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ECON 500 ECON 500 –Microeconomic Analysis and Policy Producer Theory ECON 500 Producer Theory A theory of how firms coordinate the transformation of inputs into outputs with a goal of maximizing profits in the meantime. Part I Part II Part III Production Functions Cost Functions Profit Maximization ECON 500 Part I. Production Functions The principal activity of any firm is to turn inputs into outputs. In the theory of producer behavior the relationship between inputs and outputs is formalized by a production function of the form where q rticular good during a period, k represents the machine (that is, capital) usage during the period, l represents hours of labor input, m represents represents the possibility of other variables affecting the production process. ECON 500 Part I. Production Functions Most of our analysis will involve two inputs k and l, capital and labor respectively Where q is the maximum amount of a good that can be produced by using alternative combinations of k and l. Since it is possible to produce the same amount of a good using different combinations of capital and labor, it is important to understand their respective contributions to the production process at various levels of utilization. ECON 500 Part I. Production Functions Marginal Physical Product: The marginal physical product of an input is the additional output that can be produced by employing one more unit of that input while holding all other inputs constant. Where the use partial derivatives reflect the fact that all other input usage is held constant while the input of interest is being varied. ECON 500 Part I. Production Functions Diminishing Marginal Productivity: We assume that the marginal physical product of an input depends on how much of that input is used. We also postulate that inputs cannot be added indefinitely to a production process without eventually exhibiting some deterioration in its productivity. Mathematically: ECON 500 Part I. Production Functions Diminishing Marginal Productivity Changes in the marginal productivity of labor depend not only on how labor input is growing, but also on changes in capital. Therefore, we m In most cases, this cross partial derivative is positive indicating that the marginal productivity of an input increasing in the other input. Diminishing marginal productivity of a single input can be offset by the increases in other inputs. The gloomy prediction of Malthus that led economics to be called a dismal science is misplaced. ECON 500 Part I. Production Functions Average Physical Productivity APl is easily measured and is of much empirical significance. The relationship between average and marginal physical productivity is also of significance. When APl is at its maximum, it equals MPl ECON 500 Part I. Production Functions Isoquants An isoquant shows those combinations of k and l that can produce a given level of output. Mathematically, an isoquant records the set of k and l that satisfies: ECON 500 Part I. Production Functions ECON 500 Part I. Production Functions Marginal Rate of Technical Substitution The marginal rate of technical substitution (RTS) shows the rate at which labor can be substituted for capital while holding output constant along an isoquant. ECON 500 Part I. Production Functions RTS and Marginal Productivities The total differential of the production function is Along and isoquant, we know that dq=0, therefore Hence ECON 500 Part I. Production Functions Diminishing RTS Isoquants naturally have a negative slope but they are also as convex curves. This means along any one of the isoquants, the RTS is diminishing However, it is not possible to derive a diminishing RTS from the assumption of diminishing marginal productivity alone since Marginal Physical Product depends on the level of both inputs and cross productivity effects are present. To show that isoquants are convex, we would like to show that dRTS/dl < 0. Since RTS = fl/fk, we have ECON 500 Part I. Production Functions Diminishing RTS Because fl and fk are functions of both k and l, we must be careful in taking the derivative of this expression Using the fact that dk/dl = - fl/fk along an isoquant and kl = flk), we have ECON 500 Part I. Production Functions Diminishing RTS Because we have assumed fk > 0, the denominator of this function is positive. Hence the whole fraction will be negative if the numerator is negative. Because fll and fkk are both assumed to be negative, the numerator definitely will be negative if fkl is positive. When we assume a diminishing RTS we are assuming that marginal productivities possible negative cross-productivity effects. ECON 500 Part I. Production Functions Returns to Scale If the production function is given by q = f (k,l) and if all inputs are multiplied by the same