Nov 15, 2002 - Aghion, Philippe, Nicholas Bloom, Rachel Griffith, Peter Howitt & Richard Blun- dell (2002), âCompetition and Innovation: an inverted-U ...
Competition and Economic Growth Jayasri Dutta Universityof Birmingham
Paul Seabright IDEI, Universite de Toulouse 1
15 November 2002 Abstract This paper models the interaction of spillovers and market structure in influencing investments in productive knowledge. Knowledge may be of two kinds: explicit or codifiable knowledge which is easy to copy, and tacit knowledge in which property rights can be more easily protected but which must be continually re-learned. The paper develops three models. The first is a simple model of the trade-off between investments in these two types of knowledge. We show that increased codifiability of knowledge may in some circumstances not only lower investment but even lower aggregate output, and that output may decline as competition increases.The second model considers the impact on growth of investments in a single type of knowledge generating a variable level of spillovers, when knowledge is durable though agents are not. It shows that the durability of knowledge creates an incentive for free-riding on the efforts of others, though this does not offset scale economies in equilibrium. However, strategic interaction between firms may mean that spillovers are bad for growth. The third model adds durable agents to duable knowledge. It shows that competition may generate zero-growth equilibria in which firms free-ride on each others’ efforts, but that when spillovers are sufficiently large there may be multiple equilibria in which zero-growth and positive growth equilibria co-exist. Competition may be beneficial in societies that can solve the associated coordination problems, since the maximal growth rate is increasing in the degree of competition. This is compatible with competition lowering growth in countries that cannot solve coordination problems.
1. Introduction This paper asks whether competition and spillovers are good for growth. There are two facets to the question. First, when there are spillovers, is competition good for growth? Secondly, when there is competition, are spillovers good for growth? The first question has been the subject of renewed interest, both theoretical and empirical, in recent years (see Aghion et.al. 2002). The second has received less explicit attention,
though it hovers on the margins of many contributions to the growth literature and is obviously related to questions of optimal intellectual property rights protection. In this paper we provide a simple unified framework for thinking about both questions. We do so by assuming a society of symmetric producers who face a choice between working to produce consumption goods and working to produce knowledge. Knowledge can be more or less explicit: the more explicit it is, the more cheaply it can be copied by its owner, but the more easily it can be stolen by someone who is not its owner. When knowledge is not explicit we say (following Polanyi, 1944 and many other since) that it is tacit. We begin with a static framework, in which producers can use either type of knowledge to produce consumption goods. We show that competition enables producers to benefit from each others’ use of explicit knowledge (in a way that is prima facie good for output), but that it also biases producers towards forms of production that use less explicit and more tacit knowledge than would otherwise be desirable. When there are spillovers, competition may be bad for growth, and when there is competition, spillovers are unambiguously bad for growth. Next we consider how the durability of knowledge changes things. We first consider what happens when knowledge is durable but producers are not. The fact that producers in any one period can free ride on the knowledge investments of previous generations might be thought to be bad for growth, but in fact the free rider effect is never enough to outweigh the scale economies due to durability. Including strategic interaction between producers, however, leads to more ambiguous findings. Once again, in the presence of competition spillovers are bad for growth, though now this effect may arise only when competition is particularly strong. In the presence of spillovers, competition maybe bad for growth, though this effect does not arise when spillovers are very large. Finally, we consider how things change when both knowledge and knowledgeproducers are durable. Here there may be multiple equilibria. Competition both makes multiple equilibria more likely (and thus in some sense increases the risk of coordination failures), and enhances the possibilities for growth at the best of those equilibria. One way of expressing this is to say that competition is good for growth if, and only if, the societies in which it occurs are good at resolving the coordination problems to which it may simultaneously give rise.
