# Revisiting Oligopolistic Reaction: Are FDI Decisions Strategic Complements?

Keith Head, Thierry Mayer, John Ries

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Revisiting Oligopolistic Reaction:
Are FDI Decisions Strategic Complements?

Keith HEAD

Thierry MAYER

John RIES

2000.49
Revisiting Oligopolistic Reaction:
Are FDI Decisions Strategic Complements?*

Keith Head†

Thierry Mayer‡

John Ries §

April 25, 2000

Abstract

Knickerbocker (1973) introduced “oligopolistic reaction” to explain why firms follow ri-
vals into foreign markets. We develop a model that incorporates the central features of
Knickerbocker’s story—oligopoly, uncertainty, and risk aversion—to establish the conditions
required to generate follow-the-leader behaviour. We find that rival foreign investment will
make risk-neutral firms less inclined to move abroad once its rivals have done so. We show
that risk aversion can generate the Knickerbocker prediction and derive an expression for
the minimum amount of risk aversion needed to generate oligopolistic reaction.

JEL classification: F23, R3

Keywords: Oligopolistic reaction, spatial Cournot competition, uncertainty

<footnote>*This paper received support from the Commissariat Général au Plan, convention no. 4, 1998.</footnote>

<footnote>†Faculty of Commerce, University of British Columbia, 2053 Main Mall, Vancouver, BC, V6T1Z2, Canada.
Tel: (604)822-8492, Fax: (604)822-8477, Email:keith-head@ubc.ca</footnote>

<footnote>‡TEAM, University of Paris I and CNRS, 106-112 Bd de l'hopital 75647 PARIS CEDEX 13, France. Tel/Fax: +33 1 55 43 42 67 Email:tmayer@univ-paris1.fr</footnote>

<footnote>§Corresponding Author: Faculty of Commerce, University of British Columbia, 2053 Main Mall, Vancouver, BC, V6T1Z2, Canada. Email:john.ries@ubc.ca</footnote>
# 1 Introduction

Research on foreign direct investment (FDI) has long recognized two important motives for choosing a particular country as the site for a new facility. First, firms wish to gain improved access to that country's market. A local plant can lower transport costs or circumvent barriers to trade. Second, firms want to deploy the relatively abundant factors located in the country. That is, they seek a low-cost production platform. In his 1973 book, Frederic Knickerbocker proposed a third motivation in location choice: firms might invest in a country to match a rival's move. In particular, Knickerbocker argued that firms in industries characterized by oligopoly would tend to follow each other's location decisions.

We define oligopolistic reaction (OR) as follows: *The decision of one firm to invest overseas raises competing firms' incentives to invest in the same country.* Essentially, we want to know whether, within the structure proposed in Knickerbocker, FDI decisions are strategic complements. Following the terminology of the field of industrial organization, firms' actions are strategic complements when an increase in the action of one firm raises the marginal profitability of an increase in the action for another firm.

Despite the strong recognition that Knickerbocker's hypothesis has attracted in the international business and strategy literature, there is surprisingly little formal examination of the conditions that give rise to OR. This paper develops a model that sequentially incorporates the important elements of the Knickerbocker framework—oligopoly, uncertainty, and risk aversion—to evaluate the conditions required for OR to hold. We show that in the case of certainty, an oligopolist will be less likely to establish a manufacturing facility in a foreign country once its rivals have invested there, i.e. FDI decisions are strategic substitutes. Moreover, introducing uncertainty about costs in the foreign market reinforces this “reverse” Knickerbocker effect. We proceed to establish if firms are sufficiently risk adverse, they may benefit from matching a rival's
move abroad. We are able to define the level of risk aversion necessary for the Knickerbocker hypothesis to hold in terms of the parameters of the model and show how it is influenced by changes in the number of firms as well as transport costs.

The persistent influence of oligopolistic reaction is underscored by the 89 citations to Knickerbocker (1973) in journals published between 1989 and 1999 according to the Social Science and Humanities Citation Index. A number of papers cite Knickerbocker as a motivation for adding specific covariates in regressions explaining FDI. Kogut and Chang (1991, 1996) use the 8-firm Japanese concentration index in their analyses of Japanese FDI. They interpret the positive and significant coefficient of this variable in their 1991 study as evidence of follow-the-leader behaviour.<fnref n="1" /> Hennart and Park (1994) and Yu and Ito (1988) add a measure of previous rival investment in their analyses of Japanese FDI in the United States. It has a significant positive effect in latter paper but an insignificant effect in the former.

We construct our model to reflect the main features of Knickerbocker’s argument. The first important aspect is oligopoly. We model Cournot oligopolists that choose between producing at home or abroad. A second feature of Knickerbocker’s framework is uncertainty about economic conditions in the foreign market. The follower is “uncertain of the production economies, if any, that it [the leading firm] might gain by manufacturing locally” (Knickerbocker (1973), page 26). Accordingly, we allow for uncertainty about production costs in the foreign market. Finally, Knickerbocker points to the role of risk aversion: “firms minimized their risk by matching the foreign direct investment of rivals” (Knickerbocker (1973), page 23). We consider a firm objective function in which variation of profits enters negatively.

To our knowledge, no formal mathematical models of oligopolistic reaction exist that incorporate these central features of the Knickerbocker thesis. However, some authors have worked

<footnote n="1">¹Kogut and Chang (1991) report that studies of U.S. investment in Canada yield similar findings on the influence of the degree of concentration.</footnote>
on related models. Caves (1991) introduces the idea of mergers as strategic complements, an idea formalized in Fauli-Oller (forthcoming). Fauli-Oller shows that in a Cournot oligopoly, a firm's incentive to merge is decreasing in the number of outside firms. Thus, mergers between two firms will increase the likelihood of subsequent mergers. Flaherty and Raubitschek (1990) show that follow-the-leader behaviour will occur when the leader's investment lowers the fixed costs of subsequent investment of rivals. Motta (1994) examines sequential FDI location decisions of oligopolists when products are vertically differentiated. He shows that both oligopolists in a country may choose FDI over export to serve foreign customers, a result he states is consistent with the Knickerbocker hypothesis. He acknowledges, however, that his model does not demonstrate that follower investment is *in response* to the FDI of the leader firm. Aussilloux (1998) also relates the FDI/export decision in a sequential entry Stackelberg game to the Knickerbocker hypothesis. Here again, however, the incentive to invest is not positively related to the number of firms that previously chose to invest. Graham (1998) develops a model with rivalistic behaviour that he refers to as “exchange-of-threat.” This model, however, gives rise to reciprocal FDI by oligopolists based in different countries, a phenomenon that differs from the follow-the-leader behaviour contemplated by Knickerbocker.

