Student Number

Student Number ECON212 Quiz 4

1. Identify the returns to scale (increasing, constant, or decreasing) for the following production functions. a) Q 50 K2 L 3 (1 point)

Answer 2 3 Qa  50 aK  aL 2 2 3 3 Qa  50 a K a L 5 2 3 Qa  a50 K L  5 Qa  a Q

Since an increase in all inputs by a factor of a produced a more than proportional increase in output a 5  a, the production function exhibits increasing returns to scale.

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b) Q3 K  2 L (1 point)

Answer

Qa  a3 K  a 2 L

Qa  a3 K  2 L

Qa  aQ

Since an increase in all inputs by a factor of a produced an exactly proportional increase in output a  a, the production function exhibits constant returns to scale.

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1 1 c) Q K3  L 3 (1 point)

Answer 1 1 3 3 Qa  aK   aL 1 1 1 1 3 3 3 3 Qa  a K  a L 1 1 1 Q a3 K 3  L 3 a   1 3 Qa  a Q Student Number ECON212 Quiz 4

Since an increase in all inputs by a factor of a produced a less than 1 proportional increase in output a 3  a , the production function   exhibits decreasing returns to scale.

Page Reference: 247

2. Consider a production process where capital and labor are perfect complements – two units of capital are required for each unit of labor to produce five units of output. a) Derive the production function for this production process. (1 points)

Answer This production process can be characterized with a fixed proportions production function.

Q 5 min(K , L ) g 2

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b) If the wage rate is $5 per unit of labor and the rental rate of capital is $8 per unit of capital, how much capital and labor should the firm employ to minimize the cost of producing 100 units? (1 points)

Answer With the fixed proportions production function there is no tangency condition. Simply solve the production function for capital and labor by inspection.

Q 5min(K , L ) 2 100 5min(K ,L ) 2 20 min(K ,L ) 2

This implies

K  20 2 K  40

and Student Number ECON212 Quiz 4 L  20.

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c) What will the total cost be to produce the 100 units using the quantities of capital and labor determined in part b)? (1 point)

Answer TC wL  rK TC 5(20)  8(40) TC  420

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3. In a certain market in the long-run, each firm and potential entrant has a long-run average cost curve AC10 Q2  5 Q  20 and long-run marginal cost curve MC30 Q2  10 Q  20 where Q is thousands of units per year. Market demand is given by D( P ) 39,000  2,000 P . a) In equilibrium, how many units will each firm produce? (1 point)

Answer In the long-run equilibrium, each firm will produce where P AC  MC . Thus,

10Q2 5 Q  20  30 Q 2  10 Q  20 20Q2  5 Q  0 20Q  5  0 Q  0.25

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b) What is the market equilibrium price? (1 point)

Answer Since each firm produces where P MC , price will be

P30 Q2  10 Q  20 P 30(0.25)2  10(0.25)  20 P 19.375

Page Reference: 382-383 Student Number ECON212 Quiz 4 c) What is total market demand? (1 point)

Answer To find total market demand, plug the market equilibrium price into the market demand curve. D( P ) 39,000  2,000 P D( P ) 39,000  2,000(19.375) D( P ) 250

d) What is the equilibrium number of firms in the long-run? (1 point)

Answer Since total market demand is 250 and each firm is produce 0.25 units, the total number of firms in the market in equilibrium will be

250 N  0.25 N 1,000