2. A Static Model of Tacit and Explicit Knowledge Use Suppose there are two kinds of goods produced in society, a private good X, and a public good Y , in addition to a numeraire commodity, grain (G) which grows in the garden of the representative citizen-consumer. She consumes all three, and her utility is U (X, Y, G). As an illustration, suppose U (X, Y, G) = G + 2
X 1−θ Y 1−θ + 1−θ 1−θ
with θ < 1. We note that public and private goods are similarly valued in consumption: we will also assume symmetry in their production, to illustrate our main argument as simply as possible. Consumers choose their preferred quantity of X in response to its price, P . The resulting demand function is Xd =
1 1
Pθ
;
we make use of the inverse demand function P = X −θ , noting that the elasticity of demand is constant and equal to 1θ . Private goods are produced by a number of rival firms. The public good is produced by a single firm — the state, or a cooperative of citizens. Production requires two types of knowledge: explicit knowledge, which can be codified and imitated, and tacit knowledge, which cannot. Tacit knowledge can only be used by the firm which possesses it; explicit knowledge possessed by one firm creates benefits by imitation for all others. We assume a symmetric technology for the production of X and Y : Y = F (`0 , h0 ) for the public good, and xi = F (`i , hi + γ
X
hj );
j6=i
for each firm i = 1, · · · , N producing the private good. `i is the level of tacit knowledge possessed by firm i, and hi the level of explicit knowledge, with 0 denoting the public firm. Explicit knowledge spills over to other firms in the industry, Pby a proportion γ. The stock of usable knowledge in any firm i is Hi = hi + γ j hj , whereas hi is the amount the firm acquires directly. As written, H0 = h0 , so that there are no knowledge spillovers to the public firm. We assume a Cobb-Douglas production technology characterised by constant returns to scale at the level of the individual firm: F (`, H) = `α H 1−α , 0 < α < 1. This is consistent, of course, with increasing returns to scale at the level of the economy as a whole because of the spillovers between firms. We assume, in addition, that both h and ` can be acquired at a cost by any producer, and that they cost the same per unit, say w. We now consider the incentives for firms to use knowledge in production. Consider (1−θ) the public firm, 0, first. The value of its output is V (Y ) = Y 1−θ . Input choices h0 , `0 maximize value net of cost: max V (Y ) − w(`0 + h0 ) h0 ,`0
subject to Y = F (`0 , h0 ). From the first order conditions of a maximum, we note that αy vy =w `0 3
vy where vy =
∂V ∂y
(1 − α)y = w; h0
. It follows that h0 H0 1−α = = `0 `0 α
(2.1a)
Outcomes are rather different in the private sector. A typical private producer has profits πi = P xi − w`i − whi . An increase in total quantity produced will lower prices, and producers choose their input and output levels which take account of this effect. As a result, the first-order conditions of profit maximization imply ∂P Fi,` = w; ∂X
(2.2)
X ∂P (Fi,H + γ Fj,H ) = w. ∂X i6=j
(2.3)
P Fi,` + xi
P FH + xi
Equation 2.2 represents the outcome of the usual decision of a firm with market power. In choosing the level of tacit knowledge, the producer trades off the cost of doing so against the revenue generated by the extra output produced, less any reduction in prices caused by the increase in output. The choice of explicit knowledge, in equation 2.3, involves an additional element. An increase in hi will increase firm i’s own output, for sure. In addition, it will add to the stock of knowledge of its rivals, and P so augment their productive capacity. The total increase in output is then Fi,H + γ j Fj,H , which will lower the market price by more than a corresponding increase in tacit knowledge. As a consequence, each such firm will use a lower amount of explicit knowledge, hi , than tacit knowledge, `i . The fact that hi is lower need not imply that Hi is, precisely because knowledge can be imitated and generates positive externalities. We solve explicitly in our model to show that this is indeed true. Spillovers result in aggregate underutilization of explicit knowledge. To display this property of the solution, , we consider a symmetric equilibrium with hi = h and `i = ` for i = 1, · · · , N . As a result, H = h(1 − γ + γN), x = F (`, H) and X = N x. Solving equations 2.2 and 2.3, we obtain H 1 − α [N (1 − θγ) − θ(1 − γ)] 1−α = ≡ z(N ) . ` α N −θ α
(2.4)
The extent to which private production underutilizes explicit knowledge can be observed by comparing equations 2.1a and 2.4. The quantity z(N ) decreases with N , and is less than 1 whenever N exceeds 1. A measure of the degree of underutilization 4
of explicit knowledge is given by 1 − z(N ), which increases with the number of firms. A monopoly firm does not underutilize explicit knowledge for the same reason that the public firm does not: there are no rivals to benefit from spillovers. An increase in the number of rivals reduces utilization of h; this decrease offsets the direct benefit of spillovers from an increasing number of rivals, to reduce the total stock of explicit knowledge H accessible to firms. An interesting feature of this outcome is the effect on overall productive knowledge of an increase in the degree of spillovers. From the expression for H = h(1 − γ + γN ), which is increasing in γ for N > 1 provided h is held constant, it would be tempting to conclude that spillovers increase the stock of available knowledge. However, this ignores the changes in firm behaviour (and therefore in h) that result from an increase in spillovers. In fact H/` is decreasing in γ for N > 1, as should be obvious from the fact that if here were no spillovers H/` would be equal to 1−α no matter what the value of α N . This casts an interesting light on the rationale for various types of policy designed to improve economic growth by attracting firms to locate in areas where knowledge spillovers are high: unless the policy simultaneously addresses their incentives to cut back utilization of their own knowledge the overall degree of knowledge accumulation may fall. We summarise these findings as follows: Proposition 1 (Utilization of Knowledge). At a symmetric equilibrium with N > 1, (a) h < 1−α ` α (b) H < 1−α ` α H (c) ` is decreasing in N (d) H` is decreasing in γ Efficiency in production requires balanced use of the two kinds of knowledge, proportionately to their coefficients in the production function, or H = (1−α) `. It is α not sufficient for efficiency: a monopolist will typically underutilize both types of knowledge, for reasons familiar from standard microeconomics. To see this in our illustration, we solve equation 2.2 explicitly for X. From the production function we have · ¸1−α F (`, H) H 1−α 1−α = 1−α = z(N ) ` ` α Substituting in 2.2 implies à · ¸1−α ! 1θ α θ 1−α X= .(1 − ) z(N ) q N α Here total output in the industry is determined by two countervailing influences. The competition effect makes output increase with the number of firms, while the underutilization effect tends to make it decrease. The net effect depends on the parameters of 5
the problem, and it is obviously an empirical question which parameters best capture reality. Given the constant elasticity assumption of the model and our Cobb-Douglas production technology, it would require γ to be implausibly large (> 1) for output actually to decline in N . However, constant elasticities, while a convenient simplification over certain ranges, are empirically somewhat implausible (as is attested by the Cellophane fallacy known to antitrust economists). For illustration, Figure 1 shows a series of simulations in which θ increases in N 1 . Here the initial competition effect always dominates at first but is eventually offset, this occurring most rapidly when spillovers are large and the productivity of tacit knowledge is relatively low (and therefore the productivity of explicit knowledge relatively high). Two features stand out from the figure. First, an increase in γ (the size of spillovers) unambiguously lowers output at all values of N greater than one. Secondly, holding γ constant, an increase in the relative productivity of tacit knowledge tends to lower output when there are few firms while increasing it when there are many. An interesting, if speculative comparison with real data is provided by Figure 2, taken from Carlin et.al. (2001). It shows the relative performance in terms of real sales growth over a three-year period of those firms in a sample of over 2,500 from 20 transition countries. It is strikingly clear from the figure that firms facing a small number of competitors have grown faster than those facing many competitors, and much faster than those facing none. The pattern is robust to multivariate analysis, and while there are certainly many other factors involved in generating these patterns, it is interesting that non-monotonic relationships between market structure and growth are empirically entirely plausible. Other findings with a similar flavour include those in Aghion et.al. (2002).
3. A Model of Knowledge and Growth A natural question provoked by the model of the previous section is what happens to economic growth as the degree of competition varies. To put it another way, how does the durability of knowledge affect the trade-offs discussed in the model above? To the extent that explicit knowledge, once utilized, remains available for future generations to utilize, extra competition would seem to improve the prospects for growth (by allowing greater exploitation of the economies of scale inherent in knowledge production). However, although as we shall see this intuition is a sound one, the countervailing incentive effect we described in the last section is also enhanced by competition. A long-run model also raises two other concerns. First, the inheritance of knowledge from the past may encourage free-riding in the future, as firms no longer bother to 1
In a linear Cournot model with demand P = a−bX and marginal cost c, the elasticity of demand is related to the number of firms by the expression bN(a−c) a+Nc . Setting values of (3, 0.5, 1) for (a, b, c) yields θ = 0.25 for N = 1 rising to θ = 0.7 for N = 7.