The following section outlines the set of plant location strategies available to the oligopolists and positions oligopolistic reaction in relation to the industrial organization literature on strategic substitutes and complements. Section 3 specifies the imperfect competition model we employ and calculates the profits associated with the strategy of maintaining home production and the strategy of relocating production abroad. We establish the main results about FDI being strategic complements or substitutes in three propositions in section 4. We examine equilibrium location in section 5. In the final section, we summarize our results and discuss their implications for observed patterns of FDI.
## 2 Framework

Multinational operations may have a number of different structures depending on characteristics of demand, production, and costs. For example, consider a firm that initially produces only at home and serves foreign markets by exporting. As economic conditions change, this firm may choose to relocate its single production plant to a low-cost foreign market, closing down its home plant. Alternatively, it may wish to establish production facilities in each major market to serve local demand. We refer to the first strategy as *R* to indicate “relocate” and the second strategy as *A* to represent “add location.” The alternatives are shown in Figure 1. First, home firms decide whether to stay at home, *S*, or invest abroad. If they invest, they can either choose *R* or *A*.

The focus of this paper will be on the relocation strategy *R* versus the strategy of staying at home *S*. *R* will be preferred to *A* when single-plant production is preferred to multi-plant production. We analyze strategy *A* in the appendix where we show a) the conditions on fixed production costs that make *R* preferred to *A*, and b) that the qualitative results are robust to whether *S* is compared to either *R* or *A*. Our reasons for concentrating on *R* is that, as shown in the appendix, strategy *A* contains a subset of the effects that influence strategy *R*. When a firm chooses *R*, it changes profits both at home and abroad. If the firm chooses *A*, however, only foreign profits are affected as it continues to serve home demand with home production. A model where FDI influences home profits seems most consistent with Knickerbocker's argument (p. 30) that “risk might not apply to a single market; the firm's market position might be in jeopardy in a number of foreign countries and even, perhaps, in the United States [the home country].”

The structure of the model is a two country, *N*-firm oligopoly. Firms operate a single plant and initially there are $n_h$ plants (or firms) located in the home country and $n_f$ plants in a foreign
<figure><img src="image_3.png" /><figcaption>Figure 1: Choice Structure</figcaption></figure>

country (thus a total of $N = n_h + n_f$ firms). We consider a representative firm that operates in the home country but contemplates relocating production to the foreign country. If it were to do so, as expressed in Figure 1 the number of home firms would decrease by one while the number of foreign firms would increase by one. We are interested in the benefit associated with relocating production to $f$ and how a rival's investment affects this benefit.

We begin with a certainty model where the representative home firm considers the benefit of relocating as the difference in the profits it obtains by staying at home versus the profits associated with relocating. Recall that Knickerbocker indicated that uncertainty as well as risk aversion were sources of oligopolistic reaction. Accordingly, we sequentially add these elements to the basic model. First, we introduce uncertainty about marginal costs in the foreign market and specify the benefit function defined in terms of the difference in expected profits for each alternative. Finally, we allow risk to enter negatively in utility and examine benefits in terms of differences in expected utility. Thus, in the succeeding analysis we consider the following three benefit functions which are functions of the number of firms, $N$, and the share of firms in the foreign market, $x \equiv n_f/N$:
1. $B_1 = \pi^R - \pi^S$: the difference in profits associated with choosing country $f$ instead of staying in country $h$.

2. $B_2 = E[\pi^R] - E[\pi^S]:$ the difference in *expected* profits.

3. $B_3 = EU(\pi^R) - EU(\pi^S)$: the difference in the *expected* *utility* of the profits associated with each strategy.

As mentioned earlier, oligopolistic reaction can be thought of in terms of FDI decisions being strategic complements. Bulow, Geanakoplos, and Klemperer (1985) define strategic complementarity as the case where the cross partial of the profit function with respect to each firm's strategic variable is positive, i.e. $\pi_{ij} > 0$ where $i$ is the strategic choice of firm $i$ and $j$ is that of firm $j$. The benefit functions defined above depict changes in profits associated with choosing to invest abroad. We analyze how the benefit function is influenced by an increase in the share of firms that have invested abroad. Thus, we view our approach as the discrete-choice, N-firm analogue of the condition proposed by Bulow *et al*. Oligopolistic reaction obtains when FDI decisions are strategic complements, i.e. when $\frac{\partial B}{\partial x} > 0$. "Reverse" oligopolistic reaction occurs if $\frac{\partial B}{\partial x} < 0$.

Bulow *et al.* examine a duopoly where a shock occurs that directly affects the level of a strategic variable chosen by Firm 1. They show that when the strategy variables are complements, a positive shock to one firms choice of strategic variable will increase the equilibrium level of the other firms strategy variable. In our framework, we also have in mind the existence of shocks that influence the strategic variable—whether to relocate production abroad—of particular firms. Suppose there are sunk costs of relocation that vary across firms and time. For example, information advances may reduce relocation costs and some firms may be better equipped to take advantage of these lower information costs. These changes will make it profitable for some firms to relocate, thereby increasing $x$. As $x$ changes, the expected benefits
of FDI for remaining firms will rise or fall depending on whether FDI decisions are strategic complements or substitutes.

