Intermediate Mathematical Economics
Solutions to Assignment 3
Exercise / Problems6.2 / 2
7.1 / 1, 4
7.2 / 3a, 3b, 7a, 7d, 10
7.3 / 1, 2, 3a
7.4 / 1, 2, 5
Ex 6.2, #2
a.
b.
c.
Ex 7.1, #1
(a) dy/dx = 12x11 (b) dy/dx = 0 (c) dy/dx = 35x4
(d) dw/du = −3u-2 (e) dw/du = −2u-1/2 (f) dw/du = u-3/4
Ex 7.1, #4
Ex 7.2, #3
(a) 3(27x2 + 6x − 2)
(b) 54x2 + 78x − 70
Ex 7.2, #7
a.
d.
Ex 7.2, #10
(a)
MC = dC/dQ = 6Q + 7
AC = C/Q = 3Q+7+ 12/Q
(b)
MR = dR/dQ = 10 − 2Q
AR = R/Q = 10 − Q
(c)
MP = dQ/dL = a+ 2bL − cL2
AP = Q/L = a + bL − cL2
Ex 7.3, #1
dy/dx = (dy/du)(du/dx) = (3u2 + 2)(−2x) = −2x[3(5 − x2)2 + 2]
Ex 7.3, #2
dw/dx = (dw/dy)(dy/dx) = 2ay(2bx + c) = 2ax(2b2 x2 + 3bcx + c2 )
Ex 7.3, #3a
Let w = 3x2 − 13; this implies that dw/dx = 6x.
Since y = w3, we have
dy/dx = (dy/dw)(dw/dx) = 3w2 (6x) = 18x(3x2 − 13)2
Ex 7.4, #1
(a) ∂y/∂x1 = 6x12 − 22x1 x2 ∂y/∂x2 = −11x12+ 6x2
(b) ∂y/∂x1 = 7+6x22 ∂y/∂x2 = 12x1 x2 − 27x22
(c) ∂y/∂x1 = 2(x2 − 2) ∂y/∂x2 = 2x1 + 3
(d) ∂y/∂x1 = 5/(x2 − 2) ∂y/∂x2 = −(5x1 + 3)/(x2 − 2)2
Ex 7.4, #2
(a) fx = 2x+ 5y fy = 5x − 3y2
(b) fx = 3x2 − 4x − 3y fy = −3(x − 2)
(c) fx = 5y/(x + y)2 fy = −5x/(x + y)2
(d) fx = (x2 + 1)/x2 y fy = −(x2 − 1)/xy2
Ex 7.4, #5
(a) U1 = 2(x1 + 2)(x2 + 3)3 U2 = 3(x1 + 2)2(x2 + 3)2
(b) U1(3, 3) = 2160
In addition to the above, answer the following questions:
Question 1
a. Consider the production function . Find and and interpret their meanings.
b. Consider the demand function . Find and . What is the sign of the partial derivatives?
Question 2
Consider the following supply and demand model:
Where Px is the price of a related good;
a. Solve the equilibrium price, P*
b. Find the following comparative static results:.
c. Assume the following:
Determine the sign of the comparative static results found in part b.
Question 3
Consider the national income model (7.17) of the textbook. Equation 7.18 of the book shows the equilibrium solution is given by
Find the following comparative static results and sign the partial derivatives: