Strategic Intrafirm Innovation Adoption and Diffusion.

Richard A. Jensen [*]

A theory of oligopolistic innovation adoption is developed in which intrafirm diffusions occur because the marginal cost of adoption is increasing in the rate of adoption. The equilibrium intrafirm diffusion curve is S-shaped or concave, as are empirically observed ones. This diffusion curve is more likely to be S-shaped the more competitive the industry, the larger the marginal cost of adoption or the pre-innovation unit cost of production, or the smaller the demand. The diffusion is longer, and so the extent of adoption at any date is lower the more competitive the industry, the larger the marginal cost of adoption or the pre-innovation unit cost of production, or the smaller the demand. A surprising result is that an increase in the unit cost reduction from the innovation has an ambiguous effect on diffusion. Obviously, a larger cost reduction allows each firm to earn a larger flow profit at every date from the same rate of adoption. However, a more subtle effect is that it also allows the firm to earn the same flow of profit with a slower rate of adoption, and so lower adoption costs. That is, the firms also have an incentive to spread out the diffusion over a longer period of time to save on adoption costs.

1. Introduction

This paper addresses two empirical observations that are common in the extensive literature on innovation diffusion. First, a firm usually adopts an innovation over time, not instantaneously. That is, there are intrafirm diffusions (Mansfield 1968; Nasbeth and Ray 1974; Romeo 1975; Stoneman 1981, 1983). Second, the time path of an intrafirm diffusion has a predictable shape. Diffusion curves plot the extent of adoption against time, and are generally S-shaped. That is, they are initially convex, reach an inflection point, and then are concave thereafter. Often they are also skewed in that they are initially convex for less than half of the length of the diffusion. This skewing can be so severe that the diffusion curve is essentially concave (Davies 1979; Stoneman 1983).

Given the voluminous literature on innovation diffusion, it is surprising that intrafirm diffusions have not received more attention. With few exceptions (noted below), the theoretical literature assumes adoption is a discrete choice variable, and so occurs instantaneously. Moreover, empirical studies typically assume adoption occurs at the date of first use of the innovation by the firm. To the extent intrafirm diffusions occur, these studies clearly overestimate the speed at which the innovation is put into use, and thus the speed at which its benefits accrue to the firm and society.

Intrafirm diffusions are especially common in the case of capital-embodied, new process innovations, because adoption involves adjustment costs as well as acquisition costs. In a seminal study, Mansfield (1968) found that the time interval between 10 and 90% usage of diesel locomotives ranged from three to more than 14 years for firms in the U.S. railroad industry. More recent examples abound. Most readers have undoubtedly experienced, at least once, the adoption of new computer equipment in their workplace. From my own experience, not all users are upgraded to the new equipment within a day, a week, or even a month.

Hence, in this paper I analyze the intrafirm diffusion of a new process innovation in a differential game model of an oligopoly. The model's predictions are consistent with the empirical regularities noted above. First, an intrafirm diffusion occurs in equilibrium because the marginal cost of adoption is increasing in the rate of adoption. It is not optimal to adopt instantaneously, just as it is not optimal to adjust to the desired level of a stock instantaneously, when adjustment costs are increasing at the margin. Second, the equilibrium intrafirm diffusion curve is either S-shaped or concave. Moreover, this diffusion curve is more likely to be S-shaped the more competitive the industry, the larger the marginal cost of adoption or the pre-innovation unit cost of production, or the smaller the demand. The analysis also shows a firm's diffusion is longer, and so the extent of its adoption at any date is lower the more competitive the industry, the larger the marginal cost of adoption or the pre-innovation u nit cost of production, or the smaller the demand. That is, a diffusion is longer the smaller the incentive to adopt (whether due to lower flow profit or higher adoption flow cost).

One surprising result is that an increase in the unit cost reduction from the innovation has an ambiguous effect on the intrafirm diffusion. The reason is that there are two conflicting effects. First, as is obvious, a larger cost reduction allows each firm to earn a larger flow profit at every date from the same rate of adoption. However, a more subtle effect is that it also allows the firm to earn the same flow of profit with a slower rate of adoption, and so lower adoption costs. That is, the firms also have an incentive to spread out the diffusion over a longer period of time to save on adoption costs.

These results are important for two reasons. First, they show that diffusion curves are more likely to have an S-shape for longer diffusions. A lower incentive to adopt means not only a longer diffusion, but also a lower rate of adoption at the start of the diffusion. In this model, the initial rate of adoption is so low that there are increasing returns to adoption, and so the rate of adoption is increasing, in the early stages of the diffusion. However, the rate of adoption must eventually decline as diminishing returns set in. Second, because the terminal date of the diffusion is chosen by each firm, this study provides results about the length of an intrafirm diffusion and the factors affecting it. No previous studies assume an endogenous length of intrafirm diffusion.

However, allowing a freely chosen terminal date requires a trade-off in solution concepts. Closed-loop (feedback) equilibria are usually preferred, but there are no general existence theorems for closed-loop equilibria in differential games with a freely chosen terminal date. [1] Because I am more interested in characterizing an intrafirm diffusion equilibrium than proving its existence, I focus on the symmetric, open-loop equilibrium when the terminal date of the diffusion is freely chosen. It is worth noting that the open-loop equilibrium is the appropriate solution concept in cases where the firms must commit to their adoption paths at the beginning of the diffusion. For example, the innovation's supplier may require such a commitment from adopters before making its plans to produce. Also, a union contract may force such a commitment.