## 3 Linear Cournot Segmented Markets Model

Throughout the analysis, we will assume that for $i = h, f$, inverse demand functions are given by $P_i = a_i - b_i Q_i$. If firms in the same location behave identically, we have:

$$Q_i = xNq_{fi} + (1-x)Nq_{hi},$$

the first subscript denoting location of the firm while the second subscript denotes the country where the output is sold.

We allow for different unit costs of production $c_i$ between countries and for trade to be subject to a transport cost $t$. We also assume fixed production costs per plant equal to $G$ and a sunk cost $F$ of setting up a new plant. The profit function of a representative firm depends on whether it is located in country $h$ or $f$:

$$\pi^R = (P_f - c_f)q_{ff} + (P_h - c_f - t)q_{fh} - G - F,$$

$$\pi^S = (P_f - c_h - t)q_{hf} + (P_h - c_h)q_{hh} - G.$$

Firms are Cournot competitors in segmented markets as in Brander and Krugman (1983) or Horstmann and Markusen (1992). Equilibrium quantities to each market for a firm choosing strategy $S$ are

$$q_{hh} = \frac{a_h - c_h - Nx(c_h - c_f - t)}{b_h(N + 1)}, \quad (1)$$
$$q_{hf} = \frac{a_f - c_h - t - Nx(c_h - c_f + t)}{b_f(N + 1)} \quad (2)$$

Profits earned from this strategy are

$$\pi^S = b_h q_{hh}^2 + b_f q_{hf}^2 - G. \quad (3)$$

Now consider strategy R. If a home firm were to move to the foreign country, it would increase the number of firms operating there by one. Equilibrium quantities shipped to each market by a firm producing in the foreign market are given by

$$q_{fh} = \frac{a_h - c_f - t + [N(1-x) - 1](c_h - c_f - t)}{b_h(N + 1)}, \quad (4)$$

$$q_{ff} = \frac{a_f - c_f + [N(1-x) - 1](c_h - c_f + t)}{b_f(N + 1)}. \quad (5)$$

Profits earned from this strategy are

$$\pi^R = b_h q_{fh}^2 + b_f q_{ff}^2 - G - F. \quad (6)$$

We assume that outputs $q_{hh}$, $q_{hf}$, $q_{ff}$, and $q_{fh}$ are positive for all distributions of firms, $x$, across the two countries.<fnref n="2" /> This assumption allows us to take derivatives of profits with respect to $x$ without running into corner conditions on outputs.

<footnote n="2">²The fact that all quantities are positive requires some conditions on the values that the parameters of the model can take. These conditions will always be assumed to be fulfilled during the paper and all numerical examples respect these conditions.</footnote>
# 4 FDI Decisions as Strategic Substitutes and Complements

Having specified profits associated with strategies S and R in the case of cost certainty, we can determine the benefit function's, $B_1 = \pi^R - \pi^S$, derivative with respect to the share of firms abroad, $x$. This calculation yields our first proposition:

**Proposition 1** In the Cournot segmented markets model with no uncertainty, the decisions to invest abroad are strategic substitutes, i.e. these conditions yield reverse oligopolistic reaction.

*Proof:* The mathematical version of the definition of oligopolistic reaction given above involves taking the derivative of B with respect to $x$, $\frac{\partial B}{\partial x} = \frac{\partial \pi^R}{\partial x} - \frac{\partial \pi^S}{\partial x}$, and determining its sign. The derivatives of the profit functions with respect to the locations of rivals are given by

$$ \frac{\partial \pi^S}{\partial x} = 2b_h q_{hh} \frac{\partial q_{hh}}{\partial x} + 2b_f q_{hf} \frac{\partial q_{hf}}{\partial x}, \qquad (7) $$

$$ \frac{\partial \pi^R}{\partial x} = 2b_h q_{fh} \frac{\partial q_{fh}}{\partial x} + 2b_f q_{ff} \frac{\partial q_{ff}}{\partial x}. \qquad (8) $$

Note that

$$ \frac{\partial q_{hh}}{\partial x} = \frac{\partial q_{fh}}{\partial x} = \frac{N(c_f + t - c_h)}{b_h(N + 1)}, \qquad (9) $$

$$ \frac{\partial q_{ff}}{\partial x} = \frac{\partial q_{hf}}{\partial x} = \frac{-N(c_h + t - c_f)}{b_f(N + 1)}. \qquad (10) $$

Subtracting (7) from (8) and substituting in (9) and (10) yields

$$ \frac{\partial \pi^R}{\partial x} - \frac{\partial \pi^S}{\partial x} = \frac{-2N}{N+1} [(c_f + t - c_h)(q_{hh} - q_{fh}) + (c_h + t - c_f)(q_{ff} - q_{hf})]. \qquad (11) $$

Substituting equilibrium quantities into equation (11), we have

$$ \frac{\partial B_1}{\partial x} = \frac{-2N^2}{(N+1)^2} \left( \frac{(c_f + t - c_h)^2}{b_h} + \frac{(c_h + t - c_f)^2}{b_f} \right) < 0 \qquad (12) $$
An increase in $x$ (the proportion of firms producing in $f$) makes the $R$ strategy (relocate the plant to the foreign country) less profitable relative to the $S$ strategy (maintain single home plant). This model therefore yields reverse oligopolistic reaction. •

Proposition 1 shows that the strategic interdependencies associated with oligopoly actually reduce the likelihood that firms will follow their rivals' investment decisions. Knickerbocker's original story of oligopolistic reaction involved more than oligopolistic interactions. He also identified uncertainty about the competitive advantage to be gained from investing abroad as a motivating factor. We now extend the relocation model to incorporate such uncertainty. We do so by making the cost of production in the foreign country uncertain. We assume that $c_f$ is a country-specific random variable with mean $\bar{c}_f$ and variance $\sigma^2$. This assumption is the simplest way to capture the realistic idea that firms are able to predict costs at home without much difficulty but cannot accurately predict their costs in a foreign environment. It also makes the strong assumption that firms' costs in the foreign country are perfectly correlated with each other. The latter aspect of the model is designed to make a point with minimal algebra rather than to reflect reality. However, many types of uncertainty such as real exchange rate risk or the risk of political upheaval would cause costs of all foreign firms to shift in common way. We also assume that firms learn $c_f$ following the relocation decision but prior to setting outputs.

