# Review. 1. Production function - Types of production functions - Marginal productivity - Returns to scale

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1 Review 1. Production function - Types of production functions - Marginal productivity - Returns to scale 2. The cost minimization problem - Solution: MP L (K,L)/w = MP K (K,L)/r - What happens when price of an input increases? 3. Deriving the cost function - Solution to cost minimization problem - Properties of the cost function (marginal and average costs) 1

2 Economic Profit Economic profit is the difference between total revenue and the economic costs. Difference between economic costs and accounting costs: The economic costs include the opportunity costs. Example: Suppose you start a business: - the expected revenue is \$50,000 per year. - the total costs of supplies and labor are \$35, Instead of opening the business you can also work in the bank and earn \$25,000 per year. 2 - The opportunity costs are \$25,000 - The economic profit is -\$10,000 - The accounting profit is \$15,000

3 Firm s supply: how much to produce? A firm chooses Q to maximize profit. The firm s problem max ( Q) Q TR( Q) TC( Q) Total cost of producing Q units depends on the production function and input costs. Total revenue of is the money that the firm receives from Q units (i.e., price times the quantity sold). It depends on competition and demand 3

4 Deriving the firm s supply Def. A firm is a price taker if it can sell any quantity at a given price of p per unit. How much should a price taking firm produce? For a given price, the firm s problem is to choose quantity to maximize profit. max pq TC( Q) Q s. t. Q 0 Optimality condition: P = MC(Q)

5 Short Run Supply Example. A firm has the cost function TC(Q) = Q + Q 2 and can sell each unit for a price of 30. How many units will it sell? What is the profit? Calculate the firm s profit if the market price is 40, 20, 15. Derive the firm s supply.

6 Short Run Supply A firm has the cost function TC(Q) = Q + Q 2: TFC = 100 ATC(Q)=100/Q+20+Q TVC(q) = 20Q + Q 2 AVC(q) = 20 + Q MC(q) = Q The firm s short run supply curve is: 20 \$ SMC ATC AVC Q p s - If the price is P < 20: then the firm produces nothing Q = 0 - If price is P > 20: then P = MC(Q) P = 20+2Q Q = 10 + ½P

7 Short Run Supply 7 \$ Supply SMC ATC AVC 2 0 Q p s The firm s short run supply curve is: - If the price is P < 20: then the firm produces nothing Q = 0 - If price is P > 20: then P = MC(Q) P = 20+2Q Q = ½P

8 The Firm s Decision Does the firm choose to produce a positive quantity Q>0 or to shut down and produce nothing Q=0? p s = min Q AVC(Q)

9 Short Run Supply Key Definition: A single firm s Short run supply curve specifies the profit maximizing output for each market price. The firm s short run supply curve is give by: 1. When P < Ps the firm chooses to produce nothing, Q=0 2. When P > Ps the firm chooses to produce Q>0 such that MC(Q)=p \$ Fixed costs are sunk: SMC ATC AVC P s Quantity

10 The Firm s Decision The firm s problem max ( Q) Q TR( Q) TC( Q) Optimality condition: If MR(Q)> MC(Q), the firm s profit increases if it produces more output. If MR(Q) < MC(Q), the firm s profit decreases if it produces more output. The profit maximization condition is MR(Q) = MC(Q) Since the firm is a price taker: p=mr(q)=mc(q) We must also check the shut-down condition: p>minavc(q)

11 Profit Maximization Optimality condition: 1. P = MC(Q) 2. MC(Q) increases

12 Short Run Supply Important to remember: if the firm produces output Q and sells it for a price p then: 1. When p>atc(q) the firm makes a profit. When p<atc(q) the firm loses money 2. When p>avc(q) the firm produces Q>0, when p<avc(q) the firm shuts down (assuming all fixed costs are sunk) 3. When AVC(Q)<p<ATC(Q) the firm operates at a loss

13 Short Run Market Supply Curve Definition: The short run market supply is the sum of the quantities each firm supplies at that price. Example: suppose 3 types of firms with different marginal costs and different shut down prices

14 Producer Surplus 14 p The producer surplus is the monetary benefit of a producer from a transaction= area between the supply curve and the price. Individual supply p* PS q* q

15 Perfectly Competitive Market 15 A perfectly competitive market satisfies the following conditions: 1. Fragmented industry: consists of many small buyers and sellers. 2. Buyers and sellers are price takers : - Each buyer s purchases are small and do not affect the market price. - Each seller is small and does not affect the market price. - No seller can affect the price of inputs. 3. Firms produce identical products. 4. Perfect information about prices. 5. All firms have equal access to inputs, have the same technology, and there is free entry. Implies that firms have identical long run cost functions.