6
contribute to knowledge themselves. Secondly, from a modelling point of view we can no longer ignore resource constraints (as we did implicitly above when we treated tacit and explicit knowledge as capable of being purchased at an exogenous price). We therefore incorporate an explicit resource constraint in the form of a fixed supply of labour that can be allocated either to producing output or to producing knowledge. We proceed in two stages. First, in this section, we explore the effect of durability of knowledge without considering agents who anticipate the future; indeed, agents live for one period only, but they leave their legacy for the future. In the next section, we use infinitely-lived agents who plan ahead and take legacy of their current actions explicitly into account. In both cases, the addition of a time dimension complicates the model’s calculations significantly so we simplify the per-period structure in a way that preserves the essential intuitions. We ignore the public good, and consider utility as depending solely on consumption of the private good and the numeraire. Likewise we consider only a single type of knowledge, the degree of appropriability of which is indexed by the degree of spillover. In this model, therefore, comparisons between types of good and types of knowedge are made by performing comparative statics on N and γ rather than by comparing allocations between the respective types. We shall also confine our attention to symmetric equilibria: there may exist many non-symmetric ones. The structure of the first growth model is as follows: 1. Individuals i = 1, .., N (who are also producers) live for one period, are endowed with G, produce Xt and consume both G and X. The utility function is given by X 1−θ U (G, X) = G + ; 1−θ implying P = (X)−θ as before. 2. Each individual is endowed with 1 unit of labour. 3. The production of X uses knowledge Ht and manual labour `t . The technology is xit = Hit `αi . Note that we have suppressed the exponent on Hit in the interests of simplification. 4. Each individual i inherits a knowledge level Ht , and can allocate some of his labour, hit to innovate. A fraction of innovations γ spills over to other producers. The usable stock of knowledge is X Hit = Ht (1 + hit + γ hjt ). j6=i
7
5. The aggregate knowledge inherited by each generation is Ht = Ht−1 (1 + γ
N X
hit−1 )
i=1
so the spillovers across time are the same as those across producers. Another way to express this is that a fraction (1−γ) of knowledge dies with its producers. Knowledge not communicated is lost to posterity. This implies also that there are no capital markets (therefore no markets for knowledge), inevitably since no participants from different periods are alive simultaneously to trade with each other. We use this model to explore the effects of alternative market structures on knowledge production and growth. First we consider the efficient outcome; next, one in which there are many producers who compete believing themselves to be price-takers (and therefore ignoring strategic interactions); and finally, one (corresponding most closely to the model of the previous section) in which producers take the presence of others into account. 3.1. Market structures and knowledge production Here, we find the levels of knowledge production at (symmetric) equilibrium for alternative market structures. We ignore time subscripts for notational simplicity except where this would lead to ambiguity. 1. Efficient solutions: Suppose, first, that producers can co-operate (write binding contracts on hi , xi ). They choose hi = 1 − `i to maximize total revenue : N N X X V (y) = P. xi = ( xi )1−θ i=1
where
xi = Ht−1 (1 + hi + γ
i=1
X j6=i
∗
At a symmetric solution, h , we have
hj )(1 − hi )α .
x(y ∗ ) = Ht−1 (1 + γN h∗ )(1 − h∗ )α , where γN = 1 − γ +γN . V is monotone increasing in x, and the efficient solution achieves a maximum. We obtain γN − α [EFF] h∗ (N) = . γN (1 + α) Notice that h∗ increases with N and with γ:, as does x∗ (N ). The efficient solution preserves the scale economies created by knowledge spillovers. Growth rates are γN h∗ (N ), similarly increasing in N. 8
2. The Competitive solution: In this, imagine that producers choose hi to maximize own revenues. They act as price-takers (imagining their own influence on prices to be negligible). Individual revenues are vi (yi ) = Pt At (1 + hi + γ
N X j6=i
hj )(1 − hi )α .
Individual producers choose P 1 − α − αγ j=1 hj ˆ i = min[ h , 0]. 1+α ˆ i decreasing with hj . Notice that this solution displays the “free-riding” effect, h ˆ is A symmetric solution h 1−α ˆ [COMP] h(N) = . 1 + αγN ˆ i = 0 is ruled out as α < 1.) Here, h(N ˆ ) (The trivial symmetric solution with h decreases with γ and N (the free-riding effect) but output xˆ(N) and growth rates ˆ increase. The scale effect of spillovers is preserved at equilibrium. gˆ = γN h 3. Cournot oligopoly: Suppose now that each producer evaluate the consequences of his own action on prices. In particular, we know that P X ∂xj ∂ N i=1 xi = x0i (hi ) + γ ∂hi ∂hi j6=i = x0i (hi ) + γHt
X (1 − hj )α . j6=i
Define H−i =
PN
j6=i
hj
φ(hi ; H−i ) =
From the fact that
∂P ∂X
x0i (hi ) 1 − hi − α(1 + hi + γH−i ) = . Ht (1 − hi )1−α
P = −θ X , we obtain
X ∂vi xi = Pt At (φ(hi , H−i ) − θ PN (φ(hi ; H−i ) + γ (1 − hj )α )). ∂hi i=1 xi j6=i
Let hC be the symmetric equilibrium solution. Then, hC must satisfy φ(hC ; H−i = (N − 1)hC )(1 − 9