We may now calculate the *expected* profits associated with staying at home versus those of moving abroad. Applying expectations to equations (3) and (6) and exploiting the rule that

$$E[q_{ij}^2] = V[q_{ij}] + (E[q_{ij}])^2,$$

where $E$ and $V$ are the mean and variance operators, we obtain

$$E[\pi^S] = \frac{x^2 N^2 \sigma^2}{(N+1)^2} \left( \frac{1}{b_h} + \frac{1}{b_f} \right) + \bar{\pi}^S \quad (13)$$
$$E[\pi^R] = \frac{(1-x)^2 N^2 \sigma^2}{(N+1)^2} \left( \frac{1}{b_h} + \frac{1}{b_f} \right) + \bar{\pi}^R \quad (14)$$

The new terms $\bar{\pi}^S$ and $\bar{\pi}^R$ are simply the original profit functions from equations (3) and (6) with $\bar{c}_f$ substituted in place of $c_f$. Subtracting the expected profit of locating in country $f$ from that of keeping one's plant in country $h$, we obtain the benefit ($B_2 = E[\pi^R] - E[\pi^S]$) to be gained by choosing country $f$ as

$$B_2 = \frac{N^2(1-2x)\sigma^2}{(N+1)^2} \left( \frac{1}{b_h} + \frac{1}{b_f} \right) + \bar{\pi}^R - \bar{\pi}^S. \quad (15)$$

**Proposition 2** *The introduction of cost uncertainty leads to more pronounced reverse oligopolistic reaction.*

*Proof:* Differentiating with respect to $x$ we obtain

$$\frac{\partial B_2}{\partial x} = \frac{-2N^2\sigma^2}{(N+1)^2} \left( \frac{1}{b_h} + \frac{1}{b_f} \right) + \frac{\partial \bar{\pi}^R}{\partial x} - \frac{\partial \bar{\pi}^S}{\partial x} < 0. \quad (16)$$

The first term shows that the higher the share of one's rivals that are already affected by the random foreign cost, the lower the expected benefits a prospective mover obtains by also subjecting itself to the random foreign costs. The remaining part of the equation, $\frac{\partial \bar{\pi}^R}{\partial x} - \frac{\partial \bar{\pi}^S}{\partial x}$ is just the $\frac{\partial B_1}{\partial x}$ defined in equation (12) with $\bar{c}_f$ replacing $c_f$. This result shows that, far from generating oligopolistic reaction, the introduction of cost uncertainty further reduces the incentive to follow a rival's move abroad. •

The intuition for this result comes from the convexity of the profit functions. Since profit is proportional to quantity squared, an increase in the variance in quantity (caused by increase in the variance of costs) will also increase the expected level of profits. This is because firms adjust quantity after they learn costs. If costs are high, they produce less and vice-versa when costs
are low. In general, a firm increases the variance of its output by locating in the country with fewest firms. What we are concerned about, however, is how rivals' relocation abroad affects a home firm's incentive to relocate. When $x$ increases, the foreign country has more plants and the home country has fewer plants. Thus the variance of output associated with producing in the foreign market falls and, correspondingly, the variance of home output rises. Hence, the benefit of following is reduced by the increase in $x$.

We have shown that oligopoly interactions combined with uncertainty generate reverse oligopolistic reaction. However, there is one more element of the Knickerbocker story—risk aversion—to which we now turn. We will employ a specific formulation of expected utility and utilize other assumptions that allow us to generate a closed form expression for how the benefits associated with relocation change with increases in $x$. This enables us to establish oligopolistic reaction, define the minimum degree of risk aversion that generates oligopolistic reaction, and show how this minimum level of risk aversion is influenced by changes in key parameters.

We suppose that the decision makers in the firm maximize a concave utility function of their income in which profits from the MNE's activities constitute a substantial part of income. In that case, they will care about expected profit in a positive way but view increases in the variance of profits negatively. We consider the special case of negative exponential utility defined over a normally distributed monetary wealth, i.e. suppose that utility over profits can be expressed as

$$U = -\exp(-\lambda\pi),$$

where $\lambda$ is the coefficient of absolute risk aversion. If $\pi$ were normally distributed then, expected utility (after a monotone transformation, see Varian (1992)) can be expressed as

$$EU = E\pi - \frac{\lambda}{2}V(\pi).$$
We use this result to redefine B as the gain in expected utility (rather than expected profits) associated with moving relative to staying:

$$B_3 = E[U^R] - E[U^S] = E[\pi^R] - E[\pi^S] + \frac{\lambda}{2} (V[\pi^S] - V[\pi^R]) \quad (17)$$

Since we have already calculated the first part of this expression in equation (15), we now need only calculate the difference in the variance of profits between moving and staying. To express these variances in terms of x and σ² is not easy since profits are a quadratic function of foreign costs. Greene (1993) provides a useful approximation. For Y = f(X) the variance of Y is approximately<fnref n="3" /> equal to [f'(E[X])]²V[X].