The bulk of the theoretical adoption literature assumes a firm's only options are to adopt or not, and seeks to determine the order of adoption by firms in an industry (for decision-theoretic studies, see Jensen 1982, 1983; Bhattacharya, Chatterjee, and Samuelson 1986; Jovanovic and Lach 1989; Jovanovic and MacDonald 1994; for game-theoretic studies, see Reinganum 1981; Benoit 1985; Fudenberg and Tirole 1985; Quirmbach 1986; Stenbacka and Tombak 1994; Goetz 2000; Hoppe 2000; for surveys, see Reinganum 1989; Geroski 2000). Similarly, empirical studies typically define adoption as the date at which the innovation is first used by the firm (Davies 1979; Mansfield et al. 1982; Rogers 1995; Stoneman 1995).

There are few studies of intrafirm diffusion. Mansfield (1968) studies an epidemic model in which an uninfected part of the firm is more likely to become infected (adopt) the higher the extent of infection (adoption). This is unsatisfactory because it has no apparent basis in optimizing behavior (see Davies 1979; Stoneman 1981). Stoneman (1981) derives an intrafirm diffusion from optimal behavior in an uncertainty model with learning and adjustment costs. Fine and Porteus (1989) analyze a similar problem of investment to reduce costs. However, both of these use decision-theoretic models, and so do not apply to oligopolistic industries.

The only oligopolistic studies of intrafirm diffusion are Flaherty (1980) and Gaimon (1989). Flaherty studies the open-loop equilibria of an infinite horizon oligopoly model with continuous cost-reducing investment to determine whether symmetric or asymmetric industry structures arise in locally stable steady states. She does not analyze the shape of the time path of investment (the diffusion), or the effects of changes in parameters on the diffusion itself, or the shape of the diffusion curve. Her study also cannot draw conclusions about the length of the diffusion because she explicitly assumes it never ends. Thus, one might think of her analysis as pertaining to a continuous sequence of innovations, whereas this one pertains to a single innovation. Gaimon analyzes the problem of converting capacity to a new technology in a duopoly. She compares the open-loop and closed-loop equilibria. However, she also cannot draw conclusions about the length of the intrafirm diffusion because she assumes the terminal da te is exogenously given. She also does not analyze the effects of changes in parameters on the diffusion itself or the shape of the diffusion curve.

Section 2 presents the differential game model of intrafirm diffusion, and section 3 provides the details of its equilibrium. Section 4 provides results on how the diffusion varies with demand, adoption cost, the cost reduction, and market structure. Section 5 provides numerical examples to show that the general results are not vacuous. Section 6 concludes, and the Appendix contains the proofs of the results.

2. A Differential Game of Intrafirm Diffusion

Consider an oligopoly of n identical firms, each considering adoption of an exogenously developed new process. At any date t, a firm can adopt the new process in some fraction of its current production capacity (not yet converted to the innovation). For ease of computation and exposition, specific functional forms are assumed for demand, production cost, and adoption cost that imply the resulting differential game has a linear-quadratic form. The limitations of these assumptions, and the implications of relaxing them, are discussed in detail in the conclusion.

The firms produce a homogeneous good with inverse demand P = A - [[[sigma].sup.n].sub.i=1] [q.sub.i](t), where P is price, A is a positive constant, and [q.sub.i](t) is firm i's output rate at t. Let x(t) = [x.sub.1](t),...,[x.sub.n](t)] represent the state at t when firm i has adopted in the fraction [x.sub.i](t) of its capacity. Firm i's total production cost at t is then given by

C([q.sub.i]) = (c - [x.sub.i][epsilon])[q.sub.i] (1)

where c and [epsilon] are positive constants, [epsilon] [less than] c [less than] A, and [epsilon] [less than] A - c (the new process is not drastic). That is, adopting the new process in any fraction of its capacity reduces firm i's unit cost of production. Complete adoption, [x.sub.i] = 1, reduces its unit production cost by [epsilon].

At each date, the firms simultaneously choose outputs to maximize profits. If the state is x(t), then the Nash-Cournot equilibrium profit rate for firm i at t is

[[pi].sub.i][x(t)] = [[A - c + [nx.sub.i](t)[epsilon] - y(t)[epsilon].sup.2]]/[(n + 1).sup.2] (2)

where y(t) = [[sigma].sub.j[neq]i] [x.sub.i],(t) is total rival output. As expected, firm i's profit rate is increasing in the fraction of capacity in which it has adopted, but decreasing in the fraction of capacity in which each rival has adopted.

The adoption process for each firm i is given by the state equation

[x'.sub.i](t) = [u.sub.i](t), [x.sub.i](0)=0, [x.sub.i]([T.sub.i]) = 1 (3)

where ' denotes derivative and [T.sub.i] is the terminal date at which firm i decides to complete the adoption. Notice from Equation 3 that firm i's control variable [u.sub.i](t) is its rate of adoption at t, whereas its state variable [x.sub.i](t) is the extent of its adoption at t. Hence, given an admissible control [u.sub.i](t), the solution to Equation 3 is firm i's intrafirm diffusion curve. Also assume an exogenous physical limit on the rate of adoption, so admissible controls are bounded, [u.sub.i][epsilon] [0, B], where B is a positive constant.

The flow cost of adoption for each firm i is given by K([u.sub.i]) = (k/2)[u.sub.i] (4)

where k is a positive constant. Note that the marginal flow cost of adoption K'([u.sub.i]) = [ku.sub.i], is increasing in the rate of adoption [u.sub.i]. Thus, for sufficiently large k, increasing marginal adoption costs should imply that adoption is not instantaneous, but instead takes the form of an intrafirm diffusion.