positive constant t (where t > 1), then we classify the returns to scale of the production function by ECON 500 Part I. Production Functions Constant Returns to Scale CRS production functions are homogenous of degree 1: Derivatives of a CRS function are homogenous of degree 0: And CRS functions are homothetic ECON 500 Part I. Production Functions Elasticity of Substitution If the RTS does not change for changes in k/l, we say that substitution is easy because the ratio of the marginal productivities of the two inputs does not change as the input mix changes. Alternatively, if the RTS changes rapidly for small changes in k/l, we would say that substitution is difficult because minor variations in the relative productivities. A scale-free measure of this responsiveness is ECON 500 Part I. Production Functions ECON 500 ECON 500 Part II. Cost Functions Accounting Costs vs. Economics Costs Accountants emphasize out-of-pocket expenses, historical costs, depreciation, and other bookkeeping entries. Economists on the other hand define cost by the size of the payment necessary to keep the resource in its present employment, or alternatively, by the remuneration that input would earn in its next best use. Labor costs = hourly wage = w Capital costs = rental rate of capital = v Cost of entrepreneurial services = forgone earnings ECON 500 Part II. Cost Functions Cost Minimization The firm seeks to minimize total costs given q = f (k, l) = q0. First order conditions for this constrained minimum are ECON 500 Part II. Cost Functions In order to minimize costs, the firm should choose inputs such that the rate at which k can be traded for l in production to the rate at which they can be traded in the marketplace. Alternatively, the firm should chose inputs such that the marginal productivity per dollar spent should be the same for all inputs. ECON 500 Part II. Cost Functions ECON 500 Part II. Cost Functions ECON 500 Part II. Cost Functions Total Cost Function Average Cost Function Marginal Cost Function ECON 500 Part II. Cost Functions CRS Production Function and its Costs ECON 500 Part II. Cost Functions Cubic Cost Function ECON 500 Part II. Cost Functions Properties of Cost Functions Non-decreasing in v, w, and q Homogenous of degree 1 in input prices Concave in input prices ECON 500 Part II. Cost Functions ECON 500 Part II. Cost Functions l Suppose that the price of labor (w) were to increase slightly. Costs would rise by approximately the amount of labor l that the firm was currently hiring. constant, so an increase in the wage increases costs in direct proportion to the amount of labor used. Because the true cost function is tangent to the pseudo-function at the current wage, its slope (that is, its partial derivative) also will show the current amount of labor input demanded. ECON 500 Part II. Cost Functions Lemma ECON 500 Part II. Cost Functions Short run vs. Long run In the short run, the firm is able to alter only its labor input while capital input is fixed at some level k1. With this formulation, payments to capital attain a fixed cost nature in the short run and do not vary with the amount produced. Short run cost function becomes: ECON 500 Part II. Cost Functions Short run vs. Long run In the short run, to change output, firms are forced to use input combinations that are non-optimal. Unavailability of input substitution prevents the firm from finding the input mix where RTS equals the ratio of input prices. Hence, in the short run, costs are not necessarily minimized. ECON 500 Part II. Cost Functions ECON 500 Part II. Cost Functions Long-run total costs are always less than short-run total costs, except at that output level for which the assumed fixed capital input is at the appropriate level to ensure long-run cost minimization. Therefore, long-run total cost curves constitute their respective short-run curves. A family of short run cost curves can be obtained by varying the capital level in Long run cost curve can be recovered by combining the above SC with ECON 500 Part II. Cost Functions ECON 500 Part II. Cost Functions ECON 500 Part III. Profit Maximization A profit-maximizing firm chooses both its inputs and its outputs with the sole goal of achieving maximum economic profits. That is, the firm seeks to make the difference between its total revenues and its total economic costs as large as possible. This -making unit and sweeps away all the complicated behavioral issues about the relationships (contractual or implicit) among input providers. The profit maximizing assumption has a long history in economic literature. It is plausible because firm owners may indeed seek to make their asset as valuable as possible and because competitive markets may punish firms that do not maximize profits. The assumption also yields interesting theoretical ECON 500 Part III. Profit