Tutorial 6 - Perfect Competition

March 2014

Problem 1

In a small, but perfectly competitive market for pineapples, there are 8 identical growers. Each grower has the following cost function:

C = 2 + 2q2 where q is thousands of pounds of pineapples produced. a) Write the firm’s supply curve, Si(qi) A firm in a perfectly competitive market supplies a positive quantity when the price is greater than the minimum average variable cost. If the market price is greater than the minimum average variable cost, the firm supplies the quantity that makes the marginal cost equal to the market price, in order to maximize profits (equilibrium condition).

VC 2q2 AV C = = = 2q q q Thus, the minimum average variable cost is 0. For any positive price, the firm is supplying a positive quantity. To find the supply curve: p = MC dT C MC = = 4q dq p = 4q Solving for q, we find the firm’s supply curve: 1 qS = p i 4 b) Write the equation for the market supply curve S(p). To find the market supply curve, sum horizontally the individual firms’ supply curves.

1 As firms are identical, we can multiply the individual firm’s supply curve by the number of firms in the market.

1 Q = n × qS = 8 × p = 2p S i 4 c) Suppose the (inverse) market demand curve is

D1 : p(QD) = 100 − 9.5QD

Solve for the equilibrium price and quantity.

Demand : p = 100 − 9.5QD

Supply : QS = 2p

In equilibrium, QS = QD:

p = 100 − 9.5 × 2p

p = 100 − 19p ⇒ 20p = 100 p = 5

To ﬁnd the quantity exchanged in this market in equilibrium, substitute the equilibrium price in either the supply or the demand curve.

Q = 2 × 5 = 10 Q 10 5 Each firm produces q = = = i n 8 4 d) In the long-run, will more growers enter the market or will existing growers exit? Explain. In the long run, firms can enter or exit the market, thus profits are zero. To check whether this equilibrium is a long run one, we have to check profits.

Π = TR − TC 5 25 TR = p × q = 5 × = i 4 4 52 25 TC = 2 + 2q2 = 2 + 2 = 2 + i 4 8 25 25 9 Π = − 2 − = > 0 4 8 8 Proﬁts are positive, thus the equilibrium is not a long-run one. In the long run more ﬁrms will enter the market.

2 Alternative solution. Another strategy is to compare the price with the average total cost of production.

2 4 5 8 5 41 AT C = + 2q = 2 × + 2 = + = < p q 5 4 5 2 10 The equilibrium price is higher than the average total cost. Thus proﬁts are positive. e) Suppose instead the market demand curve is

D2 : p(QD) = 60 − 9.5QD

Solve for the equilibrium price and quantity. With the new demand curve:

p = 60 − 9.5 × 2p

p = 60 − 19p ⇒ 20p = 60 p = 3

Q = 2 × 3 = 6 Q 6 3 q = = = n 8 4 f) In the long run, will more growers enter the market or will existing growers exit? Explain.

Π = TR − TC 3 9 TR = 3 × = 4 4 32 9 25 TC = 2 + 2 = 2 + = 4 8 8 9 25 7 Π = − = − < 0 4 16 8 Profits are negative, so this is not a long-run equilibrium. Some firms will exit the market. g) What is the long-run equilibrium price at which there will be no entry or exit in this market? In the long run the equilibrium price will be equal to the minimum average total cost (profits are zero). As the marginal cost intersects the average total cost at its minimum, we can find that minimum value of ATC setting MC = ATC.

3 2 AT C = + 2q q MC = 4q 2 AT C = MC; ⇒ + 2q = 4q q 2 2q = ⇒ q = 1 q Thus, the long-run equilibrium price is:

p = MC = 4 × 1 = 4

Problem 2

Suppose rice is produced under perfect competition. There are 1000 rice producers, and rice is homogeneous across all of them (think of 1000 as a really large number of producers). The market demand for rice is given by: 1 QD = 20 − P 2

where P denotes the market price of rice, which all rice producers take as given. All rice producers grow rice with the same technology, hence they all face the same cost function. 2 TC(qi) = FC + 1000qi

Suppose each individual rice producer i (notice i = 1, 2, ..., 1000 is the index of each producer), and FC denotes the ﬁxed cost. a) Find the equilibrium price and quantity in the rice market Find the individual ﬁrm’s supply curve.