Application of this rule yields

$$V[\pi^S] \approx \frac{4x^2N^2}{(N+1)^2} [\bar{q}_{hh} + \bar{q}_{hf}]^2 \sigma^2, \quad (18)$$

$$V[\pi^R] \approx \frac{4(1-x)^2N^2}{(N+1)^2} [\bar{q}_{fh} + \bar{q}_{ff}]^2 \sigma^2. \quad (19)$$

In each case above, the $\bar{q}_{ij}$ correspond to the results of substituting $\bar{c}_f$ for $c_f$ in equations (1), (2), (4), and (5). Increases in the variance of foreign costs ($\sigma^2$) also increase the variance of profits. The difference in variances is given by

$$V[\pi^S] - V[\pi^R] \approx \frac{4N^2\sigma^2}{(N+1)^2} \left( [x(\bar{q}_{hh} + \bar{q}_{hf})]^2 - [(1-x)(\bar{q}_{fh} + \bar{q}_{ff})]^2 \right). \quad (20)$$

What is the effect of shifting rivals to the foreign country on this difference? If relocation of rivals lowers the variance of profits of production in the foreign market relative to that of the home production, then for a sufficiently large amount of risk aversion we will obtain strategic

<footnote n="3">³This formula is derived from a linear Taylor series expansion where the expansion point is the average value of the uncertain variable, here $\bar{c}_f$. Using simulations (available upon request), we found the approximation of the variance to be accurate within 2%.</footnote>
complementarity. However, the following expression shows that this derivative cannot be signed
without further restrictions:

$$
\operatorname{sign}\left[\frac{\partial\left(V[\pi^{S}]-V[\pi^{R}]\right)}{\partial x}\right]=\operatorname{sign}\left[x \bar{q}_{h}^{2}+(1-x) \bar{q}_{f}^{2}+D(x^{2} \bar{q}_{h}-(1-x)^{2} \bar{q}_{f})\right],
$$

where $\bar{q}_i \equiv \bar{q}_{ih} + \bar{q}_{if}$. The first two terms in this derivative may be viewed as the “direct” effect of changing $x$ while holding outputs constant. This effect is unambiguously positive and reveals that when rivals move abroad and outputs remain constant, the variability of profits is reduced for a firm located in the foreign market and increased for a firm located in the home market. The third term considers the effect of a change in $x$ on outputs of stayers and relocators. It employs term $D$, defined as

$$
D \equiv \frac{N}{N+1}\left[(\bar{c}_{f}-c_{h})\left(\frac{1}{b_{h}}+\frac{1}{b_{f}}\right)+t\left(\frac{1}{b_{h}}-\frac{1}{b_{f}}\right)\right].
$$

This “indirect” effect cannot be signed. Furthermore, under certain extreme parameter values,
the indirect effect may be a large enough negative number so that it fully offsets the direct
effect. In that case, risk aversion does not generate OR. The circumstances under which this
occurs involve differences between the two countries in the slopes of the demand curves ($b_h$
and $b_f$) and expected costs ($c_h$ and $\bar{c}_f$). By eliminating such differences, we remove the indirect
effect and retain only the unambiguous positive direct effect in order to identify situations where
oligopolistic reaction can actually occur. Thus we proceed to add a parameter condition:

$$
\textit{Similarity Condition:} \qquad b_h = b_f = b \text{ and } \bar{c}_f = c_h = c.
$$

This condition is sufficient to yield the main proposition of the paper.

**Proposition 3** *Oligopoly, uncertainty, and large amounts of risk aversion can combine to generate oligopolistic reaction in the Cournot segmented market model. Under the Similarity Con-
dition, FDI decisions are strategic complements for a sufficiently high degree of risk aversion.

*Proof:* We now can substitute equations (15) and (20) into equation (17) and take the derivative with respect to $x$, which yields

$$ \frac{\partial B_3}{\partial x} = \frac{\partial B_2}{\partial x} + \frac{4\lambda\sigma^2 N^2}{(N+1)^2} [x\bar{q}_h^2 + (1-x)\bar{q}_f^2], \quad (21) $$

where $\frac{\partial B_2}{\partial x}$ is negative as defined in equation (16). Oligopolistic reaction will obtain if the positive second term is large enough to dominate the negative first term (carried over from proposition 2). Thus, it will occur in this framework for sufficiently large values of $\lambda$. •

The proposition shows that under particular conditions, the relocation of rival production to the foreign market reduces the variability of profits of firms located in that market and increases the variability of profits of firms remaining in the home market. With the introduction of risk aversion, there is now a reason to dislike increased variation in profits. Thus, a sufficiently large degree of risk aversion will make the negative effect on utility of profit variation dominate the positive effect of profit variation on expected profits.

We are able to define the minimum amount of risk aversion necessary to generate oligopolistic reaction by setting equation (21) equal to zero:

$$ \lambda^{\min} = \frac{b(N+1)^2(1+t^2/\sigma^2)}{(a_h + a_f - 2c - t)^2}. \quad (22) $$

Equation (22) allows us to explore how the minimum level of $\lambda$ required to generate oligopolistic reaction varies according to the exogenous model parameters. First, it is decreasing in $\sigma^2$ indicating that the level of risk aversion inducing firms to want to match rivals' moves falls when there is more uncertainty about costs in the foreign market. Second, $\lambda$ is increasing in transportation costs, $t$. High transport costs makes foreign goods relatively uncompetitive with
goods produced at home. Thus, when transport costs are high, firms are less concerned that different realizations of the random foreign cost variable will have a strong impact on profits at home. Hence, there is less incentive to match the moves of rivals. Finally, the expression shows that when the total number of firms (N) increases, the parameter measuring the degree of risk aversion must also rise to yield the same incentive to invest. If managerial risk aversion is randomly distributed in the investor populations, these comparative statics indicate that oligopolistic reaction will be more likely when there is great variability in foreign costs and less likely in highly atomistic industries.

The results of two-stage games often depend on whether the firms’ choices in the second stage of product market competition are strategic substitutes or complements. We have used Cournot competition with linear demand where outputs are strategic substitutes. This raises the issue of whether our results would generalize to Bertrand competition with differentiated products in which prices are strategic complements. We applied Ottaviano and Thisse’s (1998) model of price competition with linear demands for differentiated products to our two-stage game of location choice followed by competition in segmented product markets. Our results—available by request—are that all three propositions continue to hold in the Ottaviano-Thisse model. The intuition is that proximity of rivals is bad for profits in most imperfect competition models. As a result, even when product market choices are strategic complements, location choices are strategic substitutes unless the firm is quite risk averse. Interestingly, we found that the more differentiated the products made by each firm are, the less risk aversion is required to generate oligopolistic reaction.