Maximization Marginalism and Profit Maximization Firms perform the conceptual experiment of adjusting those variables that can be controlled until it is impossible to increase profits further. This involves, say, looking at the from producing one more unit of output. As long as this incremental profit is positive, the extra output will be produced. When the incremental profit of an activity becomes zero, the firm has pushed output far enough, and it would not be profitable to go further. ECON 500 Part III. Profit Maximization Firms choose the level of output that maximizes The first order condition for a maximum is In order to maximize economic profits, the firm should choose that output for which marginal revenue is equal to marginal cost ECON 500 Part III. Profit Maximization MR=MC condition is only a necessary condition for maximum profits. The second order condition that decreasing at the optimal level of output must also hold for sufficiency. ECON 500 Part III. Profit Maximization ECON 500 Part III. Profit Maximization Marginal Revenue If the firm can sell all it wishes without having any effect on market price, the market price will indeed be the extra revenue obtained from selling one more unit. Total revenue will be linear in output, marginal and average revenue will be equal to price. However, a firm may not always be able to sell all it wants at the prevailing market price. If it faces a downward-sloping demand curve for its product, the revenue obtained from selling one more unit will be less than the price of that unit because, in order to get consumers to take the extra unit, the price of all other units must be lowered. Marginal revenue be below price for any quantity q>1. ECON 500 Part III. Profit Maximization Marginal Revenue If price does not change as quantity increases dp/dq = 0, marginal revenue will be equal to price. In this case we say that the firm is a price taker because its output decisions do not influence the price it receives. On the other hand, if price falls as quantity increases dp/dq < 0, marginal revenue will be less than price. ECON 500 Part III. Profit Maximization ECON 500 Part III. Profit Maximization Marginal Revenue and Elasticity ECON 500 Part III. Profit Maximization Price Marginal Cost Markup Using the MR elasticity relationship and equating MR to MC The more elastic demand becomes, the lower the markup over MC ECON 500 Part III. Profit Maximization Supply Decision of a Price Taking Firm If a firm is sufficiently small such that its output choice has no market on the market price, the marginal revenue becomes equal to the market price. The profit maximizing level of output q* can be found by P = MR = MC As long as at this level of output average variable cost is below the market price, the firm would continue to operate in the short run. The short run supply curve for a price taking firm is the positively sloped -run marginal cost above the point of minimum average variable cost. ECON 500 Part III. Profit Maximization ECON 500 Part III. Profit Maximization Cobb Douglas Example ECON 500 Part III. Profit Maximization I - The supply curve is positively sloped - increases in P cause the firm to produce more because it is willing to incur a higher marginal cost II - The supply curve is shifted to the left by increases in the wage rate, III - The supply curve is shifted outward by increases in capital input IV - The rental rate of capital, v, is irrelevant to short-run supply decisions ECON 500 Part III. Profit Maximization Profit Functions We can represent depending only on the prices that the firm faces by a profit function of the form With the properties: I Homogenous of degree 0 II Non-increasing in input prices III Non-decreasing in output price IV Convex in output prices ECON 500 Part III. Profit Maximization Cobb Douglas Example ECON 500 Part III. Profit Maximization Input Demand ECON 500 Part III. Profit Maximization Comparative Statics for Input Demand Single Input ECON 500 Part III. Profit Maximization Comparative Statics for Input Demand Two Inputs ECON 500 Part III. Profit Maximization ECON 500 Demand and Marginal Revenue for a Competitive Firm DEMAND CURVE FACED BY A COMPETITIVE FIRM A competitive firm supplies only a small portion of the total output of all the firms in an industry. Therefore, the firm takes the market price of the product as given, choosing its output on the assumption that the price will be unaffected by the output choice. In (a) the demand curve facing the firm is perfectly elastic, even though the market demand curve in (b) is downward sloping. Because each firm in a competitive industry sells only a small fraction of the entire industry output, how much output the firm decides to sell will have no effect on the market price of the product. Because it is a price taker, the demand curve d