3, the firm is using only factor 2

QUESTIONS for written exam in microeconomics. Select the (unique) correct answer PRODUCTION THEORY -Varian chaps. 18 - 23

1. A competitive firm is using the factors 1 and 2, to produce output y. If the factor price ratio is (w1 / w2) > 3, the firm is using only factor 2; if (w1 / w2) < 3, the firm is using only factor 1;

finally, if (w1 / w2) = 3 it is indifferent which factor using in production. Indicate which of the following alternatives is consistent with cost minimization.

a) the production function is Cobb-Douglas.

b) the factors are perfect substitutes and MP1 = 3MP2 (MPi = marginal productivity of factor i) c) the factors are perfect substitutes and the MP1 = (1/3)MP2

d) factors 1 and 2 are perfect complements, in the proportion 3 to 1 e) none of the other answers is correct

2. A firm has production function y = f (x, z) = αx + βz, for y ≥ 0, where y is output, and x, z the factors of production. This means that returns to scale are:

a) constant b) decreasing c) increasing

d) initially increasing, then decreasing e) none of the other answers is correct

3. Consider the production function: f(x1,x2) = (x1ax2a), where a is a positive parameter. Indicate for which values of a the returns to scale in production are increasing:

a) only if a > 2 b) only if a > 1 c) only if a > 1/2

d) it is impossible to answer

e) none of the other answers given is correct

4. A firm has production function y = f (x, z) = (xz) / (x + z), for y ≥ 0, where y is output and x, z the factors of production. This means that returns to scale are:

a) constant b) decreasing c) increasing

d) initially increasing, then decreasing e) none of the other answers is correct

5. Given the production function: f(x1, x2) = (x1 αx2β), with α and β positive constants, indicate which values of α and β yield increasing returns to scale , together with decreasing marginal productivity of factors:

a) for any positive value of α and β b) α and β lie in the interval (0, ½) c) α and β lie in the interval (½, 1) d) α and β lie in the interval (1, 2)

e) none of the other answers is correct, because decreasing marginal productivity is never associated with increasing returns to scale.

6. A competitive firm has production function y = x^11/2x^21/4. Factor prices are [1, 1], respectively, and output price is p = 4. Determine the amount of y maximizing short run profit, when the quantity of factor 2 is fixed at x2 = 16.

7. A competitive firm has production function y = x^11/2x^21/4. Factors prices are [1, 1], respectively, output price is p = 4. Determine the quantity of y maximizing long run profit.

8. A competitive firm has production function: y = 2x1 + 3x². Factors prices are [1, 3], respectively.

What is the minimum total cost for producing y = 100?

9. A firm has production function y = x11/2x^21/2. Factors prices are w¹ = 4, w² = 2, respectively. The minimum total cost for producing y = 100 is:

10. A firm produces output y with the factors x1 e x2 , according to the production function y = min {2x1, x2}. Determine the minimum total cost for producing y = 100, when factor prices are

[1, 3].

11. Let C (y) be the total cost function of a firm. If C(y) = 144 + 16y². Determine the minimum average cost.

12. John has a workshop where he repairs cars (a). For all a ≥ 0 his total costs are:

c(a) = 5a² + 120a + 80. If he repairs 20 cars, his average variable costs will be:

13. The total cost function of a competitive firm is c(y) = 2 + (y²/3). At what market price the firm is producing 30 units of y?

14. In a competitive industry, a firm with marginal costs MC, average variable costs AVC, and average cost AC, chooses the short run output quantity such that:

a) p = MC and p > AC b) p = MC and p = AVC c) p = MC and p ≥ AVC d) p > MC and p > AC

e) none of the other answers given is correct

15. In a competitive industry in the long run, in the absence of incentives for entrance or exit of firms from the market, each firm with marginal costs MC (y) and average costs AC (y), produces the quantity y wherein:

a) p = MC and p > AC b) p = MC and p = AC c) p = MC and p < AC d) p > MC and p > AC

e) none of the other answers given is correct

16. In a competitive industry, there is a firm with cost function:

c(0) = 0; c(y) = 16 + 2y² for y > 0. In the long run, what is the minimum price at which the firm is prepared to produce a positive output?

17. In a competitive market, there are two firms. Due to the presence of a quasi-fixed factor they have long run cost functions c(0) = 0; c(y¹) = y¹² + 400, if y¹ > 0; c(y²) = y22 + 144, if y² > 0.

Determine the minimum price at which both firms are willing to stay in the market.

18. A good is produced by small firms with the same technology and cost function: c(0) = 0 and c(y) = 100 + y². In a long-run equilibrium of the industry, how many firms are producing a positive output, if the inverse market demand function for Y is p = 820 - 2Y?

19. A competitive firm has short-run cost function c^s(0) = F; c^s(y) = F + c_v^s(y), where F = 500 is fixed cost, and c_v^s(y) is variable cost. Knowing that c_v^s(y) = y² + 144, if y > 0, determine the minimum price at which the firm is willing to produce a positive short run output.

Questions for second partial examination: firm, oligopoly, monopoly, games

1b. A competitive firm has a constant returns to scale technology. This implies that its long-run average cost (AC) and marginal cost (MC) are:

a. constant b. increasing

c. first decreasing, then increasing d. none of the other answers Answer: a

2b. A monopolist, with cost function C(y) = 100 + (y²)/2, addresses the following inverse demand function p = 900 - y . Monopolist’s optimal output is:

Answer: 300

3b. A monopolist has inverse demand function p = 80 - q and cost function C (q) = 40. What is monopolist's total revenue R, if he produces the profit maximizing output?

