Outsourcing and vertical integration in a competitive industry.

Ciliberto, Federico ; Panzar, John C.

1. Introduction

We develop a partial equilibrium, competitive framework of a (potentially) vertically integrated industry. In our model there are three types of firms: upstream firms that use primary factors to produce an intermediate good; downstream firms that use primary factors and intermediate goods to produce a final good; and vertically integrated firms that do both. We establish conditions under which vertically integrated firms exist and outsource (part of) the production of the intermediate input. Specifically, we ask the following questions: Why do competitive firms vertically integrate? What changes in the economy and, in particular, in the demands of the intermediate and final goods can explain the vertical disintegration or integration of competitive industries?

We build on the literature of multiproduct firms in competitive markets. That literature illustrates market structures when there are two final goods: A and B. This article uses a very similar framework, with the distinction being that good B is an intermediate good in the production of good A. The results hinge on the concept of economies of scope. The comparative statics show how the industry structure changes in response to changes in the external demand of final goods and intermediate goods and also to changes in the cost structure. In particular, we show that if economies of vertical scope are present, then the vertical organization of the industry is determined by the relative ratio of demands of intermediate and final output and by the ratio of costs. As the demand external to the industry for the intermediate output changes, the equilibrium configuration of the industry changes as well. To develop our analysis we use the simplest functional forms for cost and demand functions that are necessary to demonstrate the results. We then justify how our results carry forward under more general forms. For this reason, we hope that our simplifications are considered as a point of strength rather than a point of critique.

This article formalizes George Stigler's (1951) take on vertical integration in his article "The division of labor is limited by the extent of the market." Stigler shows that when the industry grows, the vertically integrated firm outsources (part of) the production of the intermediate good, whose production process is subject to increasing costs. Stigler also shows that when one of the production processes displays increasing returns to scale, it will be turned over to specialists as the market grows. The specialists cannot charge more than the average incremental cost that the vertically integrated firm would face if it were producing the intermediate input in-house. Thus, the specialists face a negatively sloped elastic demand for the intermediate input. As the market continues to grow, the new industry will become competitive. Stigler limits the analysis to the case in which there are no economies of vertical scope and one of the production processes is subject to increasing costs.

To formalize George Stigler's (1951) take oil vertical integration we extend the competitive framework developed by MacDonald and Slivinski (1987). First, we allow for economies of vertical scope to exist, and we study when outsourcing occurs in the industry. In our model (partial) outsourcing of the production plays a critical role in determining which firms are present in equilibrium. Clearly, there is no scope for outsourcing in MacDonald and Slivinski's horizontal multiproduct industry. Second, we determine the set of output combinations that vertically competitive firms choose in equilibrium. From this set of output combinations we are able to conclude whether there are only vertically integrated firms in the industry or whether there are also upstream or downstream firms. Third, we study the industry when another industry sells the intermediate good into the market. We show how the vertical structure of the industry depends on the ability of the producers of intermediate output to compete at lower prices than the upstream firms.

Before moving on to the analysis, we want to stress how our model explains vertical integration patterns from a technological perspective. In this sense, our approach stands in contrast to the incomplete contracting literature (Williamson 1985: Hart and Moore 1988), which often dismisses technological explanations of vertical integration.

In section 2 of the article we introduce the notions of economies of vertical scope. In section 3 we introduce the notion of economies of scope. In section 4 we define a vertical competitive equilibrium. We also study the vertical structure and equilibrium of the industry when vertically integrated firms cannot outsource the production of the intermediate input and when the industry produces a surplus of the intermediate output. In section 5 of the article we still consider the industry as producing a surplus of the intermediate input, but the vertically integrated firms now outsource the production of the input. In section 6 we merge the results from section 3 and 4, and we present a comparative statics analysis to show how the industry structure changes with changes in the aggregate output demands. Finally, in section 7 we present the case when the industry buys the intermediate input. Section 8 summarizes the results and concludes the article.

2. Economies of Specialization and of Vertical Scope

The industry deals with three categories of commodities: a final good, y [greater than or equal to] 0; an intermediate product, k; and a vector of primary factors, x [less than or equal to] 0. This netput notation helps facilitate the analysis. Let p, r, and w denote the associated market prices. Firms are price takers in all markets, and there is free entry in the markets of the final good. There is free entry in the market of the intermediate product when it is an output, [p.sup.e] denotes the equilibrium price of the final good. [r.sup.e] denotes the equilibrium price of the intermediate product when it is an output.

The firm is upstream if it only produces k, and it is downstream if it only produces y. For any firm under study, the final good is always an output (y [greater than or equal to] 0), and the primary factors are always inputs (x [less than or equal to] 0). However, the intermediate product is always an output for upstream firms ([k.sup.U] [greater than or equal to] 0) and an input for downstream firms ([k.sup.D ] [greater than or equal to] 0). The net supply of the intermediate product for a vertically integrated firm may be positive, negative, or zero. That is, an integrated firm may operate a process that generates more, less, or exactly the amount of the intermediate product that it requires to produce a specified level of the final good.

More formally, firms are assumed to operate one of the three following production sets:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In this formulation, f and g are traditional production functions, and H is a production transformation function.

Next, we need to develop definitions that reflect intuitive definitions of the technological advantages of specialization and integration. First, we consider the production of the intermediate good. The notion of specialization in the production is related to the ability of firms to produce one single good more efficiently than can multiproduct firms if they only produce the same good. The following two definitions capture this notion of specialization:

DEFINITION 1. ECONOMIES OF UPSTREAM SPECIALIZATION. When none of the final good is produced, the upstream technology is at least as good as the integrated technology. That is, [Y.sub.U] [contains or equal to] [Y.sub.I] for y = 0. Equivalently, this means that f(-x) [greater than or equal to] [max.sub.K ]{k|H (x, 0, k) [less than or equal to] 0}.

