# Elementary Mathematical Techniques For Economics

Elementary Mathematical Techniques For Economics

Instructor: Ram Sewak Dubey ECON 317: Problem Set 7 November 26, 2018

1. [Marginal Concepts]

Find the marginal product of different inputs or factors of production for each of the fol- lowing production functions.

(a) Q(x,y) = 6×2 +3xy+2y2.

(b) Q(K,L) = 0.5K2 −2KL+L2.

(c) Q(x,y) = 20+8x+3×2 −0.25×3 +5y+2y2 −0.5y3.

(d) Q(x,y,z) = x2 +2xy+3y2 +1.5yz+0.2z2.

2. [Multi-product Firm’s Profit Maximization – I]

Find (i) the profit maximizing output levels x and y and (ii) the maximum profit for a firm producing two goods x and y with the profit function

(a) π(x,y) = 32x− x2 +2xy−2y2 +16y−7.

(b) π(x,y) = 320x−6×2 −4xy−4y2 +240y−36.

(c) π(x,y) = 50x−2×2 −2xy−4y2 +60y−56.

3. [Multi-product Firm’s Profit Maximization – II]

A firm produces two different types A and B of a commodity. The cost of producing x units of A and y units if B is

C(x,y) = (

4 100

) x2 +

( 1

100

) xy+

( 1

100

) y2 +4x+2y+250.

Suppose the firm sells it’s output in a perfectly competitive market with a per unit price of $15 for type A and $9 for B. Find the profit maximizing output levels x and y and also the maximum profit.

Submission deadline: December 3, 2018 in class Page 1 of 3

Instructor: Ram Sewak Dubey ECON 317: Problem Set 7 November 26, 2018

4. [Profit Maximizing Input Demand -I]

A firm produces output q using two inputs capital K and labor L using the production function

q = 6K 1 3 L

1 2 .

Assume output price p = 0.5, rent r = 0.1 and wage w = 1. Find the profit maximizing input levels K and L and also the maximum profit.

5. [Multi-product Monopoly Profit Maximization-I]

A monopoly firm sells two different brands of a product for which the demand functions are

Q1 = 25−0.5P1, Q2 = 30−P2,

and the cost function for the firm is

C(Q1,Q2) = Q21 +2Q1Q2 +Q 2 2 +20.

Find the profit maximizing output levels Q1 and Q2 and also the maximum profit.

6. [Multi-product Monopoly Profit Maximization-II]

A monopoly firm sells two different brands of a product for which the demand functions are

Q1 = 49 1 3 − 2

3 P1,

Q2 = 36−0.5P2, and the cost function for the firm is

C(Q1,Q2) = Q21 +2Q1Q2 +Q 2 2 +120.

Find the profit maximizing output levels Q1 and Q2 and also the maximum profit.

7. [Profit Maximizing Input Demand-II]

Consider a competitive firm with the following profit function

Π = PQ−wL− rK,

where P = 3, w = 1 and r = 1 and

Q(K,L) = 9K 1 3 L

1 3 .

Submission deadline: December 3, 2018 in class Page 2 of 3

Instructor: Ram Sewak Dubey ECON 317: Problem Set 7 November 26, 2018

(a) Find the input quantities K∗ and L∗ that satisfy the first order conditions for the profit maximization.

(b) Compute the profit maximizing output Q∗.

(c) Compute the maximum profit Π∗. (d) If the wage rate w changes to w = 2, determine the new L∗∗.

(e) If the rental rate r changes to r = 2, determine the new L∗∗.

(f) If the output price P changes to P = 9, determine the new L∗∗.

Hint: All other parameters remain as in the original case in last three parts.

(g) [BONUS] Also compute the profit maximizing output Q∗∗ and the maximum profit in each case Π∗∗ in each of the last three parts, i.e., (d)-(f).

Submission deadline: December 3, 2018 in class Page 3 of 3

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