The formal statement of the problem is easier with some common, game-theoretic notation. Let [x.sub.-i](t) be the extent of adoption by firm i's rivals, the (n - 1)-dimensional vector formed from x(t) by deleting [x.sub.i](t). If u(t) = [[u.sub.1](t),..., [u.sub.n](t)] and T = ([T.sub.1],..., [T.sub.n]), then define [u.sub.-i](t) and [T.sub.-i] analogously as the vector of rival adoption rates and the vector of rival completion dates. Given a discount rates r [epsilon] (0, 1), each firm's problem is to choose a rate of adoption [u.sub.i](t) and a terminal date [T.sub.i] to maximize the present discounted value of the flow of its profit, net of the adoption cost,

[J.sub.i][u(t), T] = [[[integral].sub.0].sup.[T.sub.i]] [e.sup.-rt]{[[pi].sub.i][x(t)] - K[u.sub.i](t)]} dt + [V.sub.i][x(T), T], (5)

subject to Equation 3, where the "salvage term"

[V.sub.i][x(T), T] = [[[integral].sup.[infinity]].sub.[[tau].sub.1]] [e.sup.-rt] [[pi].sub.i] [1, [x.sub.-i](t)] dt (6)

is the present discounted value of the flow of firm i's profit after its diffusion is complete.

The appropriate framework for analyzing this problem is a differential game. An open-loop strategy is a rate of adoption [u.sub.i](t) and completion date [T.sub.i] to which firm i commits at t = 0. The set of admissible adoption rates for each firm i is [S.sub.i] = {[u.sub.i](t)\[u.sub.i](t) is piecewise continuous and [u.sub.i](t) [epsilon] [0, B] for t [epsilon] [0, [T.sub.i]}. An open-loop strategy for firm i is a pair [[u.sub.i](t), [T.sub.i] where [u.sub.i](t) [epsilon] [S.sub.i] and [T.sub.i] [greater than or equal to] 0. An open-loop Nash equilibrium for this game is then a vector of strategies [[u.sup.*](t), [T.sup.*] such that, for every firm i, [[u.sup.*].sub.i] (t) [epsilon] [S.sub.i], [[T.sup.*].sub.i] [greater than or equal to] 0, and [J.sub.i][[u.sup.*](t), [T.sup.*] [greater than or equal to] [J.sub.i][[u.sub.i](t), [u.sup.*].sub.-i](t), [T.sub.i], [[T.sup.*].sub.-i] for every [u.sub.i](t) [epsilon] [S.sub.i] and [T.sub.i] [greater than or equal to] 0.

As noted above, closed-loop equilibria are usually preferred because they embody a notion of sequential rationality not necessarily present in open-loop equilibria. Unfortunately, there are no general existence theorems for closed-loop equilibria, even in linear-quadratic games, when the terminal date is freely chosen. Hence, the analysis focuses on the open-loop equilibrium. This approach has the advantage of allowing derivation of results concerning the optimal length of the firm's diffusion.

3. Equilibrium Intrafirm Diffusion

To determine the open-loop equilibrium of this differential game, the standard approach is to form, for each firm i, the Hamiltonian

[H.sub.i](t, x, [u.sub.i], [[lambda].sub.i]) = [e.sup.-rt][[[pi].sub.i](x) - K([u.sub.i])] + [[lambda].sub.i][u.sub.i] (7)

where [[lambda].sub.i] is a multiplier. As is well known, for given x and [[lambda].sub.i], the necessary condition for an interior choice of [u.sub.i] is

[partial][H.sub.i]/[partial][u.sub.i] = -[e.sup.-rt]k[u.sub.i] + [[lambda].sub.i] = 0. (8)

At each date t, the rate of adoption is increased up to the point where the present value of the flow of additional profit generated is offset by the marginal adoption cost. The necessary condition for the choice of the completion date [T.sub.i] is

[H.sub.i]([T.sub.i], x, [u.sub.i], [[lambda].sub.i]) + [partial][V.sub.i][x(T), T]/[partial][T.sub.i] = [e.sup.-r[T.sub.i]](k/2)[u.sub.i]([T.sub.i]) = 0, (9)

which holds if and only if [u.sub.i]([T.sub.i]) = 0. That is, firm i's rate of adoption must be zero at the date its diffusion is completed. The remaining necessary conditions are the state equation and the multiplier equation

[[lambda]'.sub.i](t) = -[partial][H.sub.i]/[partial][x.sub.i] = -[e.sup.-rt][2n[epsilon](A - c + [nx.sub.i][epsilon] - y[epsilon])]/[(n + 1).sup.2]. (10)

The analysis focuses on the equilibrium [[[u.sup.*].sub.1](t), ..., [[u.sup.*].sub.n](t), [[T.sup.*].sub.1], ... [[T.sup.*].sub.n]] that is interior, so [[u.sup.*].sub.i](t) [epsilon] (0, B), and symmetric, so [[u.sup.*].sub.i](t) = [u.sup.*](t), [[T.sup.*].sub.i] = [T.sup.*], [[x.sup.*].sub.i](t) = [x.sup.*](t), and [[[lambda].sup.*].sub.i](t) = [[lambda].sup.*](t) for all i = 1, ..., n. This equilibrium must satisfy Equations 3 and 8-10 of all i = 1, ..., n.

Theorem 1

If the marginal cost of adoption is sufficiently large relative to the size of the cost reduction, k [greater than] (2/[r.sup.2])[[epsilon].sup.2], then for any number of firms n [greater than or equal to] 1, the open-loop symmetric Nash equilibrium involves an intrafirm diffusion of a positive, finite length [T.sup.*]. The equilibrium extent of adoption and rate of adoption for each firm are

[x.sup.*](t) = C[e.sup.at] + D[e.sup.bt] - F and (11)

[u.sup.*](t) = [x.sup.*'](t) = aC[e.sup.at] + bD[e.sup.bt], (12)

where a = (r/2) + (1/2)[{[r.sup.2] - [[8n[[epsilon].sup.2]/k[(n + 1).sup.2]]}.sup.1/2] [epsilon] (0,1), b = (r/2) - (1/2) {[r.sup.2] - [[8n[[epsilon].sup.2]/k[(n + 1).sup.2]}.sup.1/2] [epsilon] (0, a), F = (A - c)/[epsilon], C = [[1 + F - F[e.sup.b[T.sup.*]]/[e.sup.a[T.sup.*]] - [e.sup.a[T.sup.*]] [less than] 0, and D = F[e.sup.a[T.sup.*]] - F - 1]/[e.sup.a[T.sup.*]] [greater than] 0, and the length of the diffusion is given by aC[e.sup.a[T.sup.*]] + bD[e.sup.b[T.sup.*]] = 0.

