So far we have studied the firm's pricing decision assuming that it sells its output in an outside market, i.e., to consumers or to other firms. Many firms, however, are vertically integrated-they contain several divisions, with some divisions producing parts and components that other divisions use to produce the finished product) For example, each of the major U.S. automobile companies has "upstream" divisions that produce engines, brakes, radiators, and other components that the "downstream" divisions use to produce the finished products. Transfer pricing refers to the valuation of these parts and components within the firm. Transfer prices are internal prices at which the parts and components front upstream divisions are "sold" to downstream divisions. Transfer prices must be chosen correctly because they are the signals that divisional managers use to determine output levels.

This appendix shows how a profit-maximizing firm chooses its transfer prices and divisional output levels. We will also examine other issues raised by vertical integration. For example, suppose a computer firm's upstream division produces memory chips that are used by a downstream division to produce the final product. If other firms also produce these chips, should our firm obtain all its chips from the upstream division, or should it also buy some on the outside market? Should the upstream division produce more chips than are needed by the downstream division, selling the excess in the market? And how should the firm coordinate the upstream and downstream divisions? In particular, can we design incentives for the divisions, so that the firm's profit is maximized?

We begin with the simplest case-there is no outside market for the output of the upstream division, i.e., the upstream division produces a good that is neither produced nor used by any other firm. Next we consider what happens when there is an outside market for the upstream division's output.

Transfer Pricing When There Is No Outside Market

Consider a firm that has three divisions: Two upstream divisions produce inputs to a downstream processing division. The two upstream divisions pro-

A firm is horizontally integrated when it has several divisions that produce the same product or closely related products. Many firms are both vertically and horizontally integrated.

duce quantities Q\ and Q2 and have total costs C\(Q\) and CiiQi). The downstream division produces a quantity Q using the production function

where K and L are capital and labor inputs, and Qi and Qi are the intermediate inputs from the upstream divisions. Excluding the costs of the inputs Qi and <22, the downstream division has a total production cost Cj(Q). The total revenue from sales of the final product is R{Q).

We assume there are no outside markets for the intermediate inputs Qi and Q2. (They can be used only by the downstream division.) Then the firm has two problems. First, what quantities Qi, Q2, and Q maximize its profit? Second, is there an incentive scheme that will decentralize the firm's management? In particular, is there a set of transfer prices Pi and P2, so that if each division maximizes its own divisional profit, the profit of the overall firm will also be maximized?

To solve these problems, note that the firm's total profit is tt(Q) = R(Q) - Cd(Q) - C,(Qi) - C2(Q2) (All.l)

Now, what is the level of Qi that maximizes this profit? It is the level at which the cost of the last unit of Q\ is just equal to the additional revenue it brings to the firm. The cost of producing one extra unit of Qi is the marginal cost AG/AQj = MCi. How much extra revenue results from the unit? An extra unit of Qi allows the firm to produce more final output Q of an amount AQ/AQj = MP,, the marginal product of Qi. An extra unit of final output results in additional revenue AR/AQ = MR,but it also results in additional cost to the downstream division, of an amount AQ/AQ = MQ. Thus, the net marginal revenue NMRi that the firm earns from an extra unit of Qi is (MR - MCd)MPi. Setting this equal to the marginal cost of the unit, we obtain the following rule for profit maximization:2

Going through the same steps for the second intermediate input gives

Note from equations (A11.2) and (All.3) that it is incorrect to determine the firm's final output level Q by setting marginal revenue equal to marginal cost for the downstream division, i.e., by setting MR = MG/. Doing so ignores the cost of producing the intermediate input. (MR exceeds MG/ because this cost

Using calculus,we can obtain this by differentiating equation (All.l) with respect to Q\\ cWdQ, = (dR/dQ)(3Q/3Q,) - (dQ/dQ)(3Q/3Q,) - dC./dQ, = (MR - MQMPi - MC, Setting dir/dQ = 0 maximize profit gives equation (All.2).

is positive.) Also, note that equations (A11.2) and (A11.3) are standard conditions of marginal analysis-the output of each upstream division should be such that its marginal cost is equal to its marginal contribution to the profit of the overall firm.

Now, what transfer prices Pi and P2 should be "charged" to the downstream division for its use of the intermediate inputs? Remember that if each of the three divisions uses these transfer prices to maximize its own divisional profit, the profit of the overall firm should be maximized. The two upstream divisions will maximize their divisional profits, tti and ir2/ which are given by iri = P1Q1 - Ci(Q0

Since the upstream divisions take Pi and P2 as given, they will choose Q\ and Q2 so that Pi = MCi and P2 = MC2. Similarly, the downstream division will maximize ir(Q) = R(Q) - QQ) - P1Q1 - P2Q2

Since the downstream division also takes Pi and P2 as given, it will choose Q\ and Q2 so that

Note that by setting the transfer prices equal to the respective marginal costs (Pi = MCi and P2 = MC2), the profit-maximizing conditions given by equations (All.2) and (All.3) will be satisfied. We therefore have a simple solution to the transfer pricing problem: Set each transfer price equal to the marginal cost of the respective upstream division. Then when each division is told to maximize its own profit, the, quantities Qi and Q2 that the upstream divisions will want to produce will be the same quantities that the downstream division will want to "buy," and they will maximize the total profit of the firm.