16 Perfectly Competitive Market 16 The Law of One Price: Since products are identical and there is perfect information, there is a single price at which transactions occur. Why? What happens if one firm sets price p and another sets p < p? A single firm takes the price as given and chooses Q to maximize profit. The firm s problem max Q s. t. Q pq TC( Q) 0 Optimal solution: p=mc(q)

17 Short Run Equilibrium 17 For a given number of firms N, a short run equilibrium is a pair of price and quantity (Q,P) such that: 1. Each producer maximizes profits given price P 2. Markets clear (aggregate supply equals aggregate demand) Q S N q i 1 S i ( P) ( P) Where q i S P is firm i s individual profit-maximizing output given price P. Q D

18 Short Run Equilibrium Graphically Typical firm Market with n identical firms \$ p* Supply SMC ATC AVC \$ Supply p s Demand q* q Q*=nq* 18 Q

19 Short Run Equilibrium: example 19 Suppose first that a market consists of 300 identical firms, all with the same cost curve: TC(q)= q 2. Consumers demand is given by Q d (p) = 60 p. (a) What is the equilibrium price and quantity? (b) Do firms make positive profits in the market equilibrium? Step 1: Derive individual supply curve FC = 0.1 (all are sunk, NSC= 0); AVC(q) = 150q; MC(q)=300q Since min{avc(q)}=0, the firm always produces q>0. The profit maximizing condition: MC(q)=MR(q), we have that 300q=p, or q s (p) = p/300 Step 2: derive the market supply curve Q s (p) = 300 q s (p) =300(p/300) = p Step 3: solve for equilibrium Q s (p)= Q d (p) or p = 60 p so that p 1 *= 30, Q* = 30 and each firm produces q 1 * = 30/300 = 0.1

20 Short Run Equilibrium: example 20 (b) Do firms make positive profits at the market equilibrium? Condition for positive profits: p* > ATC(q*) ATC (q)= TC(q)/q = 0.1/q + 150q. Since each firm produces q 1 * = 0.1, we have that ATC(q 1 *) = 15 < 30 = p 1 *, Therefore, p 1 * > ATC(q 1 *) and profits are positive The profit of each firm is: pq-tc(q)= ( )=1.4

21 Short Run Equilibrium: example 21 (c) What happens when the number of firms increases from 300 to 500? A single firm s supply is unchanged: MC(q)=p and q s (p) = p/300 But now, market supply increases to Q(p)=500 (p/300) and for market to clear we have that: 500(p/300)= 60 p or p 2 *=22.5 and Q 2 *= 37.5 and the individual firm produces q 2 * = 37.5/500 =0.075 (d) Does each firm earn a profit? Condition for positive profits: ATC(q*) < p* ATC (q 2 *)= 0.1/q + 150q= 0.1/ =11.25 < 22.5 = p 2 * The profit of each firm pq-tc(q)= ( )=1.3

22 Short Run Equilibrium: comparative statics 22 What happens when the number of firms increases? Market supply increases, price drops, and each firm produces less.

23 Short Run Equilibrium: example 2 23 A town has 200 identical cafes, all with the same cost curve: TC(q)= FC+VC = 4 + 2q 2. The inhabitants total demand for coffee is Q d (p) = p. (a) What is the equilibrium price and quantity? (b) What is a single cafe s profit? Calculate the town s consumer, producer and total surpluses. (c) The town s mayor closes 160 cafes in an effort to reduce the inhabitants coffee consumption. Answer (a) and (b) for n=40. How are the firms total profits and producer surplus related? Answers for n=200: p* = 2, Q* = 100, π = -3.5, CS = 500, PS = 100, TS = 600 Answers for n=40: p* = 6, Q* = 60, π = 0.5, CS = 180, PS = 180, TS = 360, PS = n (π+fc) = 40 (0.5+4)

24 Short Run Equilibrium: producer surplus 24 p Demand Individual supply Producer surplus = area between the supply curve and the price. Profit = PS FC (sunk) p* PS q* q

25 Short Run Equilibrium: market surplus 25 p Demand CS Market supply Suppose that all firms are identical (so linear supply). In equilibrium on a perfectly competitive market, total surplus is maximized. p* PS Q* Q

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