1 Chapter 11 PROFIT MAXIMIZATION

2 Profit Maximization A profit-maximizing firm chooses both its inputs and its outputs with the sole goal of achieving maximum economic profits – seeks to maximize the difference between total revenue and total economic costs

3 Market Demand Curve If we add up the demand curve of all the consumers of the market, then we will obtain the Market demand curve We will denote the market demand curve by D(p) or q(p)

4 Output Choice Total revenue for a firm is given by R(q) = p(q)  q In the production of q, certain economic costs are incurred [C(q)]->cost function Economic profits (  ) are the difference between total revenue and total costs  (q) = R(q) – C(q) = p(q)  q –C(q)

5 Output Choice The necessary condition for choosing the level of q that maximizes profits can be found by setting the derivative of the  function with respect to q equal to zero  (q) = R(q) – C(q) = p(q)  q –C(q)

6 Output Choice To maximize economic profits, the firm should choose the output for which marginal revenue is equal to marginal cost

7 Second-Order Conditions MR = MC is only a necessary condition for profit maximization For sufficiency, it is also required that This is standard maths… at the max the second derivative must be negative

8 Profit Maximization output revenues & costs R C q* Profits are maximized when the slope of the revenue function is equal to the slope of the cost function The second-order condition prevents us from mistaking q 0 as a maximum q0q0

9 Marginal Revenue R(q) = p(q)  q The second term (dp/dq) is negative for firms that face a downward-sloping curve. Typically these are large firms so that prices decreases when they produce more. The second term (dp/dq) is zero for competitive or price taking firms. Typically these are small firms so that the price does not depend on the quantity that the firm produces. In this case we have that the marginal revenue will be equal to the price.

10 Marginal Revenue Suppose that the demand curve for a sub sandwich is q = 100 – 10p Solving for price, we get p = -q/ This means that total revenue is R = pq = -q 2 / q Marginal revenue will be given by MR = dR/dq = -q/5 + 10

11 Profit Maximization To determine the profit-maximizing output, we must know the firm’s costs If subs can be produced at a constant a marginal cost of $4, then MR = MC -q/ = 4 q = 30 We should also check that the second derivative of the profit function in this point is negative

12 Marginal Revenue Curve The marginal revenue curve shows the extra revenue provided by the last unit sold MR is twice as steeper that demand

13 Marginal Revenue Curve output price D (p) MR q1q1 p1p1 As output increases from 0 to q 1, total revenue increases so MR > 0 As output increases beyond q 1, total revenue decreases so MR < 0

14 Marginal Revenue Curve When the demand curve shifts, its associated marginal revenue curve shifts as well – a marginal revenue curve cannot be calculated without referring to a specific demand curve

15 Supply Curve for a price taking firm It is the curve that tell us how much a price taking firm will produce for each possible price

16 Analysis of the supply of a price taking firm Notice that if a firm is price taker, the second derivative of the profit function is equal to minus the derivative of the marginal cost So, the condition that the second derivative of the profit function is negative is equivalent to increasing the marginal cost must be increasing) A price taker firm will only maximize profits when marginal costs are increasing

17 Supply by a Price-Taking Firm output price SMC SAC SAVC p* = MR q* Maximum profit occurs where p = SMC

18 Supply by a Price-Taking Firm output price SMC SAC SAVC p* = MR q* Since p > SAC, profit > 0 Show a situation with losses !!!!

19 Supply by a Price-Taking Firm output price SMC SAC SAVC p* = MR q* If the price rises to p**, the firm will produce q** and  > 0 q** p**

20 Short-Run Supply by a Price-Taking Firm – If p > SAVC, the firm operates because the difference between p and SAVC contributes towards covering fixed costs – If p< SACV, the firm has the fixed costs PLUS a cost of (SAVC-p) for each unit produced… It is better not to produce and only have the fixed costs Thus, the price-taking firm’s short-run supply curve is the positively-sloped portion of the firm’s short-run marginal cost curve above the point of minimum average variable cost – for prices below this level, the firm’s profit-maximizing decision is to shut down and produce no output.

21 Short-Run Supply by a Price-Taking Firm output price SMC SAC SAVC The firm’s short-run supply curve is the SMC curve that is above SAVC

22 Short-Run Supply example 11.3 in the book

23 Profit Functions for price taking firms For this, we have formulated the firms problem of maximizing profits in terms of input prices, output prices, and level of output.