DEFINITION 2. ECONOMIES OF DOWNSTREAM SPECIALIZATION. When the intermediate good is not an output, the downstream technology is at least as good as the integrated technology. That is, [Y.sub.D] [contains or equal to] [Y.sub.I] for k [less than or equal to] 0. Equivalently, g (-x, -k) [greater than or equal to] [max.sub.y] {y|H (x, y, k) [less than or equal to ] 0, k [less than or equal to ] 0}.

The notion of downstream specialization is less intuitive than that of upstream specialization. To see why, observe that only in exceptional cases is the minimum efficient scale of producing an intermediate proportional to that of producing the final good, so that the technologies are perfectly matched in terms of their outputs. Hence, a vertically integrated firm normally produces too much or too little of the intermediate output for its own use. This is why we expect economies of downstream specialization to exist.

When the vertically integrated firm can sell the intermediate good as an output, then it can fully exploit economies of scope in the production of the final and intermediate goods. We define economies of vertical scope next.

DEFINITION 3. ECONOMIES OF VERTICAL SCOPE. A collection of upstream, downstream, and integrated technologies are said to exhibit economies of vertical scope if and only if [Y.sub.I] [contains or equal to] [Y.sub.U] + [Y.sub.D]. Equivalently, given [k.sup.U] [greater than or equal to] 0, [k.sup.D] [less than or equal to ] 0, [x.sup.U] [less than or equal to ] 0, [x.sup.D] [less than or equal to ] 0, and [y.sup.D] [greater than or equal to] 0, such that [k.sup.U] [less than or equal to ] f (- [x.sup.U]) and [y.sup.D] [less than or equal to] g(- [x.sup.D], [k.sup.D]), then H ([x.sup.U] + [x.sup.D], [y.sup.D], [k.sup.U] + [k.sup.D]) [less than or equal to ] 0.

We expect to observe economies of vertical scope when producing a final good lowers the costs of producing the intermediate output. For example, if learning by doing is important in designing the best intermediate input for the final good, then producing them together can be cheaper than producing them separately. However, if the intermediate and final goods are standardized, then economies of vertical scope are less likely to exist.

3. Vertical Competitive Equilibrium

There are two questions we need to address: First, are there vertically integrated firms in the vertical competitive equilibrium? Second, can there be both vertically integrated and specialized firms in the equilibrium? The first question is easily answered. Panzar and Willig's (1981) result, which claims that economies of scope are necessary and sufficient for the existence of multiproduct firms in equilibrium, is still valid here. (1) Thus, we need to address the second question whether there are also upstream and downstream firms in equilibrium, and we need to determine conditions that explain which type of firm is present in equilibrium.

To address this question, we follow MacDonald and Slivinski (1987) and introduce some structure in the model. We denote the aggregate demand of final output by Y and the net demand of the intermediate input by K. If K > 0, then K is sold (we might say "exported" to make it simpler to understand) to another industry, which uses the same intermediate input. If K < 0, then K is bought ("imported") from another industry.

In order to simplify the analysis, we assume that K and Y are exogenous. (2) However, we could assume that Y (p) = z (c - dp) and K = z (a - br), where z is a large positive integer, and the results of this article would still carry on. This latter weaker assumption would ensure that there is always an interior solution by which all firms of the same type produce the same combination of outputs. The prices p and k are still endogenous.

Let [k.sub.p] be the amount of intermediate input that the vertically integrated firm purchases on the market, and let [k.sub.i] be the amount of intermediate output that the vertically integrated firm produces for its own use. Then [k.sub.i] + [k.sub.p] = k(y), where k(y) is the total amount of k that the vertically integrated firm needs to produce y. We define by [k.sub.s] the surplus of intermediate input that the vertically integrated firm may produce. We define by [k.sup.U] the amount of intermediate output produced by the specialized upstream firms. We define by [y.sup.D] the amount of final output produced by the specialized downstream firms.

Let [n.sup.I], [n.sup.D], and [n.sup.U] denote the number of vertically integrated, downstream, and upstream firms, respectively. Next, we define a competitive vertical equilibrium: (3)

DEFINITION 4. A competitive vertical equilibrium with free entry is given by a pair of prices for the final and intermediate goods ([p.sup.e], [r.sup.e]) and two market clearing conditions Y = [n.sup.I][y.sup.I] + [n.sup.D][y.sup.D] and K = ([K.sup.I.sub.s] - - [K.sup.I.sub.p])[n.sup.I] + [k.sup.U][n.sup.U], such that the pattern of integration of an industry satisfies the following conditions: (i) No active firm can benefit from altering its choice of the stages at which it operates; (ii) No active firm can profit by changing output levels; (iii) No potential firm finds it profitable to enter into the industry; (iv) All the firms in the industry, both incumbent and potential entrants, are price takers.

4. Equilibrium without Outsourcing

For sake of simplicity of exposition, we assume that the vertically integrated firm cannot buy intermediate input on the market. In section 5 we relax this assumption and let the vertically integrated firm buy some of the intermediate input on the market (this is what we call outsourcing), and we study how the comparative statics results change.

In this section we also maintain that [K.sub.s] > 0; That is, the industry produces a surplus of intermediate outputs. The industry sells its surplus of intermediates to another industry.