The relationship between oligopolistic reaction and market structure was a focus of the empirical analysis of Knickerbocker. Oligopolistic reaction predicts the clustering of FDI. Consequently, he derived a measure of spatial and temporal clustering of FDI by industry. There are, of course, other sources of FDI clustering. Firms would choose the same locations if cer-
tain countries were particularly attractive due to low costs or high demand. Positive spillovers between firms (agglomeration economies) would also generate concentration. To address the issue of distinguishing oligopolistic reaction from other sources of clustering, Knickerbocker related his measure of FDI concentration to industry Herfindahl indexes. His claim was that the more concentrated the industry, the more likely it would be to exhibit oligopolistic reaction.<fnref n="4" /> Knickerbocker discovered a positive but non-monotonic relationship between FDI and industry concentration. Specifically, he found that clustering increased with industry concentration except at extreme levels of concentration. “Tight” (highly concentrated) oligopolies were less likely to concentrate FDI than “loose” (less concentrated) oligopolies. He explains this finding as resulting from the collusive behaviour of tight oligopolies.<fnref n="5" /> Our model suggests a positive relationship between industry concentration and oligopolistic reaction. It does not, however, predict the non-monotonic relationship found by Knickerbocker.

# 5 Equilibrium Location Choice

Thus far, we have examined how a rival's FDI affects the expected benefit of relocating production abroad. In this section, we consider the Nash equilibrium in location choice. A share $x^*$ in country $f$ is a Nash equilibrium if there is no incentive for a firm in either country to switch locations unilaterally. For large numbers of firms, interior equilibria occur where $B(x^*) = 0$. However, we are interested in oligopoly situations and must adapt the equilibrium condition

<footnote n="4">⁴In fact, this test does not discriminate against FDI clustering due to other reasons. Head, Mayer, and Ries (1999) show that clustering increases with industry concentration in a Cournot model with complete information and asymmetric demand. This model does not contain the risk aversion which we have shown is required to generate oligopolistic reaction but yields the relationship between FDI clustering and industry concentration that Knickerbocker argues identifies OR behaviour.</footnote>

<footnote n="5">⁵This explanation is, however, not easy to justify theoretically. As we have seen in our model and as is widely accepted in location theory, avoidance of cutthroat price competition is a major explanation for spatial dispersion of firms. The clustering of firms raises the incentives to undercut the competitor and makes competition fiercer through lower prices. If firms are colluding, they are able to maintain high prices despite spatial proximity using trigger strategies. Thus, as opposed to Knickerbocker's argument, colluding firms should exhibit more concentration than non-colluding firms. This has been shown by Friedman and Thisse (1993) who reverse the usual maximal dispersion result of price competing firms by allowing them to collude in the price subgame.</footnote>
accordingly. For no firm in country h to want to relocate to country f it must be the case that $B(x^*) \le 0$. For no firm in country f to regret its relocation choice (and therefore wish to return to country h), we require $B(x^* - 1/N) \ge 0$. For a corner solution with all firms at home, i.e. $x^* = 0$, we drop the second condition and require only $B(0) \le 0$. Correspondingly, a corner solution with all firms abroad, $x^* = 1$, requires only the second condition that $B(1 - 1/N) \ge 0$.

Allowing for uncertainty and risk aversion while imposing the Similarity Condition, we solve for the difference in expected utility for a firm in h between staying in the home country and relocating to f and express it as

$$B_3 = \frac{2N}{(N+1)^2b} \left\{ [N(\sigma^2 + t^2)(1-2x) + t(a_f - a_h - t)] - \frac{\lambda\sigma^2 N(1-2x)(a_h + a_f - 2c - t)^2}{(N+1)^2b} \right\} - F. \quad (23)$$

We plot equation (23) for different levels of $\lambda$, the risk aversion parameter in figure 2. This plot sets $N = 4$, $a_h = a_f = b = 1$, $c = .25$, $t = .1$, $F = 0$, and $\sigma = .025$. The values of $\lambda$ chosen in the plot are those where the equilibrium pattern of firms location change given the parameters values stated above. In the case of risk neutrality, $\lambda = 0$, the slope is negative indicating that FDI decisions are strategic substitutes.

Figure 2 illustrates the effects of increases in risk aversion on both the magnitude and sign of oligopolistic reaction. As can be seen in (23), an increase in $\lambda$ from zero has the effect of rotating the $B(x)$ curve counterclockwise around the point where $x = 1/2$. The sign of $B(1/2)$ depends on $a_f - a_h - t$. For the identical demand curves we use in the plot, the pivot point is therefore negative. With increases in $\lambda$, $B(x)$ flattens then becomes positive. Positive slopes indicate strategic complementarity and oligopolistic reaction. For the parameters in the plot, FDI decisions become strategic complements for $\lambda$ in excess of 216.84.

The figure also indicates the types of equilibria that are possible in this four firm example. For $\lambda = 0$, the unique equilibrium occurs at $x^* = 1/2$ where it is evident that $B(x^*) < 0$ and
<figure><img src="image_4.png" /><figcaption>Figure 2: Risk Aversion and Oligopolistic Reaction</figcaption></figure>
$B(x^* - 1/4) > 0$. Thus, there will be two firms in each country. The intuition is that, under risk neutrality, firms have an incentive to operate in the less concentrated market, resulting in an equilibrium where each market is equally concentrated. This is consistent with the result in the location theory literature showing that spatial proximity of competitors reduces prices and profits (for a model with Cournot competition see Anderson and Neven (1990, 1991)).