facing an individual competitive firm is given by a horizontal line. The demand curve d facing an individual firm in a competitive market is both its average revenue curve and its marginal revenue curve. Along this demand curve, marginal revenue, average revenue, and price are all equal. Profit Maximization by a Competitive Firm A perfectly competitive firm should choose its output so that marginal cost equals price: MC(q) = MR = P Choosing Output in the Short Run Short-Run Profit Maximization by a Competitive Firm A COMPETITIVE FIRM MAKING A POSITIVE PROFIT In the short run, the competitive firm maximizes its profit by choosing an output q* at which its marginal cost MC is equal to the price P (or marginal revenue MR) of its product. The profit of the firm is measured by the rectangle ABCD. Any change in output, whether lower at q1 or higher at q2, will lead to lower profit. Output Rule: If a firm is producing any output, it should produce at the level at which marginal revenue equals marginal cost. When Should the Firm Shut Down? A COMPETITIVE FIRM INCURRING LOSSES A competitive firm should shut down if price is below AVC. The firm may produce in the short run if price is greater than average variable cost. -run Supply Curve the portion of the marginal cost curve for which marginal cost is greater than average variable cost. THE SHORT-RUN SUPPLY CURVE FOR A COMPETITIVE FIRM 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. THE RESPONSE OF A FIRM TO A CHANGE IN INPUT PRICE When the marginal cost of production for a firm increases (from MC1 to MC2), the level of output that maximizes profit falls (from q1 to q2). The Short-Run Market Supply Curve INDUSTRY SUPPLY IN THE SHORT RUN The short-run industry supply curve is the summation of the supply curves of the individual firms. Because the third firm has a lower average variable cost curve than the first two firms, the market supply curve S begins at price P1 and follows the marginal cost curve of the third firm MC3 until price equals P2, when there is a kink. For P2 and all prices above it, the industry quantity supplied is the sum of the quantities supplied by each of the three firms. Elasticity of Market Supply Es = ( Q/Q)/( P/P) Producer Surplus in the Short Run producer surplus Sum over all units produced by a firm of differences between the market price of a good and the marginal cost of production. PRODUCER SURPLUS FOR A FIRM 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 profit-maximizing 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 = Profit = PRODUCER SURPLUS FOR A MARKET The producer surplus for a market is the area below the market price and above the market supply curve, between 0 and output Q*. =R VC Choosing Output in the Long Run Long-Run Profit Maximization OUTPUT CHOICE IN THE LONG RUN The firm maximizes its profit by choosing the output at which price equals long-run marginal cost LMC. In the diagram, the firm increases its profit from ABCD to EFGD by increasing its output in the long run. The long-run output of a profit-maximizing competitive firm is the point at which long-run marginal cost equals the price. Long-Run Competitive Equilibrium ACCOUNTING PROFIT AND ECONOMIC PROFIT Economic profit takes into account opportunity costs. One such opportunity cost is elsewhere. Accounting profit equals revenues R minus labor cost wL, which is positive. Economic profit , however, equals revenues R minus labor cost wL minus the capital cost, Rk. ZERO ECONOMIC PROFIT zero economic profit A firm is earning a normal return on its investment i.e., it is doing as well as it could by investing its money elsewhere. ENTRY AND EXIT In a market with entry and exit, a firm enters when it can earn a positive longrun profit and exits when it faces the prospect of a long-run loss. long-run competitive equilibrium All firms in an industry are maximizing profit, no firm has an incentive to enter or exit, and price is such that quantity supplied equals quantity demanded. When a firm earns zero economic profit, it has no incentive to exit the industry. Likewise, other firms have no special incentive to enter. A long-run competitive equilibrium occurs when three conditions hold: 1. All firms in the industry are maximizing profit. 2. No firm has an incentive either to enter or exit the industry because all firms are earning zero economic profit. 