Following MacDonald and Slivinski (1987), we assume that k = g(x) - [square root of x], and y = min {[square root of [x.sub.y]], k}. Hence, [k.sub.i] + [k.sub.p] = y. Downstream, upstream, and vertically integrated firms face fixed costs equal to [F.sup.D], [F.sup.U], and [F.sup.I], respectively. (4) Then we can write their cost functions as follows:

[C.sup.U] (k; w) = [F.sup.U] + [wk.sup.2],

[C.sup.D] (y; w, r) = [F.sup.D] + ry + [wy.sup.2],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The only interesting case is when [F.sup.I] < [F.sup.U] + [F.sup.D], [F.sup.U] < [F.sup.I], and [F.sup.D] < [F.sup.I]. If [F.sup.I] > [F.sup.U] + [F.sup.D], then there are no economies of scope, and only specialized firms will be present in equilibrium. To see why, observe that there are marginal rivalries (as opposed to complementarities) between the production of y and k in a vertically integrated firm, which are captured by the interaction term [2wyk.sub.s]. It is the trade-off between marginal rivalries and economies of scope in the fixed costs that leads to the simultaneous presence of vertically integrated and specialized firms. This is the critical element of the article and does not depend on the functional forms we use and on the fact that we limit the analysis to economies of scope only in the fixed costs. One could envisage a situation in which economies of scope are in the marginal costs (i.e., cost complementarities), while rivalries are in the fixed costs. As long as there exists a trade-off, then more than one type of firm can exist in equilibrium.

The Equilibrium Conditions

If downstream and upstream firms are present in equilibrium, then the prices of the intermediate input k and of the final output y are uniquely pinned down and are given by re = 2[square root of [F.sup.U]w] and [p.sup.e] = 2[square root of [F.sup.U]/w] + 2 2[square root of [F.sup.D]w]. These are the average costs faced by the specialized firms at their minimum efficient scales. [k.sup.mes] = [square root of [F.sup.U]/w] and [y.sup.mes] = [square root of F.sup.D]/w] are the minimum efficient scale outputs for the upstream and downstream firms, respectively. Since the industry is perfectly competitive, firms will charge prices where their average costs are the lowest. We now lay out the conditions under which a vertically integrated firm exists.

Vertically integrated firms cannot charge higher prices than the vertically specialized firms in a perfectly competitive industry. Thus, [r.sup.e] [less than or equal to] 2 and [p.sup.e] [less than or equal to] 2[square root of [F.sup.U]w]. Moreover, competitive vertically integrated firms must produce locally at constant multiproduct returns to scale or else they could increase their profits with a marginal increase or decrease of their production outputs. Hence, the following must hold:

CRS : FI = w[([y.sup.I] + [k.sup.I.sub.s]).sup.2] + w[([y.sup.I]).sup.2].

Competitive vertically integrated firms must also choose production outputs where the product-specific economies of scale display either increasing or constant returns to scale, otherwise they would be facing decreasing returns to scale and would have an incentive to lower their output production. Hence,

SPEC : [y.sup.I] [greater than or equal to] [square root of [F.sup.I] - [F.sup.U]/2w].

Finally, the cost for a vertically integrated firm to produce a combination of outputs must not be larger than those that specialized firms jointly face if they produce at their minimum efficient scale or else the specialized firms would be able to charge lower prices than the vertically integrated firm can afford. Hence,

INT: 2([square root of [F.sup.U]w] + [square root of [F.sup.D]w])[y.sup.I] + 2 [square root of [F.sup.U]w] [k.sup.I.sub.s] [greater than or equal to] [F.sup.I] + w[([y.sup.I] + [k.sup.I.sub.s]).sup.2] + w[([y.sup.I]).sup.2]

These three conditions are very intuitive and lead to clear comparative statics predictions that depend on [F.sup.U], [F.sup.D], and [F.sup.I], as shown in Figure 1.

The curve CRS in Figure 1 corresponds to the combinations of outputs at which the vertically integrated firms are producing at constant returns to scale. Any equilibrium combination of outputs produced by vertically integrated firms must lie on this curve. All points on the left hand side of SPEC are points at which the vertically integrated firm is producing at product-specific decreasing returns to scale. Finally, only the points inside the ellipsis INT are those at which the vertically integrated firm is producing at lower costs than specialized firms. If vertically integrated firms are present in equilibrium then they must produce combinations of outputs that are in the bold portion of CRS.

[FIGURE 1 OMITTED]

Figure 1 presents the possible equilibrium outputs produced by the vertically integrated firms for different values of the ratio [F.sup.U]/[F.sup.D]. Figure la presents the set of equilibrium combinations that the vertically integrated firms can choose when the fixed costs of the upstream firm are very high relative to those of the downstream firm; Figure lb through d presents the same set of points for decreasing values of the ratio FU/FD. As the ratio FU/FD decreases, the set of combinations at which the vertically integrated firm can be in equilibrium shifts to the southeast. The last two cases in which the curve INT intersects the y-axis (Figure 1c, d) occur when [F.sup.U] + [F.sup.D] + 2 [square root of [F.sup.U][F.sup.D]] > 2[F.sup.I]. (5)

Determination of the Equilibrium Structure

In the previous section we have determined the set of combination of outputs that vertically integrated firms must produce to exist in equilibrium. The aggregate demands of the intermediate and final goods determine which of the possible combinations vertically integrated firms produce in equilibrium.

The ratio of exogenous aggregate demands, [K.sup.D.sub.s]/[Y.sup.D], can be plotted on the same Figure 1a and b. Depending on where this ratio lies, either on the marked segment or outside of it, a different market structure will occur. To see this, denote by [K.sup.A.sub.s]/[Y.sup.A] the line that goes through the point A and by [K.sup.B.sub.s]/[Y.sup.B] the line that goes through the point B in Figure 2a and b.

