Profit Maximization

Chapter 6 Profit Maximization 6.1 Introduction In this section we begin our analysis of the firm's behavior in product markets. The behavioral assumption that underlies the analysis in this chapter is that of profit maximization, but we will see that there are two ways to study profit maximization. We will study the firm's scale decision first. That is, once the firm has selected its cost minimizing factor inputs, it must look to the goods market and decide what quantity to supply. The two stage problems uses the cost function to summarize conditions in the factor markets, and in the second stage assumes firms maximize profits by selecting the optimal quantity given factor market prices and its technol-ogy as summarized by the cost function and the goods price. In this chapter we will consider perfectly competitive goods markets and we will analyze the firm's price taking assumption carefully. Then, we will study the one shot profit maximization problem directly. In this alterna-tive formulation, firms select quantities of factor inputs to maximize the difference between total revenues and total costs. These demand functions are known as factor demand functions. A key result is that under the price taking assumption the answer to both formulations of the profit max-imization problem coincide. However, the two stage cost-scale decision is more general and applies even when the goods market is not competitive. The one-shot profit maximization problem, on the other hand, can only be used when both factor and goods markets are competitive. After establish-ing the equivalence of both problems we will study the properties of the firm's factor demand fuctions. Then we will derive the profit function which tells us the maximum profit the firm can obtain, given factor market prices, and goods market prices. Don't forget to address payments to factors of production Euler's theo-rem etc. Use Euler's theorem to explain quasi-rents as being inexistent in the long run, rents however may persist in the long run like entrepreneurial ability, or uncertainty as a payment for risk. In the applications section we will study two cases in which profit maxi-mization may not be a good behavioral assumption. One instance is that of discrimination, in which firms are interested in maximizing some function not only of profits but of other variables such as race or gender as well. A second instance is that of the non-profit sector. 6.2 The Price Taking Assumption In this chapter we study the behavior of a profit maximizing firm that operates in a competitive market. The price taking assumption requires firms to take price as given when they make their output decisions. One way of formalizing this idea is to consider the elasticity of the market demand curve facing the firm, which is defined as: dy P dpy A firm that operates in a perfectly competitive market assumes that the market demand curve it faces is infinitely elastic. Essentially, this assump-tion means that the firm can not influence price by its quantity choice. A firm that operates in a perfectly competitive market assumes that the demand curve it faces is flat. Therefore, we require that dy = 0, or equiv-alently, 0. If this assumption holds, then e must be infinite since dy p both quantities are inversely related. This implies that the firm can sell any amount it wishes at the given market price, but nothing at a higher price. The firm can not take advantage of the elasticity of market demand when it makes its output decision. It is in this sense that the firm is a price taker. The assumption of perfect competition is a theoretical construct de signed to approximate the behavior of firms under certain conditions and may be inappropriate in certain industries. Therefore, it is desirable to If we totally differentiate the equilibrium condition with respect to y, we obtain: d/dy (p) = (dP)/(d * Q ^ D) * (d/dp (q) * d/dy (p) + d/dp (y) * d/dy (p)) which we can simplify to obtain: d/dy (p) = (dP)/(d * Q ^ D) * (d/dp (q) * d/dy (p) + 1) Now, solving for d/dy (p) we obtain: dp dy = dP dQ^ D 1- dP^ * d * Q ^ B * (dq)/(dp) . Under suitable mathematical assumptions we obtain: d/dp (y) = (1 - (dP)/(d * Q ^ D) * d/dp (q))/((dP)/(d * Q ^ D)) which simplifies to: d/dp (y) = d/dP (Q ^ D) - d/dp (q) Notice that the left hand side is the derivative of the demand function facing the firm, while on the right hand side we have the derivative of the demand function with respect to price, and the derivative of the supply of all other firms in the industry with respect to price. Now, we must convert the above equation into elasticity. Recall that in the text we defined epsilon ^ u = d/dp (y) * p/y therefore, multiplying both sides by p/y we obtain: epsilon ^ d = p/y * d/dP (Q ^ D) - p/y * d/dp (q) Notice that, p/y * d/dP (Q ^ D) = p/y * d/dP (Q ^ D) * (Q ^ D)/(Q ^ D)