that the output will be triple if both the capital investment and the size of the labor
force are tripled.
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