Consider first the case in which the output ratio [K.sup.D.sub.s]/[Y.sup.D] crosses the marked segment: That is to say, [K.sup.A.sub.s]/[Y.sup.A] [less than or equal to] [K.sup.D.sub.s]/[Y.sup.D] [less than or equal to ] [K.sup.B.sub.s]/[Y.sup.B] in Figure 2a or 0 [less than or equal to ] [K.sup.D.sub.s]/[Y.sup.D] [less than or equal to ] [K.sup.B.sub.s]/[Y.sup.B] in Figure 2b. If each vertically integrated firm chooses to produce a combination of outputs ([k.sup.I.sub.s], [y.sup.I]) such that [k.sup.I.sub.s], [y.sup.I] = [K.sup.D.sub.s]/[Y.sup.D] and such that [n.sup.I][K.sup.I.sub.s] = [K.sup.D.sub.s] and [n.sup.I][y.sup.I] = [Y.sup.D] where n is the number of vertically integrated firms in the market, then there would not be demand left for any specialized firm. Hence, only vertically integrated firms would be present in equilibrium. It turns out that this is the only equilibrium market structure when [K.sup.A.sub.s]/[Y.sup.A] [less than or equal to] [K.sup.D.sub.s]/[Y.sup.D] [less than or equal to] [K.sup.B.sub.s]/[Y.sup.B]:

PROPOSITION l. When max {[K.sup.A.sub.s]/[Y.sup.A]} [less than or equal to] [K.sup.D.sub.s]/[Y.sup.D] [less than or equal to] [K.sup.B.sub.s]/[Y.sup.B] then only vertically integrated firms are present in equilibrium. Each one of them produces the output combination (y, k,) that is located where the ratio KD/YD crosses the curve CRS. The prices pe and re are functions of the ratio [K.sup.D.sub.s]/[Y.sup.D].

For example, let [K.sup.D.sub.s]/[Y.sup.D] = (3/2) and let the vertically integrated firms choose the output combination ([y.sup.I], [k.sup.I.sub.s]) such that [k.sup.I.sub.s] = (3/2)[y.sup.I]. At that output ratio, [p.sup.e] = [2wk.sup.I.sub.s] + [4wy.sup.I] = [7wy.sup.I], and [r.sup.e] = [5wy.sup.I]. In equilibrium firms must make zero profit: hence, [y.sup.I] = [square root of (4/29)([F.sup.I]/w)], [p.sup.e] = 7[square root of (4w/29)[F.sup.I]], [k.sup.I.sub.s] = 3[square root of (1/29)([F.sup.I]/w)], and [r.sup.e] = 5[square root of (42/29)[F.sup.I]].

Vertically integrated firms all choose the same combination of outputs for a given set of input and output prices, and they are able to fully serve the aggregate demands [K.sup.D.sub.s] and [Y.sup.D]. In such a context, the upstream and downstream firms cannot be present in the market because they are unable to supply outputs at lower prices (since condition INT holds).

[FIGURE 2 OMITTED]

Consider now the case in which [K.sup.D.sub.s]/[Y.sup.D] crosses the CRS curve outside of the bold segment of the curve CRS. We need to address two questions: Will vertically integrated firms be present in equilibrium? Will they be able to serve both the aggregate demand of the final and intermediate outputs? The answer to the first question is always positive (as from Panzar and Willig [1981]), and the answer to the second question is always negative.

Figure 2 presents two sets of possible equilibrium output combinations for the vertically integrated firms, which are drawn for different values of the parameters [F.sup.U], [F.sup.I], and [F.sup.D]. We will only comment on the equilibrium configurations in Figure 2a, since Figure 2b is a special case of Figure 2a.

Suppose that [K.sup.D.sub.s]/[Y.sup.D] >[K.sup.B.sub.s]/[Y.sup.B], which is the situation in which there is a relatively large demand (external to the industry) for k. Upstream firms must be present to satisfy part of this demand, since an industry with only vertically integrated firms would be unable to serve the demand. To see this, observe that each vertically integrated firm must choose the same output ratio, but [k.sup.I.sub.s]/[y.sup.I] [less than or equal to] [K.sup.B.sub.s]/[Y.sup.B] < [K.sup.D.sub.s]/[Y.sup.D], so there is no number of integrated firms that can serve both [K.sup.D.sub.s] and [Y.sup.D]. There is an excess demand of intermediate output that the vertically integrated firms cannot serve in equilibrium. In order for the equilibrium to be perfectly competitive, firms must produce where their average costs are lowest. Hence, the price of the intermediate input in equilibrium must be [r.sup.e] = 2[square root of [F.sup.U]w]. Using the condition that the marginal costs must be equal to prices, 2w(y + [k.sub.s]) = 2[square root of [F.sup.U]w] and 2w (y + [k.sub.s]) + 2wy = [p.sup.e]. Using the zero profit condition, we find that the price of the final output in this equilibrium is [p.sup.e] = 2 + [square root of [F.sup.U]w] + 2 [square root of w([F.sup.I] - [F.sup.U]], which is lower than 2[square root of [F.sup.U]w] + 2[square root of w[F.sup.D]], the price that a downstream firm would set.

The proofs in this article use this recursive approach: When specialized firms exist in equilibrium, they must set prices equal to the lowest average cost. Then, we use the profit maximization condition and the free entry condition to determine the price of the other good that the vertically integrated firms must charge in equilibrium.

The following proposition characterizes the full set of equilibria for this market for different values of the parameters [F.sup.U], [F.sup.I], and [F.sup.D]:

PROPOSITION 2. There are two general cases.

Case 1. Let 2[F.sup.U] > [F.sup.I]. Then (a) if [K.sup.D.sub.s]/[Y.sup.D] > [K.sup.B.sub.s]/[Y.sup.B], then vertically integrated firms and upstream firms are present in equilibrium. Vertically integrated firms produce a surplus of k, [k.sub.s] > 0, to be sold in the market. In particular, [r.sup.e] = 2[square root of [F.sup.U]w] and [p.sup.e] = 2[square root of [F.sup.U]w] + 2[square root of w([F.sup.I] - [F.sup.U])]; and (b) If [K.sup.D.sub.s]/[Y.sup.D] < [K.sup.A.sub.s]/[Y.sup.A], then vertically integrated firms and downstream firms are present in equilibrium. Vertically integrated firms produce a surplus of k, [k.sub.s], > 0, to be sold in the market. In particular, [r.sup.e] = 2[square root of w([F.sup.I] - [F.sup.D])]and [p.sup.e] = [r.sup.e] = 2[square root of w([F.sup.I] - [F.sup.D])] + 2[square root of [F.sup.D]w].

