ME II Classes 5-6 Topic 4 Profit maximization EA1 2019-2020 PDF

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Topic 4 Profit maximization and competitive supply

Microeconomics II Term III, 2019-2020

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Outline 1. Profit maximization in the short run 2. Marginal revenue and elasticity of demand. Revenue maximization 3. Profit maximization and supply for a competitive firm in the short run 4. Producer surplus in the short run 5. Profit maximization in the long-run 6. Duality in production

Literature: N: Chapter 13. Profit maximization and supply V_IM: Chapter 19. Profit maximization, Chapter 21. Firm supply

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Previous lecture: Review • • •

If a firm with strictly quasiconcave production function is producing where MP1/w1>MP2/w2, what can it do to reduce costs but maintain the same output? What is the general form of cost function and conditional input demand functions for a homogeneous production function of degree α>0? Define the corresponding production function for the typical total cost functions: r w  1. C  q  α 1  α 2 

2. C  q  min  r α ,w α  2  1 3. C  q α 1 (r / α 1 )  α 2 (w / α 2 ) g

where

g 

4. C  B  q

1

g



1

g

,

ρ ρ 1

α1  α2

α1

r

α1  α 2

α2

w

α 1 α 2

 α B   2  α 1 

α1

  

α 1 α 2

α   2  α1

  

α 2

α 1 α 2

   

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Previous lecture: Review •

•

•

A firm whose production function displays increasing returns to scale will have a total cost curve that is A. a straight line through the origin. B. a curve with a positive and continually decreasing slope. C. a curve with a positive and continually increasing slope. D. a curve with a negative and continually decreasing slope. For a constant returns to scale production function A. MC are constant but the AC curve has a U-shape. B. both MC and AC are constant. C. MC curve has a U-shape whereas AC are constant. D. both MC and AC curves are U-shaped. As long as marginal cost is below average cost, average cost will be A. falling. B. rising. C. constant. D. changing in a direction that cannot be determined without more information.

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Profit maximization The assumption of profit maximization is frequently used in microeconomics because it predicts business behavior reasonably accurately and avoids unnecessary analytical complications. Firms that do survive in competitive industries make long-run profit maximization one of their highest priorities.

The profit-maximization problem includes two main questions (stages): 1) how to minimize the costs of producing any desired level of output q, 2) which level of output is indeed a profit-maximizing level of output.

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Profit maximization •

•

•

•

Firms as profit maximizers make decisions in a marginal way: when the incremental profit of an activity becomes zero, profits are maximized. The firm chooses the level of output that generates the largest level of economic profit. Economic profit is defined as π = TR (q) – TC (q) where TR (q) is the amount of total revenues received and TC (q) are the total economic costs incurred, both depending upon the level of output q produced. Remember that the economic definition of profit requires that all inputs are measured at their opportunity cost (i.e. what an input would cost if purchased now). We examine the profit-maximization problem with a single output, a single period of time and certainty about a firm’s stream of profits.

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Profit maximization Profit maximization problem and its solution: π = TR (q) – TC (q) → max FOC: dπ/ dq = dTR/ d q – dTC/ dq = 0 or MR = MC SOC: d2π/ dq 2 < 0 Profit maximization principle: In order to maximize profits, a firm should produce that output level for which the marginal revenue from selling one more unit of output is exactly equal to the marginal cost of producing that unit of output, given that the second derivative of profit with respect to output is negative at the optimal level of output.

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Profit maximization in the short run Profit maximization, general case (graphical illustration): Revenue R(q) is a curved line which accounts for the possibility that, accordi the law of demand, an increased output be achieved only with a lower price. Total costs C (q ) is a sum of fixed and variable costs (also a curved line in the general case). C is positive when q=0 because of fixed costs in the short run. Profit, the difference between revenue R cost C, is negative when q=0 because o fixed costs. A firm chooses output q*, so that profit is maximized. At that output, marginal revenue (the slope of the revenue curve) is equal to marginal cost (the slope of the cost curve).

MR(q) = MC(q)

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Profit maximization in the short run The application of the rule that marginal revenue should equal marginal cost depends on a manager’s ability to estimate marginal cost.