Intrafirm diffusions occur whenever adoption costs are increasing at the margin and sufficiently large. Firms do not adopt instantaneously, but they also do not extend the diffusion indefinitely. Recall that [u.sup.*(t)] is each firm's equilibrium rate of adoption and [x.sup.*(t)] is its equilibrium diffusion curve. Notice from Equation 12 that the rate of adoption, which is the slope of the diffusion curve, also can be written (see the Appendix) as

[u.sup.*](t) = [ab/(a - b)](F + 1)[e.sup.-r([T.sup.*] - t)][[e.sup.a([T.sup.*]-t)] - [e.sup.([T.sup.*] -t)]] (13)

It is apparent from Equation 13 that the intrafirm diffusion curve has one essential property. Recalling that a [greater than] b [greater than] 0, the rate of adoption is positive until the diffusion ends, [u.sup.*](t) [greater than] 0 for all t [epsilon] [0, [T.sup.*]), which implies that the firm's diffusion curve increases monotonically from [x.sup.*](0) = 0 to [x.sup.*]([T.sup.*]) = 1.

It is also important to determine what this theory predicts about the curvature of the intrafirm diffusion curve. This depends on its second derivative, which is just the slope of the rate of adoption, [x.sup.*"] = [u.sup.*'](t). That is, the diffusion curve is convex when the rate of adoption is increasing, and concave when the rate of adoption is decreasing. Using Equation 13, one can show that the sign of [u.sup.*'](t) is the same as the sign of

f(t) = b[e.sup.a([T.sup.*]-t)] - a[e.sup.b([T.sup.*]-t)] (14)

for all t [epsilon] [0, [T.sup.*], so [x.sup.*](t) is (strictly) convex if f(t) [greater than] 0 and concave if f(t) [less than] 0. [2]

The diffusion curve should always be concave in its final stages because the rate of adoption decreases to zero at [T.sup.*]. This is confirmed by noting that f([T.sup.*]) = b - a [less than] 0. And because f'(t) [less than] 0, there are only two possibilities. If the rate of adoption is decreasing initially, f(0) [less than or equal to] 0, then the diffusion curve is concave throughout. But if the rate of adoption is increasing initially, f(0) [greater than] 0, then the diffusion curve is initially convex but eventually concave, or S-shaped. That is, f(0) = b[e.sup.a[T.sup.*]] - a[e.sup.b[T.sup.*]] [greater than] 0 is necessary and sufficient for an intrafirm diffusion curve to be S-shaped. This gives the next result.

Theorem 2

Each firm's equilibrium intrafirm diffusion curve [x.sup.*](t) must be either S-shaped or concave. If b[e.sup.a[T.sup.*]] [greater than] a[e.sup.b[T.sup.*]], the diffusion curve is S-shaped: there exists a unique [t.sup.o] [epsilon] (0, [T.sup.*]) such that [x.sup.*"](t) [greater than] 0 for all t [epsilon] [0, [t.sup.o]), [x.sup.*"]([t.sup.o]) = 0, and [x.sup.*"](t) [less than] 0 for all t [epsilon] ([t.sup.o], [T.sup.*]]. But if a[e.sup.b[T.sup.*]] [greater than or equal to] b[e.sup.a[T.sup.*]], the diffusion curve is concave: [x.sup.*"](t) [less than] 0 for all t [epsilon] (0, [T.sup.*]].

This theory of intrafirm adoption and diffusion does predict that a firm's diffusion curve must have the same shape as those empirically observed. The shape of the diffusion curve depends on the equilibrium rate of adoption, which is chosen to balance the present value of the flow of marginal profit from that rate of adoption to its marginal cost. Marginal profit from adoption can be increasing in the early stage of the diffusion, in which case the rate of adoption is increasing and the diffusion curve is convex. However, diminishing returns implies that marginal profit from adoption must eventually decline as the diffusion proceeds. Therefore, the rate of adoption must be decreasing and the diffusion curve must be concave in its final stage at least. It is worth noting that Theorem 2 holds for any finite number of firms, including a monopoly. Moreover, Theorem 2 holds for the industry-wide diffusion curve [nx.sup.*](t) as well as each firm's diffusion curve.

Two final remarks about Theorem 2 are noteworthy. First, a, b, and [T.sup.*] are all functions of the same parameters in equilibrium. Therefore, it is reasonable to ask whether or not both S-shaped and concave diffusion curves can, in fact, occur in this model. A general proof of this would require the imposition of several additional assumptions that are otherwise unnecessary. Instead, section 5 presents several numerical examples that show that both types of diffusion curves can arise from this model.

Second, note that the necessary and sufficient condition for an S-shaped diffusion curve, f(0) [greater than] 0, can be rewritten as [T.sup.*] [greater than] [T.sup.o] = [ln(a) - ln(b)]/(a - b), where 0 [less than] b [less than] a [less than] 1 implies [T.sup.o] [greater than] 0. Writing this condition in terms of [T.sup.*] and [T.sup.o] shows that the shape of the diffusion curve and the length of the diffusion are related in equilibrium. In particular, this suggests that diffusion curves are more likely to be S-shaped the longer the diffusion (this is discussed in detail in the next section). Moreover, this also shows why it is important to allow the firm to freely choose the terminal date for its diffusion, rather than fixing it (as in Gaimon 1989) to find a closed-loop equilibrium.