Case 2. Let 2[F.sup.D] > [F.sup.I] and [F.sup.U]+ [F.sup.D] + 2[square root of [F.sup.U][F.sup.D]] > [2F.sup.I]. Let [K.sup.D.sub.s]/[Y.sup.D] > [K.sup.B.sub.s]/[Y.sup.B]. Then upstream and vertically integrated firms are present in equilibrium. The prices are [r.sup.e] = 2[square root of [F.sup.U]w] and [p.sup.e] = 2[square root of w[F.sup.I]].

5. Equilibrium with Outsourcing

In the previous section we described the case in which vertically integrated firms might produce a surplus of intermediate good. Now we consider the opposite case, when they purchase (part of) the intermediate input they need to produce the final output, which occurs when the upstream fixed costs are small relative to the downstream fixed costs. We still maintain that [K.sub.s] > 0: That is, the industry produces a surplus of intermediate outputs that it sells to another industry.

If the size of [K.sup.D.sub.s]/[Y.sup.D] > [K.sup.B.sub.s]/[Y.sup.B], then the vertically integrated firm prefers to outsource part of the production to the upstream firm and exploit its economies of vertical scope to produce y at a lower cost than the downstream firm requires to produce it. Only the cost function of the vertically integrated firm changes and is described as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [k.sub.i ]represents the amount of intermediate input produced in-house, y - [k.sub.i] is the amount of production of input [k.sub.i] that is outsourced to specialized firms.

We rewrite the cost function in terms of [k.sub.p] = y - [k.sub.i]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus, the combinations ([k.sup.I.sub.p], [y.sup.I]) that are compatible with a vertical competitive equilibrium must be such that:

CRS : [F.sup.I] = w[(y - [k.sub.p]).sup.2] + [wy.sup.2],

INT: 2([square root of ([F.sup.U]w) + [square root of ([F.sup.D]w)] y [greater than or equal to] [F.sup.I] + w[(y - [k.sub.p]).sup.2] + [wy.sup.2] + 2[square root of ([F.sup.W][wk.sub.p])]

[FIGURE 3 OMITTED]

As before, if vertically integrated firms are present in equilibrium, then they must produce combinations of outputs that are in the bold portion of CRS. Also, only the points inside the ellipsis INT are those at which the vertically integrated firm is producing at lower costs than specialized firms. Notice that the condition that vertically integrated firms produce at product-specific decreasing returns to scale (SPEC) is always satisfied when the vertically integrated firm does not produce a surplus of intermediate good.

When vertically integrated firms outsource part of the production of the intermediate input to specialized firms, their internal cost of producing k must equal to the market price for k. Hence, the marginal cost of an additional unit of intermediate good must be equal to the market price for it, thus:

OUT : y = [k.sub.p] + [square root of ([F.sup.W]/w]).

For a vertically integrated firm to be in equilibrium, it must produce a combination of outputs that is inside the ellipsis INT; it lies along the curve CRS and is on the line OUT. Figure 3 shows there is only one point that satisfies all three conditions, where OUT intersects CRS. This observation leads to the second set of possible market structures:

PROPOSITION 3. Let [2F.sup.D] > [F.sup.1] and [F.sup.U] + [F.sup.D] + 2[square root of ([F.sup.U][F.sup.D]]) < 2[F.sup.1]. Then (outsourcing) vertically integrated firms and upstream firms are present in equilibrium. The combination of outputs chosen by the vertically integrated firm is given by [y.sup.I] = ([F.sup.I] - [F.sup.U])/w and [k.sup.I.sub.p] = [square root of ([F.sup.I] - [F.sup.U]/w)] - [square root of ([F.sup.U]/w))]. The equilibrium prices are given by [r.sup.e] = 2[square root of ([F.sup.U]w)] and [p.sub.e] = 2[square root of (w([F.sup.I] - [F.sup.U])] + 2[square root of (w[F.sup.U])].

[FIGURE 4 OMITTED]

6. Equilibrium and Comparative Statics

In this section, we illustrate the full set of equilibrium market structures for an industry that sells (e.g., "exports") the intermediate good to another industry. To do this, we first give the exact definition of the cost function of the vertically integrated firm:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Notice that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (y, y + [k.sub.s]; w, r) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (y, y + [k.sub.p]; w, r) when [k.sub.s] = 0 and [k.sub.p] = 0--the cost function is continuous when the vertically integrated firm consumes all the intermediate output that it produces in-house.

Equilibria

Depending on the values taken by [F.sup.U], [F.sup.D], and [F.sup.I], there are three possible equilibrium configurations for the industry, and they are presented in Figure 4 (where, as usual, [k.sub.p] = -[k.sub.i], for simplicity of exposition). Consider first the cases illustrated by Figure 4a and b. Here, there are three possible industry structures: when there is a relatively large demand for the intermediate good, [K.sup.D.sub.S/[Y.sub.D] > [K.sup.B.sub.S]/[Y.sub.B], then we have upstream and vertically integrated firms in equilibrium. Vertically integrated firms produce a surplus of intermediate output if the upstream costs are very high, while they outsource part of the production of the intermediate to specialized firms if the downstream costs are high. The intuition here is that the vertically integrated firms are more flexible than specialized firms and can dedicate their production to the good with the highest costs.

Comparative Statics

A Change in the Demand

Consider the effect of a change in the aggregate demands. Suppose that we start with 2[F.sup.D] > [F.sup.I] > 2[F.sup.U] and an output ratio [K.sup.A.sub.S]/[Y.sup.A] [less than or equal to] [K.sup.D.sub.S]/[Y.sup.D] [less than or equal to] [K.sup.B.sub.S]/[Y.sup.B]. We are then in the economy depicted by Figure 4a, where only vertically integrated firms serve the demands [Y.sup.D] and [K.sup.D.sub.S]. As [K.sup.D.sub.S] increases (for example, because another industry is growing and demands k), then the output ratio [K.sup.D.sub.S]/[Y.sup.D] increases as well, and at some point it becomes greater than [K.sup.B.sub.S]/[Y.sup.B]. As soon as that happens, upstream firms start entering into the market to serve part of the (external) demand of intermediate output. When [K.sup.D.sub.S] decreases and becomes smaller than [K.sup.A.sub.S]/[Y.sup.A] we will observe entry of downstream firms into the industry. Notice that vertically integrated firms will always be present in the industry as long as economies of vertical scope exist.