1000q2 AV C = i = 1000q q i The minimum value is 0. The ﬁrm produces a positive quantity for all possible prices. To maximize proﬁts, p = MC

MC = 2000qi

p = 2000qi

4 Individual ﬁrm’s supply curve: 1 qS = p i 2000

To ﬁnd the market supply curve, multiply this curve by the number of ﬁrms: 1 1 QS = n × qS = 1000 p = p i 2000 2

To find the market equilibrium, set QS = QD 1 1 p = 20 − P 2 2 p = 20 Q = 10 b) How much rice does each individual producer end up producing? The quantity exchanged in the market is Q = 10 There are 1000 identical firms, thus the quantity supplied by the individual firm is: 10 q = = 0.01 i 1000 c) Calculate the consumer surplus in the rice market

Figure 1: Problem 2, point c)

(40 − 20)10 CS = = 100 2

5 d) Suppose FC = 1. Is this market in a long-run equilibrium? Justify your answer. If FC = 1, total cost function is TC = 1 + 1000q2

TR = 20 × 0.01 = 0.2 TC = 1 + 1000(0.01)2 = 1.1 Π = 0.2 − 1.1 = −0.9 < 0

Profits are negative, thus this market is NOT in a long-run equilibrium. With negative profits, some firms will exit the market.

Problem 3

Consider a perfectly competitive market. In this market there are 10 ﬁrms, characterized by the following cost function

TC(qi) = 10qi

The production capacity of these firms is limited: the maximum amount produced with the capacity installed is 10 units per firm. The good can also be supplied by other 5 firms, less efficient, and characterized by the following cost function:

TC(qj) = 15qi and again these ﬁrms are able to produce 10 units each. a) Suppose that the demand curve is:

QD = 180 − 12P

Find the equilibrium price and quantity.

MC1 = 10 The first group of firms has a marginal cost of 10 (constant!). Each firm can offer 10 units at that price. Total supply at this price is thus 10 × 10 = 100. The second group of firms is less efficient. Those firms are characterized by a constant marginal cost equal to 15. These firms can supply 10 units each and they are 5. Thus, total supply at the price of 15 is 50. We have derived the market supply curve (see Figure 2). To find the equilibrium draw the demand curve. The demand curve crosses the supply curve at a price = 10.

6 Figure 2: Problem 3

The equilibrium price is p = 10, the quantity demanded at this price is QD = 180 − 12 × 10 = 60. Only the firms of the first group (most efficient ones!) are serving the market. The equilibrium price is equal to the marginal cost of these firms. b) Suppose now that the demand curve becomes:

0 QD = 300 − 12P

Find the new equilibrium. The new demand intersects the supply curve at p = 15.

At this price the quantity demanded is QD = 300 − 12 × 15 = 120. The equilibrium price is 15. 100 units are supplied by the most efficient firms, 20 units are supplied by the least efficient ones. c) Compare the two situations in points a) and b). How is the price changed? Why? The price is set at the level of the marginal cost of the least productive firms.

Problem 4 a) Is the following statement true or false? Explain. In the short run, the supply curve will be upward sloping in perfect competition The statement is true. The supply curve is made up of the ﬁrms’ marginal cost curves. In the short run marginal cost will be upward sloping because of the principle of diminishing returns

7 b) Is the following statement true or false? Explain. If firms in a perfectly competitive industry are earning positive profits, the industry is not in long run equilibrium This statement is true. If firms in the industry are earning positive profits, in the long run there will be entry. Entry will continue until the price has fallen to the point that profits have been eliminated. If firms are earning losses, there will be exit. Exit will continue until the price has risen to the point where losses have been eliminated. c) Assume the beer industry is perfectly competitive. What will happen to the equilibrium price and quantity of beer and the firms’ profits in the short run if the drinking age is lowered from 21 to 18? Explain your answer. The market demand for beer increases. In the short run, equilibrium price increase and quantity increases. The firm increases its output, profits become positive (P > AC). The industry is not in a long run equilibrium because firms are earning positive profits. d) Is the following statement true or false? Explain. In the long run, the supply curve will always be upward sloping The statement is false. In long run, equilibrium price must be equal to the minimum average cost because firms must earn zero profits. As a result the long run supply curve, if the industry is characterized by constant costs, is perfectly elastic at that price (horizontal). e) Nate’s Car Wash is one firm in a perfectly competitive industry. What will happen to price and output in the car wash industry if price of water increases (assume the car wash industry has constant costs)? The increase in the price of water shifts firms’ MC and AC curves up. This causes the industry supply curve to shift inwards. Equilibrium price will increase and output will decrease.