4. Factors Influencing Intrafirm Diffusion

This section focuses on the effects of market structure, the marginal cost of adoption, the pre-innovation unit cost of production, the strength of the demand, and the magnitude of the cost reduction on intrafirm diffusions. The first results show how changes in these variables affect the length and extent of an intrafirm diffusion.

Theorem 3

The length of equilibrium intrafirm diffusion [T.sup.*] is longer the more competitive the industry (larger n), the larger the marginal cost of adoption k or the pre-innovation unit cost of production c, or the smaller the demand A. However, a change in the magnitude of the cost reduction [epsilon] has an ambiguous effect on the length of the intrafirm diffusion.

Theorem 4

At any date during the diffusion t [epsilon] (0, [T.sup.*]), the equilibrium extent of adoption [x.sup.*](t) is smaller the more competitive the industry, the larger the marginal cost of adoption or the preinnovation cost of production, or the smaller the demand. However, a change in the magnitude of the cost reduction has an ambiguous effect on the extent of adoption.

To see the intuition underlying these results, note from Equation 2 and Theorem 1 that in equilibrium each firm's profit rate is

[Pi]*[x(t)] = [[A - c + [x.sup.*](t)[epsilon]].sup.2]/[(n + 1).sup.2] (15)

Greater competition reduces each firm's Cournot equilibrium output rate (by shifting back its residual demand curve), and thus reduces its equilibrium profit rate at every date during and after the diffusion. This implies a smaller incentive to adopt, ceteris paribus, and so a longer diffusion and a smaller extent of adoption at any date during the diffusion. That is, the diffusion curve shifts down and elongates, as shown in the change from [x.sup.*](t) to [x.sup.**](t) in Figure 1. Intrafirm diffusions, and so industry-wide diffusions, are slower in more competitive industries because greater competition means each firm earns lower profit during and after the diffusion.

An increase in the pre-innovation unit cost of production has the same effect because it also reduces each firm's equilibrium profit rate at every future date. Similarly, an increase in the marginal cost of adoption reduces each firm's incentive to adopt, thereby slowing the diffusion and reducing the extent of adoption at each date during the diffusion.

However, an increase in demand increases each firm's equilibrium profit rate at every date (by shifting out its residual demand curve), and thus increases its incentive to adopt. This implies a faster diffusion and a larger extent of adoption at any date during the diffusion. The diffusion curve shifts up and contracts, as in the change from [x.sup.**](t) to [x.sup.*](t) in Figure 1.

The most surprising result is that the effect of an increase in the magnitude of the cost reduction [epsilon] from complete adoption is ambiguous. The reason for this ambiguity is that an increase in [epsilon] has two conflicting effects. First, obviously, it increases each firm's flow profit at any date from the same rate of adoption, just as an increase in demand does. Second, a more subtle effect is that an increase in [epsilon] also allows each firm to earn the same flow profit at any date with a slower rate of adoption. That is, if e increases, then one can see from Equation 15 that, at any date [tau] [greater than] 0, flow profit [pi]*([tau] can be held constant with a smaller extent of adoption [x.sup.*][tau]. Because the marginal cost of adoption is increasing, this smaller extent of adoption can be achieved with lower adoption costs at every preceding date, t [epsilon] [0, [tau]) Thus, a firm's total discounted profit net of adoption cost is larger at [tau]. That is, as the magnitude of the cost reduction increases, firms also have an incentive to spread out the diffusion over a longer period of time to save on adoption costs. The net result is ambiguity. Indeed, the proof shows that the effect of a change in e can be decomposed into a weighted average of the effect of a change in demand A and a change in the adoption cost k.

Theorems 3 and 4 also show that the effect of a change in any of these parameters on the equilibrium rate of adoption is necessarily ambiguous. Suppose an increase in competition lengthens the diffusion and reduces the extent of adoption at all dates, as in the change from [x.sup.*](t) to [x.sup.**](t) in Figure 1. Then the rate of adoption at the start of the new diffusion must be lower (otherwise the extent of adoption would not be lower). However, because the new diffusion is longer, adoption continues after the terminal date of the old diffusion. The rate of adoption at the end of the new diffusion must be higher.

Finally, as noted above, the shape of the diffusion curve is also affected by these economic factors. Recall that [x.sup.*](t) is concave if f(0) [less than or equal to] 0 and S-shaped if f(0) [greater than] 0. Hence, if an increase (decrease) in a parameter increases f(0) = b[e.sup.a[T.sup.*]] - a[e.sup.b[T.sup.*]], then one can say that the diffusion curve is more likely to be S-shaped the larger (smaller) that parameter.

Theorem 5

An equilibrium intrafirm diffusion curve is more likely to be S-shaped the more competitive the industry, the larger the marginal cost of adoption or the pre-innovation unit cost of production, or the smaller the demand. However, a change in the magnitude of the cost reduction [epsilon] has an ambiguous effect on the curvature of the intrafirm diffusion.

This result essentially says the following. Suppose parameters are chosen so that the diffusion curve is concave. Then, for example, increase the number of firms or the cost of adoption. By Theorem 3, the length of the diffusion increases and the diffusion curve shifts down. However, by Theorem 5, this also increases the likelihood that the diffusion curve is now S-shaped, as in the change from [x.sup.*](t) to [x.sup.**](t) in Figure 2. Hence, any change that reduces the incentive to adopt increases the length of the diffusion, and so makes the diffusion curve more likely to be S-shaped.

The intuition for this result is simple. Each firm adopts as rapidly as economically feasible. If the incentive to adopt initially is large enough, adoption begins at a rate high enough that diminishing returns apply throughout the diffusion. In this case the rate of adoption declines throughout as well, so the diffusion curve is concave. However, if the initial incentive to adopt declines enough, adoption may begin at a rate low enough that increasing returns exist in the early stage of the diffusion. Note from Equation 15 that the possibility of increasing returns from adoption comes from the convexity of a firm's equilibrium profit rate in the extent of adoption. In this case the diffusion curve is S-shaped because the rate of adoption begins at a low level, increases during the early stage of the diffusion, but eventually declines as diminishing returns set in.

