<p>Student Number ECON212 Quiz 4</p><p>1. Identify the returns to scale (increasing, constant, or decreasing) for the following production functions. a) Q 50 K2 L 3 (1 point)</p><p>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</p><p>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.</p><p>Page Reference: 247</p><p> b) Q3 K  2 L (1 point)</p><p>Answer</p><p>Qa  a3 K  a 2 L</p><p>Qa  a3 K  2 L</p><p>Qa  aQ</p><p>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.</p><p>Page Reference: 247</p><p>1 1 c) Q K3  L 3 (1 point)</p><p>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</p><p>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.</p><p>Page Reference: 247</p><p>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)</p><p>Answer This production process can be characterized with a fixed proportions production function.</p><p>Q 5 min(K , L ) g 2</p><p>Page Reference: 278</p><p> 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)</p><p>Answer With the fixed proportions production function there is no tangency condition. Simply solve the production function for capital and labor by inspection.</p><p>Q 5min(K , L ) 2 100 5min(K ,L ) 2 20 min(K ,L ) 2</p><p>This implies</p><p>K  20 2 K  40</p><p> and Student Number ECON212 Quiz 4 L  20.</p><p>Page Reference: 278</p><p> 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)</p><p>Answer TC wL  rK TC 5(20)  8(40) TC  420</p><p>Page Reference: 278</p><p>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)</p><p>Answer In the long-run equilibrium, each firm will produce where P AC  MC . Thus,</p><p>10Q2 5 Q  20  30 Q 2  10 Q  20 20Q2  5 Q  0 20Q  5  0 Q  0.25</p><p>Page Reference: 382-383</p><p> b) What is the market equilibrium price? (1 point)</p><p>Answer Since each firm produces where P MC , price will be</p><p>P30 Q2  10 Q  20 P 30(0.25)2  10(0.25)  20 P 19.375</p><p>Page Reference: 382-383 Student Number ECON212 Quiz 4 c) What is total market demand? (1 point)</p><p>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</p><p>Page Reference: 382-383</p><p> d) What is the equilibrium number of firms in the long-run? (1 point)</p><p>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</p><p>250 N  0.25 N 1,000</p><p>Page Reference: 382-383</p>