A Change in the Costs

Now consider changes in the fixed costs. Suppose that both [F.sup.D] and [F.sup.I] increase while [F.sup.U] remains unchanged, so that we still have [F.sup.I] < [F.sup.U] + [F.sup.D]. Suppose, moreover, that [K.sup.D.sub.S]/[Y.sup.D] is smaller than [K.sup.A.sub.S]/[Y.sup.A] so that we are in Figure 4a, with vertically integrated and downstream firms serving the markets. As [F.sup.D] increases relative to [F.sup.U], the set of combinations that the vertically integrated firms can choose in equilibrium moves to the southeast, and when FD becomes larger than [F.sup.U] (with [F.sup.I] < [F.sup.U] + [F.sup.D] still holding), the industry attains the configuration given in Figure 4b, with outsourcing and vertically integrated and upstream firms serving the markets.

7. Equilibrium and Comparative Statics in an Importing Industry

To close the model, we consider an industry (e.g., its vertically integrated firms) that is a net buyer of the intermediate output: That is, [K.sub.p] = -[K.sub.s] > 0. For this to occur, it has to be true that another industry that produces k can produce it at a lower cost. Call the price of the intermediate good k produced by the other industry [r.sup.f]. Thus, [r.sup.f] < 2[square root of ([F.sup.U]w]), and upstream firms cannot be present in equilibrium because their lowest average cost is higher than the market price for the intermediate input.

The relevant issue is which type of firms--vertically integrated and specialized downstream--exist in equilibrium. Not surprisingly, it depends on the relative magnitude of the fixed costs faced by the downstream and by the vertically integrated firms. When the price of the intermediate good produced by another industry is very low, then vertically integrated firms cannot be in the market, regardless of how strong the economies of vertical scope are. Hence, we will observe downstream specialization when the imported intermediate goods are very cheap relative to the prices that the domestic upstream producers would charge.

[FIGURE 5 OMITTED]

When the price at which another industry is selling the intermediate good is not too low, then whether we observe vertically integrated firms or downstream specialized firms depends on the magnitude of the economies of vertical scope. If the economies of vertical scope are very strong, then the vertically integrated firms will outsource some of their demand of intermediate good to firms in another industry. However, when the economies of vertical scope are not strong enough, then again, only downstream firms will be serving the market for the final good.

The following proposition formalizes these results:

PROPOSITION 4. There are three possible industry configurations: If [r.sup.f] > 2[square root of (w[F.sup.D])] and [F.sup.D] + [[([r.sup.f]).sup.2]/4w] - [F.sup.I] < 0, or if [r.sup.f1] l< 2[square root of (w[F.sup.D]]), then only downstream firms are present in equilibrium, and [p.sup.e] = [r.sup.f]+ 2[square root of ([F.sup.D]w)]. If [r.sup.f] > 2[square root of (w[F.sup.D])] and [F.sup.D] + [[([r.sup.f]).sup.2]/4w] - [F.sup.I] > 0, then only vertically integrated firms are present in equilibrium. The prices are functions of the output ratio [K.sup.D.sub.p]/[Y.sup.D].

8. Summary and Conclusions

We summarize the results of this article ill Figure 5:

* When the fixed costs for the upstream producers are small relative to those of the downstream producers, vertically integrated firms exploit their economies of vertical scope in the production of the final good and possibly outsource part of the production of the intermediate input to specialized upstream firms.

* When the fixed costs for the upstream producers are large relative to those of the downstream producers, then the industry structure is determined by the ratio of the demand for the final good and the demand for the intermediate inputs by another industry.

* Finally, when the industry is a net buyer (e.g., an "importer") of the intermediate input, then vertically integrated firms can only exist in equilibrium if the economies of vertical scope are very strong.

While the results are for specific functional forms, the economic trade-off underlying them is very general, and, thus, the results are likely to hold if we extend the model to allow for cost complementarities or for more general specifications of demand functions. The critical idea is that there must exist some trade-off in the production of the intermediate and final goods. In the model that we have considered, vertically integrated firms can exploit economies of vertical scope in the fixed costs, but they face rivalries (rather than complementarities) in the marginal costs. Hence, both types of firms can exist.

Economies of vertical scope in the fixed costs are likely to exist when firms are still learning how to produce final goods and how to design the best intermediate inputs for the final goods. Firms can then benefit from producing both at the same time. As time goes on, the process of learning by doing will standardize the production processes, and economies of scope will slowly disappear.

We would like to conclude with a discussion of the limitations of our analysis and with a suggestion for future research topics. First, our article offers no explanation of the production activities in the industry that buys the surplus of intermediate input. That is, where does the external demand and supply come from for the intermediate good k? Future work might draw interesting conclusions by intertwining our ideas of scope economies with those in the trade literature. Second, our article takes the existence of economies of scope as a "black box." Future work might look at what originates economies of scope. For example, one might look at the role of "learning by doing" and investigate how growth affects industry structure or, vice versa, how endogenizing vertical structure affects growth.