5. Explicit Examples

Two numerical examples are presented below to clarify the results regarding the shape of the diffusion curve. Each shows how reducing the incentive to adopt can change a concave diffusion curve into an S-shaped one. Notice the decline in the initial rate of adoption [u.sup.*](0) that accompanies both of these changes.

Let r = 0.9, A - c = 4, [epsilon] = 1, k = 2.5, and n = 1 (monopoly). Then [x.sup.*](t) = -(7.948)[e.sup.0.5t] + (11.948)[e.sup.0.4t] - 4, [u.sup.*](0) = 0.805, [T.sup.*] = 1.845, and [T.sup.o] = 2.233, which implies a concave diffusion curve. But if the cost of adoption increases to k = 10, then [x.sup.*](t) = - (0.006)[e.sup.0.841t]+ (4.006)[e.sup.0.059t] - 4, [u.sup.*](0) = 0.231, [T.sup.*] = 4.96, and [T.sup.o] = 3.398, which implies an S-shaped diffusion curve. Now increase the number of firms to n = 4. Then [x.sup.*](t) = -(0.27l)[e.sup.0.723t] + (4.271)[e.sup.0.177t] - 4, [u.sup.*](0) = 0.560, [T.sup.*] = 2.477, and [T.sup.o] = 2.579, which implies a concave diffusion curve. But if the number of firms increases to n = 5, then [x.sup.*](t) = -(0.157)[e.sup.0.752t] + (4.157)[e.sup.0.148t] - 4, [u.sup.*](0) = 0.497, [T.sup.*] = 2.731, and [T.sup.o] = 2.692, which implies an S-shaped diffusion curve.

6. Conclusion

This paper has developed a theory of intrafirm innovation adoption and diffusion in oligopolistic industries where the length of the diffusion is freely chosen. Each firm adopts the new process over time, not instantaneously, because of increasing marginal cost of adoption. The equilibrium intrafirm and industry-wide diffusion curves have the same property as empirically observed diffusion curves: They are either S-shaped or concave. They are more likely to be S-shaped the longer the diffusion. The diffusion is longer, and the extent of adoption at any date is lower, the more competitive the industry, the larger the marginal cost of adoption or the pre-innovation unit cost of production, or the smaller the demand. A surprising result is that an increase in the cost reduction from the new process need not speed up the diffusion. The reason is that, although a larger cost reduction does increase the incentive to adopt, it also increases the incentive to lengthen the diffusion to reduce the total cost of adopti on (due to increasing marginal adoption costs).

Whether or not these results also hold if different functional forms are chosen for the demand, production cost, and adoption cost functions is an open question. A referee, noting the work of Seade (1980), predicted that the results would differ for a constant elasticity of substitution demand function. Seade's point was that if the elasticity of the slope of the demand curve is large enough (so the demand curve is very convex), then the entry of new firms could increase the marginal revenue of the incumbents without violating the usual stability condition. Indeed this can occur for a CES demand function. However, it is difficult to see how this would change the comparative statics. What does seem evident is that, as the adoption process begins and rivals expand their output, it is possible that a firm's marginal revenue will increase. This increases the incentive to adopt, and surely hastens the intrafirm diffusion, compared to the linear demand case, but should have no qualitative effect on the comparative statics. An increase in the strength of demand should hasten this new intrafirm diffusion even more, whereas increased adoption cost should still slow it down, and an increase in the magnitude of the cost reduction still has conflicting effects on the incentive to adopt. One possible change, of course, is that the increased marginal revenue might increase the incentive to adopt so much that complete adoption is instantaneous, and there is no intrafirm diffusion. But this can occur in the linear demand case also (if the marginal adoption cost k is low enough, so the marginal cost of complete adoption is low enough, as noted above). Conversely, for some forms of demand the firm's equilibrium profit might be concave in the extent of adoption, rather than convex as in Equation 2. This would imply that the incentive to adopt at any given date would decline more rapidly, and thus the diffusion would be slower. Again, it is difficult to see how this would change the comparative statics. Nevertheless, whether the sh ape of the diffusion curve would remain the same for these different forms is less obvious.

The form of adoption cost used might also seem restrictive. A common assumption in the adoption literature is that adoption costs decline through time because of learning-by-doing or economies of scale in the production of the innovation. This could be incorporated in the analysis by making the adoption cost parameter a decreasing function of time, k(t) where k'(t) [less than] 0. The reduction in marginal adoption cost over time would surely hasten the intrafirm diffusion, but again it is difficult to see how it would alter the comparative statics (assuming, of course, an intrafirm diffusion still occurs). Again, whether the shape of the diffusion curve would remain the same for this different form is not obvious.

The assumption that unit production cost declines uniformly for all levels of output with conversion to the new technology is restrictive. That is, as a referee has noted, a vintage capital approach might be more appropriate given that an intrafirm diffusion necessarily takes time. The current analysis, therefore, assumes that technological progress is slow enough that the next innovation does not arrive before the current diffusion terminates. If technological progress is more rapid, then the arrival date of the next innovation and its magnitude may affect the current decision on the extent as well as the length of adoption (Balcer and Lippman 1984 provide a decision-theoretic analysis of this case when both features of the new generation are uncertain). The effect of this generalization is less obvious because now both the terminal date and the terminal extent of adoption are also endogenous.

Finally, whether or not these results also hold in closed-loop equilibria of such a game is an open question. It is also a difficult question since it is not clear such equilibria exist in a differential game with a freely chosen terminal date. Nevertheless, as discussed above, most of the results, especially the comparative statics, are so plainly intuitive that it seems highly unlikely they would not hold more generally.