To obtain useful measures of cost, managers should keep three guidelines in mind: 1. Average variable cost should not be used as a substitute for marginal cost. When AVC and MC are increasing sharply, the use of AVC can be misleading when deciding how much to produce. 2. The marginal cost of increasing production is not the same as the savings in marginal cost when production is decreased (e.g. because of additional firing costs=severance payment to laid off workers). 3. All opportunity costs should be included in determining marginal cost.

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Marginal revenue and elasticity of demand Total revenue TR ( q) = P (q) ∙q Marginal revenue

MR ( q) = d TR / dq = d[P (q ) ∙q] / dq = q ∙ dP / dq + P = =P (q/P ∙ dP / d q + 1)= P (1+ 1/ε),

where ε is the price elasticity of market demand. If demand is Elastic |ε |>1

Marginal revenue positive

Unit elastic ε = –1

zero

Inelastic |ε| P ∙ q – VC – FC). • So, the shutdown condition in the short-run is P < AVC. • If the price is equal to average variable cost, the firm is indifferent between producing the profit-maximizing quantity (i.e. loss-minimizing quantity) and shutting down operations. • Therefore, the firm will opt to continue operating, i.e. q > 0, providing that P ≥ AVCmin. Shut-down rule: The firm should shut down if the price of the product is less than the average variable cost of production at the profit-maximizing output.

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Profit maximization for a competitive firm in the short run • If

P >ATCmin, the firm will earn

profits in the short run (profit zone). • If P =ATCmin , the firm earns zero profits. This zero-profit point is also called break-even point as it corresponds to the case when AR=AC and therefore TR=TC. • If P > AVCmin but P < ATCmin, the firm continues to produce in the short-run, making economic losses. • If P =AVCmin, the firm is indifferent between operating and shutdown (shutdown point). • If P < AVCmin, then the firm stops producing and only incurs its fixed costs (shutdown zone).

Source : https://courses.lumenlearning.com/microeconomics/chapter/the-shutdownpoint/.

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Profit maximization and supply for a competitive firm in the short run The firm’s short-run supply function is the relationship between price and quantity supplied by a firm in the short-run: q = f (P). For a price-taking profit-maximizing firm, MC (q) =P. The equation P = MC (q) gives us the inverse supply function: price as a function of output. What do SOC (second-order conditions) for profit maximization for a competitive firms in the short run tell us?

d2π/ dq2 < 0 d2(P ∙ q – VC – FC )/ dq 2 < 0 – MC ’ < 0

MC ’ > 0

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Profit maximization and supply for a competitive firm in the short run From SOC and the shut-down rule follows that, a short-run supply function is the positively sloped portion of the short-run marginal cost curve that lies above the average variable cost curve. In the short run, the firm chooses its output so that marginal cost MC is equal to price as long as the firm covers its average variable cost. The short-run supply curve is given by the crosshatched portion of the marginal cost curve. If the market price is less than the firm’s shutdown price (P < AVCmin), the firm produces no output.

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Important relationship between costs and productivity measures for a price-taking firm in the short run Assume that labor is a variable input and its market price is w. Then VC=wL and • MC = ∂VC/ ∂q = ∂(wL)/ ∂q = w /(∂q /∂L) = w /MP L, • AVC = VC/q = wL / q = w /(q/ L) = w /APL.

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Important relationship between costs and productivity measures for a price-taking firm in the short run The U-shape of the marginal cost curve is closely related to the hump-shape of the marginal product curve: •

the increasing portion of the marginal product curve (reflecting increasing marginal returns to labor) corresponds with the decreasing portion of the marginal cost curve;

•

the decreasing portion of the marginal product curve (due to decreasing marginal returns to labor) corresponds with the increasing portion of the marginal cost curve;

•

the peak of the marginal product curve corresponds with the minimum of the marginal cost curve;

•

the peak of the average product curve corresponds with the minimum of the average variable cost curve. Source : http://www.raybromley.com/notes/AveMargApp.html.

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Important relationship between costs and productivity measures for a price-taking firm in the short run The main conclusion is that the positively-sloped portion of the marginal cost curve (= SR supply curve) is directly attributable to the law of diminishing marginal returns to variable inputs. Shutdown point P = AVCmin= w /APL max.

