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## Profit Maximization

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**Profit Maximizing Assumptions**• Firm: Technical unit that produces goods or services. • Entrepreneur (owner and manager) • Gains the firm’s profits and suffers losses and has the goal of maximizing profit. • Transforms inputs (aka factors) into outputs through the technology of the production function. • Decides how much of each input to use and what quantity to produce.**Realistic?**• For this class, we assume corporation’s with shareholders and boards and executives function like the entrepreneur. • Obviously, managers and executives have other incentives besides maximizing profit. • But there is a more developed theory of the firm for corporations that we will not get into. • Profit maximization? • Libraries and water utilities are strictly non-profit. • Most hospitals claim to be non-profit, but they act like for-profit hospitals.**Why Firms?**• You could ask why even have firms? • Why don’t entrepreneurs outsource EVERYTHING? • Transactions costs make that infeasible. Or not. • In the 1870s-1950s vertical integration was the norm (River Rouge plant included a steel mill and processed rubber). • Since then, outsourcing has been growing. • New communication technology has driven this more towards the entrepreneur-only model.**Profit**• Profit maximization could easily be 1/2 of the text as we assume firms maximize profit under a host of situations: perfectly competitive, monopoly, price discrimination, oligopoly, monopsony, etc. • However, the basics are perfect competition (price taker) and monopoly (price setter).**Price Taker vs Price Setter**• Profit is TR-TC • Price Taker (competitive firm) • Treat the price they face as given when choosing quantity • Price Setter (single price, no strategic behavior) • Price is chosen along with quantity. • Short and long run options are the same • Short Run, quantity decision includes the shut down option • Long Run, consideration of returns to scale and entry and exit**Profit**• Profit = TR-TC • Total costs include all implicit and explicit costs (unlike accounting cost that would only include explicit costs). • In our model, we assume the firm rents capital at a rate of v. But that is exactly the same as if the firm owned the capital but could rent it out to another firm at a rate of v. • Accounting profit using the firm’s owned resources in the next best alternative use • Includes Value of the entrepreneur’s time • Selling off owned factors and investing elsewhere • For us, SC = VC + FC = wL + vK**Price Takers**• The rest of this lecture focuses on price takers. • Homogeneous output • No barriers to entry/exit in long run • Many sellers • Perfect price information**Revenue: Price Taker**• Price Taker • If a firm charges p > Pm , they will sell q = 0 • Demand for firm’s output is p = Pm, the firm can sell as many as it wants, until q = 3,000,000, and then need to lower the price to sell more. P Market Demand However, price taker assumption is that no firm is big enough to be able to affect the market price. p = Pm q 1,000 2,000 3,000 3,000,000**Revenue: Price Taker**• So for price taker, we assume decreasing returns to scale precludes getting large enough to have production influence price: • So R = p·q, where p = Pm P Demand for firm: Pm = MR = AR p = Pm q 1,000 3,000 5,000**Cost and Short Run Supply**• Let’s for the moment assume production exhibits IRS and then DRS. • Firms will, in the LR, choose a level of K commensurate with getting the lowest possible SAC. • The price will be driven to the low point of AC, the break even price. • At this starting point, the firm’s SAC, AVC, and SMC are relevant in the short run • Other returns to scale options considered in the long run. Exhibits IMR, DMR C C SRC SRC AC SAC SMC Exhibits IMR, DMR Exhibits IMR, DMR MC SMC AC SAC pbe Ehhibits IRS, DRS AVC q q**Cost and Short Run Supply**• But for the moment, we don’t care which level of K the firm has or what the AC curve looks like. • In the SR, here is what we have to work with. • To determine the profit maximizing level of q = q*, and whether shut down and produce q = 0, MC and SAVC are most important. C SC SC AC SRAC SRMC Exhibits IMR, DMR Exhibits IMR, DMR SAC AVC SMC q q**Profit Max**• Maximize π = R(q) - C(q) • FOC for this yields q where slope of π function = 0 SC R=p·q SC π=R-SC MR=P slope of R π maximized at q where MR=SMC FOC, derivative of π function is zero SMC = slope of SC q q**Profit Max**• Checking the SOC too. R SC π also minimized at q where MR=SMC (FOC satisfied here too) Which is why we check SOC, to make sure profit is falling where MR = SMC (i.e. MR is falling relative to SMC) SC π=R-SC MR=P slope of R SMC = slope of SC q q**Profit Max**• The more common graph • Maximize π = R(q) - C(q) • FOC for this yields MR=SMC, which it does twice. • SOC ensures MC is rising relative to MR R C SC SC AC SAC SMC Exhibits IMR, DMR SMC q q**Price Taker Profit Max**• So long as you are better off producing than not, • As you increase q, the change in profit = MR-SMC. • Produce until MR = SMC and marginal profit (change in profit as q increases) is falling.**To Maximize Profit**I pulled these starting values out of the air**To Maximize Profit**Note, at π max, AR > ATC. When AR = ATC, π = 0**MC and qs: If price was $16, then the firm would produce**106.**MC and qs: If price was $10.40, then the firm would produce**103.