24 Profit Functions for price taking firms A firm’s economic profit can be expressed as a function of inputs  = pq - C(q) = pf(k, l ) - vk - w l Only the variables k and l are under the firm’s control – the firm chooses levels of these inputs in order to maximize profits treats p, v, and w as fixed parameters in its decisions

25 The firm solves the following problem: The first-order conditions for a maximum are  /  k = p[  f/  k] – v = 0  /  l = p[  f/  l ] – w = 0 A profit-maximizing firm should hire any input up to the point at which its marginal contribution to revenues is equal to the marginal cost of hiring the input

26 Profit Maximization and Input Demand These first-order conditions for profit maximization also imply cost minimization – they imply that RTS = w/v

27 Profit Maximization and Input Demand To ensure a true maximum, second-order conditions require that  kk = f kk < 0  ll = f ll < 0  kk  ll -  k l 2 = f kk f ll – f k l 2 > 0 – capital and labor must exhibit sufficiently diminishing marginal productivities so that marginal costs rise as output expands

28 Input Demand Functions The first-order conditions can be solved to yield input demand functions Capital Demand = k(p,v,w) Labor Demand = l (p,v,w) These demand functions are unconditional – they implicitly allow the firm to adjust its output to changing prices

29 Substitution and Output effect Now we are equipped to study how input choices changes with input prices… When w falls, two effects occur – substitution effect if output is held constant, there will be a tendency for the firm to want to substitute l for k in the production process – output effect the firm’s cost curves will shift and a different output level will be chosen

30 Substitution Effect q0q0 l per period k per period If output is held constant at q 0 and w falls, the firm will substitute l for k in the production process Because of diminishing RTS along an isoquant, the substitution effect will always be negative

31 Output Effect Output Price A decline in w will lower the firm’s MC MC MC’ Consequently, the firm will choose a new level of output that is higher P q0q0 q1q1

32 Output Effect q0q0 l per period k per period Thus, the output effect also implies a negative relationship between l and w Output will rise to q 1 q1q1

33 Cross-Price Effects No definite statement can be made about how capital usage responds to a wage change – a fall in the wage will lead the firm to substitute away from capital – the output effect will cause more capital to be demanded as the firm expands production

34 Substitution and Output Effects We have two concepts of demand for any input – the conditional demand for labor, l c (v,w,q) – the unconditional demand for labor, l (p,v,w) At the profit-maximizing level of output l c (v,w,q) = l (p,v,w)

35 Substitution and Output Effects Differentiation with respect to w yields substitution effect output effect total effect

36 Profit Functions for price taking firms A firm’s profit function shows its maximal profits as a function of the prices that the firm faces We use the unconditional input demands to obtain the profit function

37 Properties of the Profit Function If output and input prices change in the same percentage, the profits change in the same percentage – with pure inflation, a firm will not change its production plans and its level of profits will keep up with that inflation

38 Properties of the Profit Function Nondecreasing in output price Nonincreasing in input prices

39 Properties of the Profit Function Convex in output prices – the profits obtainable by averaging those from two different output prices will be at least as large as those obtainable from the average of the two prices

40 Graphical analysis of profits Because the profit function is non-decreasing in output prices, we know that if p 2 > p 1  (p 2,…)   (p 1,…) The welfare gain to the firm of this price increase can be measured by welfare gain =  (p 2,…) -  (p 1,…)

41 Graphical analysis of profits output price SMC p1p1 q1q1 If the market price is p 1, the firm will produce q 1 If the market price rises to p 2, the firm will produce q 2 p2p2 q2q2 A B Increase in Total Revenue= p 2 Aq 2 q 1 Bp 1 !!!!!!!!!!!!!!!

42 The increase in costs is the area: Aq 2 q 1 B Remember that:

43 The firm’s change in profits rise by the shaded area Graphical analysis of profits output price SMC p1p1 q1q1 p2p2 q2q2

44 Producer Surplus in the Short Run Let’s measure how much the firm values producing at the prevailing price relative to a situation where it would produce no output This is given by: This difference in profits is known as the producer surplus

45 Producer surplus at a market price of p 1 is the shaded area Producer Surplus in the Short Run output price SMC p1p1 q1q1 p0p0 Shut down price

46 Producer Surplus in the Short Run Producer surplus is the extra return that producers make by making transactions at the market price over and above what they would earn if nothing was produced – the area below the market price and above the supply curve

47 Producer Surplus in the Short Run Because the firm produces no output at the shutdown price,  (p 0,…) = -vk 1 – profits at the shutdown price are equal to the firm’s fixed costs This implies that producer surplus =  (p 1,…) -  (p 0,…) =  (p 1,…) – (-vk 1 ) =  (p 1,…) + vk 1 – producer surplus is equal to current profits plus short-run fixed costs