Appendix

Proof of Proposition 1

PROOF. Let the equilibrium price of k be [r.sup.e] and the equilibrium price of y be [p.sup.e.] Each vertically integrated firm must choose [k.sup.I] and [y.sup.I] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = [p.sup.e] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = [r.sup.e]. Because of the assumptions on the cost functions (they are U-shaped) this is a system of two equations ill two unknowns with a unique solution (if it exists). Hence, all vertically integrated firms choose the same combination ([y.sup.I], [k.sup.I.sub.s]) such that [k.sup.I.sub.s]/[y.sup.I] = [K.sup.D/[Y.sup.D].] Thus, 2[wy.sup.I][1 + ([K.sup.D]/[Y.sup.D])] = [r.sup.e] and 2[wy.sup.I][2 + ([K.sup.D.sub.S]/ [Y.sup.D])] = [p.sup.e]. In equilibrium firms must make zero profit: hence. [F.sup.I] = w[([y.sup.I]).sup.2] {1 + [1 + [([K.sup.D.sub.S]/[Y.sup.D]).sup.2]]} and so [y.sup.I] = [square root of ([F.sup.I]/ {1 + [1 + (([K.sup.D.sub.S]/[Y.sup.D])].sup.2]}]). From [y.sup.I] we can determine [k.sup.I] and the equilibrimn prices.

Proof of Proposition 2

Proof. The first case, when 2[F.sup.U] > [F.sup.I], is already proved in the text. Consider the second case, when [K.sup.D.sub.S/[Y.sup.D] < [K.sup.A]/[Y.sup.A]. Downstream firms must be present in equilibrium because the vertically integrated firms cannot serve all of the demand of y on their own. The price of the final good must then be equal to [p.sup.e] = [r.sup.e] + 2[square root of ([F.sup.D]w)]. Hence, 2w([y.sup.I] + [k.sup.I.sub.S]) = [r.sup.e]. This implies that [y.sup.I] = [square root of ([F.sup.D]w)] and [r.sup.e] = 2w([square root of ([F.sup.D]/w)] + [k.sup.I.sub.S]) Using the zero profit condition, we find [k.sup.I.sub.S] = [square root of ([F.sup.I]-[F.sup.D]/w)] - [square root of ([F.sup.D]/w)] [r.sup.e] = 2[square root of ([w[F.sup.I] - [F.sup.D])] and [p.sup.e] = 2[square root of (w[F.sup.I] - [F.sup.D]) + 2[square root of ([F.sup.D]w)]. [r.sup.e] = 2[square root of (w([F.sup.I] - [F.sup.D]))] is less than 2[square root of ([F.sup.U]w)], the average cost at the minimum efficient scale of production of the upstream firms.

Now consider the cases in which 2[F.sup.D] > [F.sup.I] and [F.sup.C] + [F.sup.D] + 2[square root of ([F.sup.U][F.sup.D])] > 2[F.sup.I.]. [K.sup.D.sub.S]/[Y.sup.D] > [K.sup.B.sub.S]/[Y.sup.B]-- there is again a relatively large demand for k, and upstream firms must be present to satisfy at least part of this demand. Vertically integrated firms are still present. To see this, first observe that [r.sup.e] = 2[square root of ([F.sup.U]w)], since upstream firms must be in equilibrium.

The only way in which the optimization problem of tile vertically integrated firm could have an interior solution is if [k.sup.I.sub.S] = [square root of ([F.sup.U]/w)] - [square root of (([F.sup.I] - [F.sup.U])w)], which is negative. We assumed that the vertically integrated firm cannot purchase intermediate input; hence, the best that the vertically integrated firm can do is to set [k.sup.I.sub.s] - 0. Then, using [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = [p.sup.e] and [k.sup.I.sub.s] = 0. we find 4wy = [p.sup.e.] Using the CRS condition we have [F.sup.I] = 2[wy.sup.2], y = [square root of ([F.sup.I]/2w)]. Thus, [p.sup.e] = 2[square root of (w[F.sup.I])], which is smaller than the price that the downstream firm would set, 2[square root of ([F.sup.U]w)] + 2[square root of ([F.sup.D]w)].

Proof of Proposition 3

PROOF. To show that a market structure with vertically integrated and upstream firms would be an equilibrium configuration, observe that the price of the intermediate input has to be equal to r = 2[square root of ([F.sup.U]w)]. The vertically integrated firm chooses [k.sup.I.sub.i] and [y.sup.I] to maximize its profits. Thus, [y.sup.I] = [square root of ([F.sup.I] - [F.sup.U])/w)] and [k.sub.i] = [square root of ([F.sup.U]/w)], or [k.sup.I.sub.p] = [square root of (([F.sup.I] - [F.sup.U])/w)] - [square root of ([F.sup.U]/w)]. The price of the final output is then [p.sup.e] = 2[square root of (w([F.sup.I] - [F.sup.U])] + 2[square root of (w[F.sup.U])], lower than the price the downstream firms would be able to set. The profits of both types of firms are zero.

Proof of Proposition 4

PROOf. Suppose that downstream firms arc present in equilibrium and vertically integrated firms are not outsourcing any of the production of k. Then 2w([y.sup.I] + [k.sup.I.sub.S]) = [r.sup.f] and 2w([y.sup.I] + [k.sup.I.sub.S] + 2[wy.sup.I] = [r.sup.f] + 2[square root of ([F.sup.D]w)]. Then [y.sup.I] = [square root of ([F.sup.D]/w)] and [k.sup.I.sub.s] = ([r.sup.I]/2w) - [y.sup.I] or [k.sup.I.sub.s] = ([r.sup.f]/2w) - [square root of ([F.sup.U]/w)]. This can be an equilibrium if [r.sup.f] > 2[square root of (w[F.sup.D])]. However, it is not an equilibrium because tile vertically integrated firm makes positive profits. To see this, observe that the profit can be written as (2[square root of (w[F.sup.D])] + [r.sup.f]) [square root of ([F.sup.D]/w)] [r.sup.f][[([r.sup.f]/2w) - [square root of ([F.sup.D/w)]] - [F.sup.I] - w[([r.sup.f]/2w).sup.2] - ([F.sup.D], which turns out to be equal to [F.sup.D] + [[([r.sup.f]).sup.2]/ 4w] - [F.sup.I]. Only downstream firms are present in equilibrium if [F.sup.D] + [[([r.sup.f).sup.2]/4w] - [F.sup.I] < 0, while only vertically integrated firms are present in equilibrium if [F.sup.D] + [[([r.sup.f]).sup.2/4w] - [F.sup.I] > 0.