Answer: 1600

4b. In a Cournot duopoly, firms 1 and 2, produce y1 and y2, respectively. They have identical total cost function C(yi) = 2 yi, i = 1, 2. The market demand function is y = 20 - p, where y = y₁ + y₂. The market price p in a Cournot equilibrium is:

Answer: 8

5a. Define and graphically represent the concept of ‘monopoly deadweight'

5b. At output y = 100, firm’s marginal cost is MC (y) < AC (y), where AC is average cost. This implies that at y = 100:

a) AC is constant in y b) AC is increasing in y c) AC is decreasing in y

d) AC can be increasing or decreasing in y, depending on the value of fixed cost e) none of the other answers is correct

Answer: c

6a. Discuss the circumstances affecting long-run profits in a perfectly competitive industry.

6b. A monopolist operates in a market with demand function: y = 10 - p. His cost function is C (y)

= 10, for y > 0. Producer’s net surplus, and monopoly deadweight loss are:

Answer: net surplus 15 deadweight loss 25/2

7b. The firms of a competitive industry operate with a constant returns to scale technology

described by the production function y = x1 ½ x2 ½. The long-run equilibrium price of output is p = 4, and market demand is y^D = 8000 – 2p. The output produced by each firm, and the number of firms in the industry are, respectively:

a. 10, 100

b. 100, 10

c. undetermined, undetermined d. None of the other answers

Answer: c

8b. A firm has technology described by production function q = f (x, y) = 2x + y. If the unit factor prices are px = 1 and py = 2, respectively, then the minimum cost C (100) for producing q = 100 is:

Answer: 50

9b. A firm produces output q with technology described by production function:

q = f(x, y) = x^1/2 y^1/2 . If the unit factor prices are p_x = 1 e p_y = 4, the minimum cost C (20) to produce q = 20 is:

Answer: 80

10b. In a Cournot oligopoly, 9 identical firms produce the same output y_i , i = 1, …, 9. They have identical total cost function C (yi) = 2. Market demand function is y = 400 - p, where y = ∑ yi . The market price p in a Cournot equilibrium is:

Answer: 40

11b. In a Stackelberg duopoly, firms 1 (leader) and 2 (follower) take decisions on quantities and produce, respectively, y1 e y2. They have identical total cost function C(yi) = 2 yi, i = 1, 2. The market demand function is y = 400 - p, where y = y₁ + y₂. Quantity y in a Stackelberg equilibrium is:

Answer: 300

12b. An industry has inverse demand function p = 100 - 0.2 y. At current factor prices the only existing technology generates total cost function C (0) = 0, C (y) = 40 + 2y , for y > 0. In a partial equilibrium, the industry output is lower if the industry is:

a. a collusive duopoly b. a Cournot duopoly

c. a Stackelberg duopoly with quantity leadership d. a Bertrand duopoly

e. none of the other answers

Answer : a

13a. Define the concept of ‘reaction curve’ in a Cournot duopoly, in the light of game theory

14a. Provide the example of a simultaneous game in strategic form, with two players and two strategies, reproducing a 'prisoner's dilemma'. What are the distinguishing features of this game?

14a. Provide the example of a simultaneous game in strategic form, with two players and two strategies, such that a Nash equilibrium in pure strategies fails to exist. Is there, in this case, an equilibrium in mixed strategies?

14b. Provide the example of a simultaneous game in strategic form, with two players and two strategies, with two Pareto rankable Nash equilibria. What can prevent selection of the efficient equilibrium?

15a. Provide the example of a sequential game in extensive form, with two players and two strategies, with two Nash equilibria, but only one 'perfect subgame equilibrium'.

Hint: non-credible threat of a monopolist towards a potential entrant

monopolistic competition (extended solution)

In a 'monopolistic'-competition market, the inverse demand function p (y) for the differentiated output y of any single firm in the industry is p = (A / n) - b · y, where n is the number of firms in the market. The cost function of each firm is: C (y) = Q + y². Determine the number n of firms in a long-run equilibrium.

Solution:

The number n and the quantity y in long-run equilibrium are such that p (y) = AC (y), where AC is the average cost, and the average cost curve AC (y) is tangent to the inverse demand curve p(y) at output y.

AC = (Q / y) + y (1)

AC = (Q / y) + y = (A / n) – b · y = p (2)

(Q / y) + (b + 1) y = (A / n) (3)

Tangency condition: ∂AC / ∂y = ∂p / ∂y

− (Q / y²) + 1 = − b (4)

(Q / y²) = b + 1

y² = Q / (b + 1) that is y = Q / (b + 1)y (5)

y = Q^1/2 / (b + 1)^1/2 (6)

use (5) to obtain:

(b + 1) y = Q / y

Substituting Q / y in (3) we obtain:

2(b + 1)·y = (A / n)

2(b + 1)· Q^1/2 / (b + 1)^1/2 = (A / n) 2(b + 1)^1/2 · Q^1/2 = (A / n)

n = A / [2(b + 1)^1/2 · Q^1/2]