We can illustrate this graphically with the following example. Race Car Motors, Inc., has two divisions. The upstream Engine Division produces engines, and the downstream Assembly Division puts together automobiles, using one engine (and a few other parts) in each car. In Figure All.l, the average revenue curve AR is Race Car Motors' demand curve for cars. (Note that the firm has monopoly power in the automobile market) MCa is the marginal cost of assembling automobiles, given the engines (i.e., it does not include the cost of the engines). Since the car requires one engine, the marginal product of the engines is one, so that the curve labeled MR - MCa is also the net marginal revenue curve for engines: NMRs = (MR/ - MC/i)MP/ = MR - MCa.

The profit-maximizing number of engines (and number of cars) is given by the intersection of the net marginal revenue curve NMRe with the marginal

FIGURE A11.1 Race Car Motors, Inc. The firm's upstream division should produce a quantity of engines Qe that equates its marginal cost of engine production MO: with the downstream division's net marginal revenue of engines NMRi. Since the firm uses one engine in every car, NMRi is the difference between the marginal revenue from selling cars and the marginal cost of assembling them, i.e., MR - MGi. The optimal transfer price for engines Pe equals the marginal cost of producing them. Finished cars are sold at price Pa.

cost curve for engines MC/;. Having determined the number of cars it will produce, and knowing its divisional cost functions, the management of Race Car Motors can now set the transfer price Pe that correctly values the engines used to produce its cars. It is this transfer price that should be used to calculate divisional profit (and year-end bonuses for the divisional managers).

Now suppose there is a competitive outside market for the intermediate good produced by an upstream division. Since the outside market is competitive, there is a single market price at which one can buy or sell the good. Therefore, the marginal cost of the intermediate good is simply the market price. Since the optimal transfer price must equal marginal cost, it must also equal the competitive market price.

To see this, suppose there is a competitive market for the engines that Race Car Motors produces. If the market price is low. Race Car Motors may want to buy some or all of its engines in the market; if it is high, it may want to sell engines in the market. Figure All.2 illustrates the first case. For quantities below Qej, the upstream division's marginal cost of producing engines MC/; is below the market price Pe,m, and for quantities above Qej it is above the market price. The firm should obtain engines at least cost, so the marginal cost of engines MC*e is the upstream division's marginal cost for quantities up to Qe,i and the market price for quantities above Qe,i. Note that Race Car Motors uses more engines and produces more cars than it would have had there not been an outside engine market. The downstream division now buys Qe,2 engines and produces an equal number of automobiles. However, it "buys" only Qe,i of these engines from the upstream division, and buys the rest on the open market.

FIGURE A11.2 .Race Car Motors Buys Engines in a Competitive Outside Market.

The firm's marginal cost of engines VIC*/: is the upstream division's marginal cost for quantities up to Qej and the market price PE,M for quantities above Qe,i. The downstream division should use a total of Qe,2 engines to produce an equal number of cars; then the marginal cost of engines equals net marginal revenue. Qe,2- Qej of these engines are bought in the outside market. The upstream division "pays" the downstream division the transferprice Pe,m for the remaining Qej engines.

FIGURE A11.3 Race Car Motors Sells Engines in a Competitive Outside Market. The optimal transfer price is again, the market price Pem This price is above the point at which MCe intersects NMRe so the upstream division sells some of its engines in the Outside market. The upstream division produces Qe,i engines, the quantity at which MCe equals Pe,m. The downstream division uses only Qe,2 of these, the quantity at which NMRe equals Pe,m. Compared with Figure All.l, in which there is no outside market, more engines but fewer cars are produced.

It might appear strange that Race Car Motors should have to go into the open market to buy engines, when it can make those engines itself. If it made all its own engines, however, its marginal cost of producing engines would exceed the competitive market price, and although the profit of the upstream division would be higher, the total profit of the firm would be lower.

Figure A11.3 shows the case where Race Car Motors sells engines in the outside market. Now the competitive market price Pe,m is above the transfer price that the firm would have set had there not been an outside market. In this case the upstream Engine Division produces Qe,i engines, but only Qe,2 engines are used by the downstream division to produce automobiles. The rest are sold in the outside market at the price Pe,m.

Note that compared with a situation in which there is no outside engine market. Race Car Motors is producing more engines but fewer cars. Why not produce this larger number of engines, but use all of them to produce more cars? Because the engines are too valuable. On the margin, the net revenue that can be earned from selling them in the outside market is higher than the net revenue from using them to build additional cars.

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