(*.) Department of Economics, University of Notre Dame, Notre Dame, IN 46556-0783, USA; E-mail rjensenl @ nd.edu.

I am grateful to Mort Kamien, the Editor, and two anonymous referees for helpful comments. This work was supported in part by a Summer Research Grant from the Gatton College of Business and Economics of the University of Kentucky. The grant was made possible by a donation of funds to the College by Ashland Oil, Inc. It was also conducted in part while the author was visiting at the Department of Managerial Economics and Decision Sciences, Kellogg School of Management, Northwestern University, whose support is gratefully acknowledged.

(1.) Because the game studied has a linear-quadratic structure, a closed-loop equilibrium does exist if the terminal date is fixed. However, it is well worth noting that the existence theorems for these games rely on using numerical methods to solve a system of ordinary differential equations (Friedman 1971; Mehlmann 1988). Although numerical estimates of the closed-loop equilibrium can be found, they are of no use in characterizing it in any meaningful way.

(2.) In the following, when I refer to the derivative of [x.sup.*](t) and [u.sup.*](t) at t = 0 or t = [T.sup.*], I mean the right-hand derivative at t = 0 and the left-hand derivative at t = [T.sup.*] (both of which are well-defined in this problem).

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Appendix

1. Proof of Theorem 1

Totally differentiate Equation 8 and rearrange terms to obtain [[lambda]'.sub.i](t) = k[e.sup.-rt]([u'.sub.i] - [ru.sub.i]). Equate this with Equation 10 to eliminate [[lambda]'.sub.i], use Equation 3 to substitute [x'.sub.i] for [u.sub.i] and [x".sub.i] for [u'.sub.i], and rearrange terms to obtain

[x".sub.i] - [rx'.sub.i] + [(2n[[epsilon].sup.2]([nx.sub.i] - y)/k[(n + 1).sup.2]] = -[2n[epsilon](A - c)/k[(n + 1).sup.2]]. (A1)

If the equilibrium is symmetric, then y = (n - 1)[x.sub.i], so Equation A1 can be rewritten as

[x".sub.i] - [rx'.sub.i] + [2n[[epsilon].sup.2][x.sub.i]/k[(n + 1).sup.2]] = -[2n[epsilon](A - c)/k[(n + 1).sup.2]]. (A2)

Using the standard technique, one can show that the solution to Equation A2, given [x.sup.*](0) = 0 and [x.sup.*](T) = 1, is Equation 11 providing that [r.sup.2] [greater than] 8n[[epsilon].sup.2]/[(n + 1).sup.2]. One can readily verify that k [greater than] 2[[epsilon].sup.2]/[r.sup.2] implies [r.sup.2] [greater than] 8n[[epsilon].sup.2]/k[(n + 1).sup.2] for all n, and so Equation 11 solves Equation A2 for all n. That [u.sup.*](t) is given by Equation 12 then follows from Equations 3 and 11, and Equation 8 implies [[lambda].sup.*](t)= [e.sup.-rt]k[u.sup.*](t). The condition for [T.sup.*] in Equation 9 follows from this and [H.sub.i][[T.sub.i], x([T.sub.i]), [u.sub.i]([T.sub.i]), [[lambda].sub.i]([T.sub.i])] + ([partial][V.sub.i]/[partial][T.sub.i]) = [u.sub.i]([T.sub.i])[[[lambda].sub.i]([T.sub.i]) - [e.sup.-r[T.sub.i]] (k/2)[u.sub.i]([T.sub.i])]. Hence, Equations 9 and 12 imply [u.sup.*]([T.sup.*]) = aC[e.sup.a[T.sup.*]] + bD[e.sup.b[T.sup.*]] = 0. Now let g(T) = (F + 1)(a[e.sup.-bT] - b[e.sup.-aT]) - F(a - b). Then one can show that [u.sup.*](T) = 0 if and only if g(T) = 0. Because g(0) = a - b [greater than] 0, [lim.sub.T[right arrow][infinity]] g(T) = -F(a - b) [less than] 0, and g(T) is strictly concave, it follows that [T.sup.*] [epsilon] (0, [infinity]).

2. Proof of Theorem 2

The terminal condition aC[e.sup.a[T.sup.*]] + bD[e.sup.b[T.sup.*]] = 0, the definitions of C and D, and a + b = r can be used to rewrite [u.sup.*](t) as Equation 13. Hence, [u.sup.*]'(t) = ab(F + 1)[e.sup.-([T.sup.*] - 1)](t), where f(t) is defined by Equation 14. As noted above, a [greater than] b [greater than] 0 implies f'(t) = ab[[e.sup.b([T.sup.*] - 1)] - [e.sup.a([T.sup.*] - 1)]] [less than] 0 for all t [epsilon] [0, [T.sup.*]) and f([T.sup.*]) = b - a [less than] 0, but f(0) = b[e.sup.a[T.sup.*]] - a[e.sup.b[T.sup.*]] is ambiguous. Hence, if f(0) [less than or equal to] 0, then [u.sup.*]'(t) [less than] 0 for t [epsilon] (0, [T.sup.*]]. However, if f(0) [greater than] 0, then there exists a unique [t.sup.o] [epsilon] (0, [T.sup.*]), defined by f([t.sup.o]) = 0, such that [u.sup.*]'(t) [greater than] 0 for t [epsilon] [0, [t.sup.o]) and [u.sup.*]'(t) [less than] 0 for t [epsilon] ([t.sup.o], [T.sup.*]].

3. Proofs of Theorems 3, 4, and 5

Recall C, D, and [T.sup.*] are defined by [x.sup.*](0) = 0, [x.sup.*]([T.sup.*]) = 1, and [u.sup.*]([T.sup.*]) = 0, which imply

C + D = F, (A3)

C[e.sup.a[T.sup.*]] + D[e.sup.b[T.sup.*]] = F + 1, and (A4)

aC[e.sup.a[T.sup.*]] + bD[e.sup.b[T.sup.*]] = 0. (A5)

Totally differentiate these with respect so C, D, T, ,t, k,(A - c), and G to obtain the relevant partial derivatives.