Now consider [r.sup.f] < 2[square root of (w[F.sup.D])]. If vertically integrated firms and downstream firms are in equilibrium, then 2[wk.sup.I.sub.i] - [r.sup.f] = [r.sup.f] and 2[wy.sup.I] + [r.sup.f] = [r.sup.f] + 2[square root of ([F.sup.D]w)], with [y.sup.I] = [square root of ([F.sup.D]/w)] and [k.sup.I.sub.p] = [r.sup.f]/2w. The profit of the vertically integrated firm is now given by [F.sup.D] - [F.sup.I] - [[([r.sup.f]).sup.2]/2w], which is again less than zero, and so only downstream firms are present in equilibrium.

References

Baumol, William J.. John C. Panzar. and Robert D. Willig. 1982. Contestable markets and the theory of industry structure. New York: Harcourt Brace Jovanovich.

Eaton. Curtis. and S. Q. Lemche. 1991. The geometry of supply, demand, and competitive market structure with economies of scope. American Ecomomic Review 81:901-1011.

Hart, Oliver. and John Moore. 1988. Incomplete contracts and renegotiation. Econometrica 56:755-85. MacDonald, Glenn, and Alan Slivinski. 1987. The simple analytics of competitive equilibrium with multiproduct firms. American Economic Review 77:941-53.

Panzar, John C., and Robert D. Willig. 1981. Economies of scope. American Economic Review 71:268-72. Perry. Martin. 1989. Vertical integration. In Handbook of industrial organization. North Holland: pp. 185-255. Stigler, George. 1951. The division of labor is limited by tile extent of the market. Journal of Political Economy 56:129-41.

Williamson, O. 1985. The economic institutions of capitalism: Firms. markets and vertical contracting. New York: Free Press.

(1) Proposition 1 in Panzar and Willig (1981) states that economies of scope are sufficient for the existence of multiproduct firms in a multiproduct competitive equilibrium, and weak economies of scope are necessary for such existence of a multiproduct firm. Notice that their proposition does not refer to the nature (e.g., final or intermediate) of the goods. Here, a multiproduct firm is a vertically integrated firm.

(2) See Eaton and Lemche (1991) for an extension of MacDonald and Slivinski (1987) that allows for endogenous demands. The analysis could be developed with inverse demand schedules r = A - [BK.sup.s] and P = C - D Y instead of a fixed-demands [K.sup.D.sub.s] and [Y.sup.D]. Consider the case in which only vertically integrated firms are in the market. Then they would choose 2w( [y.sup.I] + [k.sup.I.sub.s]) = A - [Bn.sup.I][k.sup.I.sub.s] and 2w([y.sup.I] + [k.sup.I.sub.s] + [2wy.sup.I] - C [Dn.sup.I][y.sup.I]. Once we add the zero profit condition [[F.sup.I] = w[([y.sup.I] + [k.sup.I.sub.s).sup.2] + w[([y.sup.I ]).sup.2]], these are three equations in three unknowns ([y.sup.I], [k.sup.I.sub.s], and n), and the analysis is analogous to the one in the text. When we do the comparative statics exercises, we would study changes in the parameters A, B, C, and D rather than [K.sup.D.sub.s]. For the sake of simplicity we have decided to present the model with [K.sup.D.sub.s] exogenous. See Baumol, Panzar, and Willig (1982) for more on this.

(3) Perry (1989) first defined a vertical equilibrium to be a pattern of integration in the industry such that no firm would alter its choice of the stages at which it operates. Perry also provides a superb review of the early literature on vertical integration.

(4) The analysis in this article, as in MacDonald and Slivinski (1987), assumes that the source of economies of scope is differences in the fixed costs across firms. However, we could develop the analysis assuming that there are cost complementarities in the production and no differences in the fixed costs. For example, assume [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The term denotes the extent to which there are cost complementarities in the production of y and k. In particular, observe that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], there are cost complementarities in the production of the final and intermediate output. The vertically integrated firm saves money even if 0 < d < 1, although there are cost rivalries in such a case. The equilibrium analysis would be analogous, but the results would depend on the magnitude of d rather than on the relationship among [F.sup.U], [F.sup.D], and [F.sup.I]. Interestingly. Stigler (1951) talks about complementary and rival production processes in his article.

(5) To see why, observe that when INT and the y-axis intersect, [k.sub.s] = 0, and so 2[([square root of [F.sup.U]w] + [square root of [F.sup.D]w]).sup.2][y.sup.I] = [F.sup.I] + 2w[([y.sup.I]).sup.2]. This equation has a solution when [F.sup.U] + [F.sup.D] + 2 [square root of [F.sup.U][F.sup.D]] > 2[F.sup.I].

Federico Ciliberto, Department of Economics, University of Virginia, P.O. Box 400182, Charlottesville, VA 22904-4182, USA; Email ciliberto@virginia.edu; corresponding author.

John C. Panzar, The University of Auckland Business School, Owen G. Glenn Building, 12 Grafton Road, Auckland, New Zealand; E-mail j.panzar@auckland.ac.nz.

We would like to thank two referees and the Editor for their comments, as well as seminar participants at the 2001 European Association for Research in Industrial Economics Conference in Dublin, the 2001 Northeast Universities Development Consortium Conference in Boston, and the 2001 Southeastern Economic Theory and International Trade Conference in Miami. Federico Ciliberto acknowledges financial support from the Center for the Studies of Industrial Organization at Northwestern University. This article comprises a revised version of the third chapter of the PhD dissertation of Federico Ciliberto at Northwestern University.

Received July 2009; accepted January 2010.