As risk aversion increases, it will eventually be the case that $B(1/4) < 0$, resulting in a unique equilibrium of three firms at home and one abroad. Further increases in $\lambda$ will generate $x^* = 0$ as the unique equilibrium and then cause the slope to switch from negative to positive. Eventually, higher $\lambda$s will make the slope steep enough to set $B(3/4) \ge 0$, resulting in an equilibrium of $x = 1$. In this case, if a firm believed that its three rivals would choose country $f$, then it would want to do so as well. Thus, two different equilibria are possible. In the first, all firms remain at home while, in the second, all firms migrate to the foreign country. This is the most interesting parameter range since it predicts either a large cluster of FDI in the foreign country or none at all. Two otherwise equal countries might have very different levels of success in attracting investment. This is despite the fact that there are no agglomeration economies—the direct effect of clustering is still to lower expected profits.

How realistic are the values of $\lambda$ required to generate OR? Kocherlakota's (1996) survey article on the equity premium puzzle discusses the plausible range for the coefficient of relative risk aversion (CRRA). He argues that there is a consensus that the CRRA should be less than 10 but that only values of 18 or more can explain why equities have historically earned a 6% premium over risk-free Treasury bills. In the mean-variance expected utility function we have used, $\lambda$ is the coefficient of absolute risk aversion. We can obtain the implied CRRA by multiplying $\lambda$ by profits. For the numerical example in the figure, OR obtains for a CRRA of about 10 and two equilibria ($x = 0$ and $x^* = 1$) exist for a CRRA above 15. We also simulated the model using the utility function $(1-\gamma)^{-1}\pi^{1-\gamma}$ where $\gamma$ is the CRRA. We calculate expected
utility based on 10000 draws of $c_f$. For 20 simulations, we find on average that $\gamma \ge 17$ yields OR for all $x$ and $\gamma = 22$ is the minimum value of the CRRA required for two equilibria. These values, while large, are in line with values required to explain the equity premium puzzle.

# 6 Conclusion

Knickerbocker's oligopolistic reaction hypothesis can be formalized in terms of FDI decisions being strategic complements, a concept that is clearly defined in the industrial organization literature. The idea that the principal elements of the Knickerbocker story—oligopoly, uncertainty, and risk aversion—can combine to generate follow-the-leader investment behaviour has intuitive appeal.

The Cournot, segmented market model we develop shows that the requirements for oligopolistic reaction are rather strict. The cases of certainty and uncertainty with risk neutrality generate the opposite result—the benefits of relocating a plant to a foreign location are reduced when rivals invest there first. Only when a large amount of risk aversion is introduced does oligopolistic reaction occur. Even with risk aversion, we show that parameter values in the model must be restricted to guarantee the result.

Our investigation of equilibrium patterns of location reveals that under reverse oligopolistic reaction, there will be a tendency for investments of rival firms to be dispersed across the two locations. Oligopolistic reaction, on the other hand, will tend to cause the clustering of all investment in one location or the other. On the surface, this suggests that observations of concentrations of FDI in particular locations lend empirical support for the Knickerbocker hypothesis. However, such a claim would require distinguishing clustering due to oligopolistic reaction from other sources of clustering such as agglomeration economies.
Appendix: Relocation vs. Additional Location

The analysis in the body of the paper focuses on the potential relocation of a single plant, thereby abstracting from the option of establishing a new plant while retaining production at the old plant. Here, we begin by determining the critical level of fixed production costs, G, around which establishing an additional location (strategy A) or the relocation of the home plant (strategy R) is optimal. We then proceed to show that our three propositions continue to hold when FDI is associated with strategy A rather than strategy R.

Strategy A only makes sense when $t < |c_h - c_f|$. If this condition does not hold, a firm will only produce in the low cost location. Thus, we maintain this restriction throughout the analysis in this appendix which implies that a firm with plants in each country will serve each market with the local plant.

Under both strategy A and R, a firm will serve foreign consumers with goods produced in the foreign market. Thus, profits in the foreign market will be the same under either strategy. The choice between these alternatives, therefore, involves whether it is more profitable to serve home consumers with foreign production or maintain the home plant, thereby saving transport costs but incurring fixed costs of production both at home and abroad. The difference in profits is

$$ \pi^A - \pi^R = (P_h - c_h)q_{hh} - (P_h - c_f - t)q_{fh} - G $$

Suppose that $xN$ firms already invested in country $f$ and all kept their $h$ plant active. This implies the following equation for the difference in profits:

$$ \pi^A - \pi^R = \frac{(a_h - c_h)^2 - [a_h - c_h - N(c_h - c_f + t)]^2}{b_h(N + 1)^2} - G $$

For the considered firm to prefer keeping its plant in $h$ we therefore need sufficiently small fixed costs, precisely:

$$ G < \frac{N(c_h - c_f + t)[2(a_h - c_h) - N(c_h - c_f + t)]}{b_h(N + 1)^2}. \quad (24) $$

The condition on fixed costs is independent of $x$, the proportion of firms that already chose strategy A. We therefore know that if this condition is met all firms will prefer an additional location to a relocation and conversely. The model developed in body of the paper considers fixed costs that are high enough to justify treating only the relocation case. We now proceed to show that if firms choose to keep their home plants, our propositions continue to hold.

If inequality (24) is met, the firms will consider the choice between strategy S and strategy A. As mentioned above, the firms serve consumers in $h$ with their most proximate producing unit, their plant in $h$. Assuming that plants of a same firm do not compete with each other, that markets are segmented and that marginal production costs are constant, the only variable profits relevant in the choice are therefore the profits made in country $f$.

$$ B = \pi^A - \pi^S = (P_f - c_f)q_{ff} - (P_f - c_h - t)q_{hf} - (F + G) $$

Differentiating, we obtain the following negative slope for the certainty case:

$$ \frac{\partial B_1}{\partial x} = \frac{-2N^2}{(N+1)^2} \left( \frac{(c_h + t - c_f)^2}{b_f} \right). $$
Thus, we have reverse oligopolistic reaction and Proposition 1 obtains.

We must consider the timing of decisions when firms are uncertain about foreign production costs, $c_f$. As shown by inequality (24), the choice between $R$ and $A$ depends on foreign costs, which are unknown until after the firm makes the investment decision. Thus, the decision is based on the expected value of $c_f$. Having invested and learned foreign costs, a firm decides whether to shut its home plant down and then sets outputs. In what follows, we focus on cases where inequality (24) holds once foreign costs are known.