3. The price of the product is such that the quantity supplied by the industry is equal to the quantity demanded by consumers. LONG-RUN COMPETITIVE EQUILIBRIUM Initially the long-run equilibrium price of a product is $40 per unit, shown in (b) as the intersection of demand curve D and supply curve S1. In (a) we see that firms earn positive profits because long-run average cost reaches a minimum of $30 (at q2). Positive profit encourages entry of new firms and causes a shift to the right in the supply curve to S2, as shown in (b). The long-run equilibrium occurs at a price of $30, as shown in (a), where each firm earns zero profit and there is no incentive to enter or exit the industry. FIRMS HAVING IDENTICAL COSTS To see why all the conditions for long-run equilibrium must hold, assume that all firms have identical costs. Now consider what happens if too many firms enter the industry in response to an opportunity for profit. The industry supply curve will shift further to the right, and price will fall. Only when there is no incentive to exit or enter can a market be in longrun equilibrium. FIRMS HAVING DIFFERENT COSTS Now suppose that all firms in the industry do not have identical cost curves. Perhaps one firm has a patent that lets it produce at a lower average cost than all the others. In that case, it is consistent with long-run equilibrium for that firm to earn a greater accounting profit and to enjoy a higher producer surplus than other firms. If the patent is profitable, other firms in the industry will pay to use it. The increased value of the patent thus represents an opportunity cost to the firm that holds it. It could sell the rights to the patent rather than use it. If all firms are equally efficient otherwise, the economic profit of the firm falls to zero. THE OPPORTUNITY COST OF LAND There are other instances in which firms earning positive accounting profit may be earning zero economic profit. Suppose, for example, that a clothing store happens to be located near a large shopping center. The additional flow of customers can substantially increase historical cost. When the opportunity cost of land is included, the profitability of the clothing store is no higher than that of its competitors. Economic Rent economic rent Amount that firms are willing to pay for an input less the minimum amount necessary to obtain it. A payment for the services of an economic resource which is not necessary as an incentive for its production In competitive markets, in both the short and the long run, economic rent is often positive even though profit is zero. Producer Surplus in the Long Run In the long run, in a competitive market, the producer surplus that a firm earns on the output that it sells consists of the economic rent that it enjoys from all its scarce inputs. FIRMS EARN ZERO PROFIT IN LONG-RUN EQUILIBRIUM In long-run equilibrium, all firms earn zero economic profit. In (a), a baseball team in a moderate-sized city sells enough tickets so that price ($7) is equal to marginal and average cost. In (b), the demand is greater, so a $10 price can be charged. The team increases sales to the point at which the average cost of production plus the average economic rent is equal to the ticket price. When the opportunity cost associated with owning the franchise is taken into account, the team earns zero economic profit. Long-Run Supply Curve Constant-Cost Industry constant-cost industry Industry whose long-run supply curve is horizontal. LONG-RUN SUPPLY IN A CONSTANT COST INDUSTRY In (b), the long-run supply curve in a constant-cost industry is a horizontal line SL. When demand increases, initially causing a price rise, the firm initially increases its output from q1 to q2, as shown in (a). But the entry of new firms causes a shift to the right in industry supply. Because input prices are unaffected by the increased output of the industry, entry occurs until the original price is obtained (at point B in (b)). The long-run supply curve for a constant-cost industry is, therefore, a horizontal line at a price that is equal to the long-run minimum average cost of production. Long-Run Supply Curve Increasing-Cost Industry increasing-cost industry Industry whose long-run supply curve is upward sloping. LONG-RUN SUPPLY IN AN INCREASING COST INDUSTRY In (b), the long-run supply curve in an increasing-cost industry is an upward-sloping curve SL. When demand increases, initially causing a price rise, the firms increase their output from q1 to q2 in (a). In that case, the entry of new firms causes a shift to the right in supply from S1 to S2. Because input prices increase as a result, the new long-run equilibrium occurs at a higher price than the initial equilibrium. In an increasing-cost industry, the long-run industry supply curve is upward sloping. The Effects of a Tax EFFECT OF AN OUTPUT TAX OUTPUT marginal cost curve by the amount of the tax. The firm will reduce its output to the point at which the marginal cost plus the tax is equal to the price of the product. EFFECT OF AN OUTPUT TAX ON INDUSTRY OUTPUT An output tax placed on all firms in a competitive market shifts the supply curve for the industry upward by the amount of the tax. This shift raises the market price of the product and lowers the total output of the industry.
