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Chapter 19: Problem 4

Suppose a firm's costs for dollars spent on product differentiation (or advertising) activities \((z)\) and quantity \((q)\) can be written as \\[ T C=g(q)+z \quad g^{\prime}(q)>0 \\] and that its demand function can be written as \\[ q=q(P, z) \\] Show that the firm's profit-maximizing choices for \(P\) and \(z\) will result in spending a share of total revenues on \(z\) given by \\[ \frac{z}{P q}=-\frac{e_{q, z}}{e_{q, P}} \\]

Short Answer

Expert verified
In this exercise, we are given a firm's cost function and demand function, and our goal is to find the profit-maximizing choices for price (P) and dollars spent on product differentiation or advertising (z). We found that the firm's profit-maximizing choices will result in spending a share of total revenues on z given by the ratio of the demand elasticities, which can be represented by the equation: \\[ \frac{z}{Pq} = -\frac{e_{q,z}}{e_{q,P}} \\]

Step by step solution

01

Write the profit function

The profit function can be written as the total revenue minus total cost: \\[ \pi(P, z) = P q(P, z) - g(q(P, z)) - z \\]
02

Find the first-order conditions for profit maximization

To find the profit-maximizing choices for \(P\) and \(z\), we need to find the first-order conditions of the profit function with respect to \(P\) and \(z\). To do this, we will differentiate \(\pi(P, z)\) with respect to \(P\) and \(z\), and then set both derivatives equal to zero: \\[ \frac{\partial \pi}{\partial P} = q(P, z) + P\frac{\partial q(P, z)}{\partial P} - \frac{\partial g(q(P, z))}{\partial q(P, z)}\frac{\partial q(P, z)}{\partial P} = 0 \\] \\[ \frac{\partial \pi}{\partial z} = P\frac{\partial q(P, z)}{\partial z} - \frac{\partial g(q(P, z))}{\partial q(P, z)}\frac{\partial q(P, z)}{\partial z} - 1 = 0 \\]
03

Solve for the desired ratio

Divide the second first-order condition by the first one to obtain: \\[ -\frac{1}{z} = \frac{\frac{\partial q(P, z)}{\partial z} / \frac{\partial q(P, z)}{\partial P}}{q(P, z) / \frac{\partial q(P, z)}{\partial P}} \\] Now, use the definition of the elasticity of demand to rewrite the ratio as: \\[ \frac{z}{Pq} = -\frac{e_{q,z}}{e_{q,P}} \\] This equation shows that the firm's profit-maximizing choices for \(P\) and \(z\) will result in spending a share of total revenues on \(z\) given by the ratio of the demand elasticities.

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Most popular questions from this chapter

Assume for simplicity that a monopolist has no costs of production and faces a demand curve given by \\[ Q=150-P \\] a. Calculate the profit-maximizing price-quantity combination for this monopolist. Also calculate the monopolist's profits. b. Suppose a second firm enters the market. Let \(q_{1}\) be the output of the first firm and \(q_{2}\) the output of the second. Market demand is now given by \\[ q_{1}+q_{2}=150-P \\] Assuming this second firm also has no costs of production, use the Cournot model of duopoly to determine the profit-maximizing level of production for each firm as well as the market price. Also calculate each firm's profits. c. How do the results from parts (a) and (b) compare to the price and quantity that would prevail in a perfectly competitive market? Graph the demand and marginal revenue curves and indicate the three different price-quantity combinations on the demand curve.

A monopolist can produce at constant average (and marginal) costs of \(A C=M C=5 .\) The firm faces a market demand curve given by \\[ Q=53-P \\] a. Calculate the profit-maximizing price-quantity combination for this monopolist. Also calculate the monopolist's profits. b. Suppose a second firm enters the market. Let \(q_{1}\) be the output of firm 1 and \(q_{2}\) the output of firm 2. Market demand now is given by \\[ q_{1}+q_{2}=53-P \\] Assuming firm 2 has the same costs as firm \(1,\) calculate the profits of firms 1 and 2 as functions of \(q_{1}\) and \(q_{2}\) c. Suppose (after Cournot) each of these two firms chooses its level of output so as to maximize profits on the assumption that the other's output is fixed. Calculate each firm's "reaction function," which expresses desired output of one firm as a function of the other's output. d. On the assumption in part (c), what is the only level for \(q_{1}\) and \(q_{2}\) with which both firms will be satisfied (what \(q_{1}, q_{2}\) combination satisfies both reaction curves)? e. With \(q_{1}\) and \(q_{2}\) at the equilibrium level specified in part (d), what will be the market price, the profits for each firm, and the total profits earned? f. Suppose now there are \(n\) identical firms in the industry. If each firm adopts the Cournot strategy toward all its rivals, what will be the profit- maximizing output level for each firm? What will be the market price? What will be the total profits earned in the industry? (All these will depend on \(n .\) ) g. Show that when \(n\) approaches infinity, the output levels, market price, and profits approach those that would "prevail" in perfect competition.

Suppose a firm is considering investing in research that would lead to a cost- saving innovation. Assuming the firm can retain this innovation solely for its own use, will the additional profits from the lower (marginal) costs be greater if the firm is a competitive price taker or if the firm is a monopolist? Develop a careful graphical argument. More generally, develop a verbal analysis to suggest whether competitive or monopoly firms are more likely to adopt cost-saving innovations. (For an early analysis of this issue, see W. Fellner, "The Influence of Market Structure on Technological Progress," Quarterly Journal of Economics [November 1951]: \(560-567\).)

One way of measuring the size distribution of firms is through the use of the Herfindahl Index, which is defined as \\[ \boldsymbol{H}=\sum \boldsymbol{\alpha}_{i}^{2} \\] where \(\alpha_{i}\) is the share of firm \(i\) in total industry revenues. Show that if all firms in the industry have constant returns-to-scale production functions and follow Cournot output decisions (Equation 19.10 ), the ratio of total industry profits to total revenue will equal the Herfindahl Index divided by the price elasticity of demand. What does this result imply about the relationship between industry concentration and industry profitability?

Suppose demand for crude oil is given by \\[ Q=-2,000 P+70,000 \\] where \(Q\) is the quantity of oil in thousands of barrels per year and \(P\) is the dollar price per barrel. Suppose also that there are 1,000 identical small producers of crude oil, each with marginal costs given by \\[ M C=q+5 \\] where \(q\) is the output of the typical firm. a. Assuming each small oil producer acts as a price taker, calculate the market supply curve and the market equilibrium price and quantity. b. Suppose a practically infinite supply of crude oil is discovered in New Jersey by a wouldbe price leader and can be produced at a constant average and marginal cost of \(\$ 15\) per barrel. Assuming the supply behavior of the competitive fringe described in part (a) is not changed by this discovery, how much should the price leader produce in order to maximize profits? What price and quantity will now prevail in the market? c. Graph your results. Does consumer surplus increase as a result of the New Jersey oil discovery? How does consumer surplus after the discovery compare to what would exist if the New Jersey oil were supplied competitively?
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