After some manipulation, and liberal substitution from Equations A3-A5, one can show that the sign of [partial][T.sup.*]/[partial]n is given by the sign of [a.sub.n]h([T.sup.*]), where [a.sub.n] = [partial]a/[partial]n = -[partial]b/[partial]n = (1/4)[{[r.sup.2] - [8n[[epsilon].sup.2]/k[(n + 1).sup.2]}.sup.-1/2][8(n - 1)[[epsilon].sup.2]/k[(n + 1).sup.3]] [greater than] 0 and h(t) = (a - b)t(a[e.sup.at] + b[e.sup.bt]) - r([e.sup.ut] - [e.sup.bt]). One can also show that h(t) has its minimum for t [greater than or equal to] 0 at t - 0, where h(0) = 0. Hence, h([T.sup.*]) [greater than] 0, whence [partial][T.sup.*]/[partial]n [greater than] 0. Moreover, observe that [partial][T.sup.*]/[partial]k = ([partial][T.sup.*]/[partial]n)([a.sub.k]/[a.sub.n]), where [a.sub.k] = [partial]a/[partial]k = -([partial]b/[partial]k) = (1/4)[{[r.sup.2] - [8n[[epsilon].sup.2]/k[(n + 1).sup.2]]}.sup.-1/2][8n[[epsilon].sup.2]/[k.sup.2][(n + 1).sup.2]] [greater than] 0, so [partial][T.sup.*]/[partial]k [greater than] 0 also. Next, observe that [partial][x.sup.*](t)/[partial]n = [a.sub.n]H(t), where H(t) = [[T.sup.*](D[e.sup.b[T.sup.*]] - [Ce.sup.a[T.sup.*]])([e.sup.at] - [e.sup.bt]/([e.sup.a[T.sup.*]] - [e.sup.b[T.sup.*]])] + t(C[e.sup.at] - D[e.sup.bt]). One can show H(0) = H([T.sup.*]) = 0 and [H.sup.n](s) [greater than] 0 at any s [epsilon] (0, [T.sup.*]) such that H'(s) = 0, which implies H(t) [less than] 0, and so [partial][x.sup.*](t)/[partial]n [less than] 0, for all t [epsilon] (0, [T.sup.*]). Therefore, [partial][x.sup.*](t)/[partial]k = [[partial][x.sup.*](t)/[partial]n]([a.sub.k]/[a.sub.n]) [less than] 0 for all t [epsilon] (0, [T.sup.*]) as well. One can show that the sign of [partial][T.sup.*]/[partial](A - c) is given by the sign of a[e.sup.a[T.sup.*]] - b[e.sup.b[T.sup.*]] - (a - b)[e.sup.r[T.sup.*]], and [partial][x.sup.*](t)/[partial](A - c) = M(t)/[epsilon]([e.sup.a[T.sup.*]] - [e.sup.b[T.sup.*]]), where M(t) = ([e.sup.b[T.sup.*]] - 1)(1 - [e.sup.at]) + ([e.sup.a[T.sup.*]] - 1)([e.sup.bt] - 1). Because M(0) = M([T.sup.*]) = 0 and [M.sup.n](s) [less than] 0 at any s [epsilon] (0, [T.sup.*]) such that M'(s) = 0, it follows that M(t) [greater than] 0, and so [partial][x.sup.*](t)/[partial](A - c) [greater than] 0, for all t [epsilon] (0, [T.sup.*]). Moreover, M'([T.sup.*]) = a[e.sup.a[T.sup.*]] - b[e.sup.b[T.sup.*]] - (a - b)[e.sup.r[T.sup.*]] [less than] 0, so [partial][T.sup.*]/[partial](A - c) [less than] 0.

Differentiating with respect to [epsilon], [partial][T.sup.*]/[partial][epsilon] = -F[[partial][T.sup.*]/[partial](A - c)] + ([partial][T.sup.*]/[partial]k)([a.sub.[epsilon]]/[a.sub.k]), where [a.sub.[epsilon]] = [partial]a/[partial][epsilon] = -([partial]b/[partial][epsilon]) = (1/4)[{[r.sup.2] - [8n[[epsilon].sup.2]/k[(n + 1).sup.2]]}.sup.-1/2][-16n[epsilon]/k[(n + 1).sup.2]], which is ambiguous because [partial][T.sup.*]/[partial]k [greater than] 0 [greater than] [partial][T.sup.*]/[partial](A - c) and [a.sub.k] [greater than] 0 [greater than] [a.sub.[epsilon]]. Similarly, [partial][x.sup.*](t)/[partial][epsilon] = -F[[partial][x.sup.*](t)/[partial](A - c)] + [[partial][x.sup.*](t)/[partial]k]([a.sub.[epsilon]]/[a.sub.k]) is ambiguous because F [greater than] 0, [partial][x.sup.*](t)/[partial]k [less than] 0 [less than] [partial][x.sup.*](t)/[partial](A - c), and [a.sub.k] [greater than] 0 [greater than] [a.sub.[epsilon]].

Finally, because a and b do not depend on A or c, the results in Theorem 5 for A and c follow from Theorem 3 and [partial]f(0)/[partial][T.sup.*] [greater than] 0. It is straightforward to show that [partial]f(0)/[partial]n and [partial]f(0)/[partial]k have the same sign. The results of Theorem 5 for n and k then follow from the fact that [partial]f(0)/[partial]n [greater than] 0 when evaluated at f(0) = 0. Ambiguity in the effect of [epsilon] of f(0) follows from the ambiguity in its effect on [T.sup.*].

[Graph omitted]

[Graph omitted]