**MC and qs: If price was $8.00, then the firm would produce**101.**MC and Supply**• Price Takers • The MC curve tells us the profit maximizing qs by the firm at any price. • Since it is the MC curve that determines the relationship between p and the quantity to supply, the SMC curve IS the firm’s short run supply curve. • Important caveat, if suffering a loss, firm might want to shut down if the loss is larger than FC. • Side note: Price Setters • set the price, they do not respond to it, so they have no supply curve.**Shut Down Option(price takers and price setters)**• Shut down: Short run situation where the firm produces a quantity of 0 while remaining in the industry. It is still considered to be in the industry as long as it cannot rid itself from its fixed inputs. • The firm could start producing very easily by employing some of the variable input.**Intuition**• A firm bearing a loss can produce qs = q*, (where MR=MC) or can shut down, produce qs = 0. • If qs = q*: π = R – VC – FC • If the firm shuts down: π = -FC (that is, has a loss = FC) • If FC is greater than the loss from producing, qs = q*. If FC is less than the loss from producing qs = q*, better to shut down and produce qs=0. • Decision Rule: Shut down if – FC > R – VC – FC R = 10,000 VC = 8,000 FC = 4,000 π = -2,000 -4,000 < -2,000 qs = q* R = 7,000 VC = 8,000 FC = 4,000 π = -5,000 -4,000 > -5,000 qs =0 Profit from shut down Profit from q=q***Decision Rule**• Shut down if: –FC > R – VC – FC 0 > R – VC R > VC • So long as revenue covers all variable cost, the loss will be less than FC so q = q* .**Side note: Which can change the firm’s output decision, a**change in Variable Cost and/or a change in Fixed Cost? • Shut down if: – FC > • – FC > R – VC – FC • FC is on both sides, so a change in FC does not affect the relationship or the decision. • Ok, yes, fixed costs are fixed (don’t vary with output) • health insurance premiums rise • Tony Romo signs a $108m extension. • But a change in either R or VC could change the decision.**Price Taker in the Short Run**• Simple, just MR = MC • Maximize profit w.r.t. q • Maximize profit w.r.t. L**Profit Max 1**• Simple, supply is SMC, find q where SMC = p.**Profit Max 2, MR = MC**• Optimize by choosing q.**Profit Max 3, MRPL=w**• Optimize by choosing L.**Producer Surplus**• Producer surplus is the amount by which a firm is better off than shut down (q=0) • If shut down, loss is = FC • By producing, the firm covers this potential loss, plus gains profit. • If profit = 0, then better off by the amount of FC • PS = π + FC • PS = R-VC-FC+FC • PS = R-VC**Producer Surplus**• If π = 0, then producer surplus = FC • If π < 0, but -π < FC, producer surplus > 0 • If π < 0, and -π = FC, producer surplus = 0 • Shut down point • If π < 0, and -π > FC, producer surplus < 0 • Will shut down, so PS = 0 and profit = -FC.**Profit MaximizationPrice Taker, Long Run**• Returns to Scale Matter • IRS, LMC falling • CRS, LMC constant • DRS, LMC rising**Increasing Returns to Scale**• IRS only: incompatible with competition as the biggest firm has the lowest average cost… natural monopoly results C AC MC C AC MC q q**Decreasing Returns to Scale**• DRS only: an infinite number of infinitely small firms. MC C AC MC C AC q q**Constant Returns to Scale**• CRS only: any size firm can produce at the same AC. AC = AVC = MC (Firm LRS is horizontal at MC). C AC MC C AC=MC q q**IRS, DRS**• IRS, DRS: MC rising. Firm LRS = MC above AC (exit otherwise. C C AC MC MC AC Exhibits IRS, DRS q q**IRS, CRS, DRS**• IRS, CRS, DRS: MC rising, but flat spot while CRS. • Perhaps most realistic, but not easy to solve – or find a production function that creates this. C C AC MC DRS MC Firm LRS = MC for p ≥ pBE CRS AC IRS q q**Profit Max. vs. Perfect Comp.**• We will eventually assume that in the long run K will be fixed to yield this SAC and the market price will be pbe (so in the LR q* will be at the low point of SAC) • But in this chapter, we want to explore the possibility that price will exceed pbe for a while. So we need a firm LRS curve. AC MC SAC SMC MC SMC AC SAC Firm exit if p < pBE q**Production and Exit**• Essentially, a firm’s long run supply curve will be its long run MC curve… • While shut down is a viable option in the short run, in the long run all costs can be avoided by exiting the market. • If p < pbe, (minimum value of AC curve), the firm should exit the industry. • So firm long run supply is MC above pbe.**Price Taker in the Long Run**• Simple, just MR = MC • Maximize profit w.r.t. q • Maximize profit w.r.t. K, L**Profit Max 1**• Simple, set MC = P, find q.**Profit Max , MR=MC**• Optimize by choosing q**Profit Max, MRPL=w; MRPK=v**• Optimize by choosing inputs**Profit Max, choose K and L**• Ratio of FOC**Profit Max, choose K and L**• Profit function, maximal profits for a given w, v, and p.**Properties of the Profit Function**• Homogeneous of degree one in all prices • with inflation, K*, L*, and q* are the same profit will keep up with that inflation • Nondecreasing in output price • Δ profit ≥ 0 with Δ p > 0 • Nonincreasing in input prices • Δ profit ≤ 0 with Δ w or Δ v > 0 • Convex in output prices • profits from averaging those from two different output prices will be at least as large as those obtainable from the average of the two prices**Envelope Results**• Long run supply To maximize profit when there is a change in price , q=q*= f(K*, L*), continue producing such that L* = L(w, v, p) K* = K(w, v, p)