In the uncertainty case with risk neutrality, we obtain

$$ \frac{\partial B_2}{\partial x} = \frac{-2N^2}{(N+1)^2} \left( \frac{\sigma^2 + (c_h + t - c_f)^2}{b_f} \right) < 0, $$

and Proposition 2 holds.

For the risk aversion case, we can re-express the difference in variances as

$$ V[\pi^S] - V[\pi^A] \approx \frac{4N^2\sigma^2}{(N+1)^2} \left( [x\bar{q}_{hf}]^2 - [(1-x)\bar{q}_{ff}]^2 \right) $$

The derivative of the benefit function is now

$$ \frac{\partial B_3}{\partial x} = - \frac{4N^2}{(N+1)^2} \left[ \frac{\sigma^2 + t^2}{2b} - \lambda \sigma^2 \left( x \bar{q}_{hf}^2 + (1-x) \bar{q}_{ff}^2 + \frac{tN}{b(N+1)} [x^2 \bar{q}_{hf} + (1-x)^2 \bar{q}_{ff}] \right) \right]. $$

We will thus obtain oligopolistic reaction in this $A$ framework for sufficiently large values of $\lambda$, namely:

$$ \lambda > \frac{b(N+1)^2(1+t^2/\sigma^2)}{2[6(Ntx)^2 - 6tN(a_f - c_h + t(N-1))x + (a_f - c_h + t(N-1))(a_f - c_h + t(2N-1))]}. $$

Thus, we have shown that the choice of strategy $A$ or $R$ depends on the level of fixed production costs $G$. Regardless of which strategy is adopted by firms, reverse oligopolistic reaction obtains in the case of certainty as well as the case of uncertainty with risk neutrality. A critical level of risk aversion is necessary to obtain oligopolistic reaction.

## References

* ANDERSON, SIMON AND DAMIEN NEVEN, 1990, “Spatial Competition la Cournot: Price Discrimination by Quantity-Setting Oligopolists” *Journal of Regional Science* 30(1): 1–14.

* ANDERSON, SIMON AND DAMIEN NEVEN, 1991, “Cournot Competition Yields Spatial Agglomeration” *International Economic Review* 32(4): 793–809.

* AUSSILLOUX, VINCENT, 1998, “Investissement direct à l’étranger et entrée séquentielle sur le marché” *Revue Économique* 49(3): 607–615.

* BRANDER, JAMES AND PAUL KRUGMAN, 1983, “A Reciprocal Dumping Model of International Trade,” *Journal of International Economics* 23: 313–321.
BULOW JEREMY, JOHN GEANAKOPLOS, AND PAUL KLEMPERER, 1985, “Multimarket Oligopoly: Strategic Substitutes and Complements” *Journal of Political Economy* 93:488-511.

CAVES, RICHARD, 1991, “Corporate Mergers in International Economic Integration,” in A. Giovannini and C. Mayer, eds., *European Financial Integration*, Cambridge: Cambridge University Press, 136–160.

FAULI-OLLER, Ramon, forthcoming, “Takeover Waves,” *Journal Economics and Management Strategy*.

FLAHERTY, M. THERESE AND RUTH RAUBITSCHEK, 1990, “Local Presence and International Manufacturing in Technology-Intensive Industries” *Japan and the World economy: International Journal of Theory and Policy* 2: 301–326.

FRIEDMAN J.W. AND JACQUES-FRANÇOIS THISSE, 1993, “Partial Collusion Fosters Minimum Product Differentiation” *Rand Journal of Economics* 24(4): 631–645.

GRAHAM, EDWARD M., 1998, “Market Structure and the Multinational Enterprise: A Game-Theoretic Approach,” *Journal Of International Business Studies* 29(1): 67–83.

HEAD, KEITH, THIERRY MAYER, AND JOHN RIES, 1999, “Market Size Effects and Agglomeration” University of British Columbia manuscript.

HENNART, JEAN-FRANÇOIS AND YOUNG-RYEOL PARK, 1994, “Location, Governance, and Strategic Determinants of Japanese Manufacturing Investment in the United States,” *Strategic Management Journal* 15: 419–436.

HORSTMANN, IGNATIUS AND JAMES MARKUSEN, 1992, “Endogenous Market Structures in International Trade (Natura Facit Saltum),” *Journal of International Economics* 32: 109–129.

KNICKERBOCKER, FREDERIC, 1973, *Oligopolistic Reaction and Multinational Enterprise*, Cambridge, MA: Harvard University Press.

KOCHERLAKOTA, NARAYANA, 1996, “The Equity Premium: It’s Still a Puzzle,” *Journal of Economic Literature* XXXIV(1): 42–71.

KOGUT, BRUCE AND SEA JIN CHANG, 1991, “Technological Capabilities and Japanese Direct Investment in the United States,” *Review of Economics and Statistics* 73(3): 401–413.

KOGUT, BRUCE AND SEA JIN CHANG, 1996, “Platform Investments and Volatile Exchange Rates: Direct Investment in the U.S. by Japanese Electronic Companies,” *Review of Economics and Statistics* 78(2): 221–231.

MOTTA, MASSIMO, 1994, “International Trade and Investments in a Vertically Differentiated Industry,” *International Journal of Industrial Organization* 12: 179–196.

OTTAVIANO, GIANMARCO AND JACQUES-FRANÇOIS THISSE, 1998, “Agglomeration and Trade Revisited,” Center for Economic Policy Research Discussion Paper # 1903.

VARIAN, HAL, 1992, *Microeconomic Analysis*, New York: W.W. Norton.

YU, CHWO-MING J. AND KIYOHIKO ITO, 1988, “Oligopolistic Reaction and Foreign Direct Investment: The Case of the U.S. Tire and Textiles Industry,” *Journal Of International Business Studies* 19(3): 449–460.