Source : http://www.raybromley.com/notes/AveMargApp.html.

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Producer surplus in the short run Producer surplus is a measure of producer welfare. It is defined as the difference between the amount that a producer receives from the sale of a good and the lowest amount that producer is willing to accept for that good. The producer surplus for a firm is measured by the yellow area below the market price and above the marginal cost curve, between outputs 0 and q*, the profitmaximizing output. Alternatively, it is equal to rectangle ABCD because the sum of all marginal costs up to q* is equal to the variable costs of producing q* . Producer surplus versus Profit: Producer surplus = PS = TR − VC Profit = π = TR − VC − FC

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Profit maximization in the long-run 2 inputs: K and L Assume that a firm is a price taker on both output and input markets. The market price for its output is P , and w and r are input prices. The firm faces technological constraints represented by its production function q =f (K, L) satisfying assumption about desired properties, where K and L are amounts of inputs to produce an output level q.

The firm needs to decide: • •

what level of output to produce; how much of which factors to use to produce it.

The firm will answer these questions solving the profit-maximization problem: π  Pq  rK  wL  max (4.1)

s .t .

f (K , L )  q

K , L ,q  0

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Profit maximization in the long-run Replacing the inequality in the constraint by an equality (because the production function is strictly increasing), we may rewrite the maximization problem (4.1) in terms of a choice over the input vector as

π  Pf (K , L)  rK  wL  max

(4.2)

K ,L  0 Assume that this profit-maximization problem has an interior solution at K*> 0, L*> 0. Then the profit-maximizing amount of output produced is q ∗=f (K*, L* ).

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Duality in production If instead of maximizing profits in one step as was done above, we consider the two-step procedure: 1) calculate for each possible level of output the minimum cost of producing it and get the cost function C (w, r, q) and 2) choose the level of output that maximizes the difference between the revenues it generates and its costs to produce:

π  Pq  C (w ,r ,q )  max, q 0 we get the FOC that we have obtained before: P = MC.

(4.3)

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Duality in production Problems (4.2) and (4.3) are equivalent and provide the same solutions: • the optimal choice of output q*=q (P, w, r ) that is called the firm’s output supply function, • the optimal choice of inputs L *=L( P, w, r ) and K*=K( P, w , r), i.e. firm input demand functions. Unlike the conditional input demands derived from the cost minimization problem that depend partly on output, functions derived from the extended profit maximization problem are fully-fledged input demand functions because they maximize the firm’s profit.

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Duality in production Example/exercise: Derive the output supply and input demand functions for a firm using two inputs (K, L) and having a DRS Cobb-Douglas technology (q= K 0.25 L0.5). Step 1. Write the objective function π =P K 0.25L 0.5 – wL – rK and then FOC

P  MPL  w  P  MPK  r  0.25 0.5 q  K L  Step 2: Solving the system, find conditional input demand functions for both inputs: L =f 1(w , r, q ) and K=f2 (w, r, q). Step 3: Put them into one of the first two equations of the above system and find the output supply function q*= q (P, w, r). Step 4. Then substitute q* into conditional input demand functions to obtain the input demand functions K* and L*.

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Duality in production Alternative approach of getting the output supply function in the above example/exercise (following from duality in production): Steps 1-2: The same as above to get conditional input demand functions: L=f 1(w, r, q) and K= f2 (w, r, q). Step 3: Derive the cost function C =C (w , r, q). Step 4: Find marginal cost from the cost function and derive the output supply function q*=q (P, w , r), given that MC = P.

Step 5: Then substitute q* into conditional input demand functions to obtain the input demand functions K* and L*. Can we use any of these approaches to get the output supply function for the technology (production function) with constant or increasing returns to scale?

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Profit maximization by a competitive firm: does the rule P =MC always apply? •

Increasing returns to scale:  MC is decreasing  Interior solution of P =MC (q*) minimizes profit  Optimal output is a corner solution – to produce “infinite” output as long as price is above marginal cost (the more output is produced, the lower the costs and the higher the profits)

•

Constant returns to scale:  MC is constant  Optimal output is a corner solution with 3 scenarios: i. Produce zero if MC > P ii. Produce “infinite” output if MC 1): decreasing output supply function q (P)...