θ θ ) = γ(N − 1)(1 − hC )α . N N
Obtain from this and routine substitutions that (1 − hC ) − α(1 + γN hC ) =
θ γ(N − 1)(1 − hC ). N −θ
This solves as [COURN] hC (N ) = where qN ≡ γθ
1 − α − qN 1 + αγN − qN
N −1 . N −θ
ˆ ). Here, hC (N) typically decreases with N and γ, and is bounded above by h(N It is a complicated object and best studied in numerical solutions. It is possible to get an idea for large N. Notice that lim qN = γθ;
N→∞
and so, 1 − α − γθ . N α For large N, growth rates decrease with γ (unlike g ∗ but also gˆ). In simulations, the response of x and g to variations in N and γ are often non-monotone. This requires θ to be high (otherwise, the solutions are nearly competitive). For example, with θ = 0.9 and α = 0.25, we have xC (N) and g C (N ) to be Ushaped in N (fix γ at 0.75 say) while they are inverse U-shaped in γ (fix N at 20 say). The non-monotone behaviour is typically explained by the incentive effect overwhelming the scale effect. Finally, the scale effect effect outweighs the incentive effect for large N when γ is sufficiently large. lim g C (N) =
3.2. Implications of the Results These results have confirmed that, on its own, the free-rider effect of long-lived knowledge is not enough to outweigh the effect of scale economies on growth. In the competitive solution (where free-rider effects operate but strategic incentives do not) spillovers are good for growth. However, in the Cournot solution, where strategic incentives operate, spillovers may be bad for growth. The result that competition between a few firms may lead to higher growth than competition between many itself depends on γ not being too large - Schumpeterian outcomes depend, one might say, on there being some sufficient degree of property rights protection. Or to put it another way, in this model, Schumpeterian policy prescriptions are complementary to rather than a substitute for a degree of intellectual property rights protection. We now turn to a model that incorporates explicitly the anticipations of agents for whom the future matters. 10
4. A Model with Infinitely-Lived Agents We look at the impact of competition between agents who care about the future and anticipate the impact of their decisions on future output and profits. Here something very striking happens. Because investment in knowledge in any one period by one producer builds on the whole of the knowledge inherited from the past (namely that which has spilled over from the decisions of others, as well as the producer’s own past investments), the returns to investment today depend very sensitively upon expectations about future growth. In particular, each producer’s optimal choice of knowledge investment depends positively upon other producers’ expected levels of knowledge investment. This property (known in the game theory literature as supermodularity) is a well known source of multiple equilibria, and this model no exception. We derive three results. The first is that competition increases the likelihood of a zero growth equilibrium. The intuition is straightforward. When a producer expects that others will not invest in knowledge either today or in the future, his own returns to knowledge investment depend purely on the amount of extra output that knowledge will eventually produce, the cost of producing it and the price at which this can be sold. Increased competition means that others will produce output that lowers the price, without investing in knowledge from which the producer himself can benefit. The impact of competition on the marginal returns to that producer is therefore unambiguously negative, and is eventually great enough to ensure that the producer prefers to devote all of his labour investment to producing output today (provided there is some discounting of the future). The second result is that when spillovers to other firms are large enough there may exist equilibria with positive growth as well. This is because when other firms are expected to invest in knowledge, the complementarity between their knowledge investment and the producer’s own investment can outweigh the substitutability between the respective outputs of consumption goods that the knowledge eventually creates. So provided spillovers per firm (modelled by γ) are sufficiently great, competition creates the possibility for positive growth. The third result is that, in such a positive growth equilibrium (or in at least one, if more than one exist), the more competition there is, the higher the growth rate. This is because competition is the only force in the model offsetting diminishing returns to knowledge (in keeping with the model of the previous section we have retained constant product returns to knowledge in the production finction, but diminishing marginal utility of consumption leads to diminish marginal revenue). Provided the knowledge complementarities created by competition outweigh the substitutability of the outputs, therefore, the more competition there is, the more it is possible to offset diminishing returns to knowledge investment and the higher the long run growth rate. The main components of the model are summarised in Table 1. Preferences are
11
represented as follows: VC ((G0 , X0 ), · · · (Gt , Xt ), · · · ) =
∞ X
β t U (Xt , Gt )
t=0
with U(G, X) defined as in the previous section. ¯ and of labour is N every period.2 • Endowments: the aggregate supply of Gt is G, • Technology : the production of X requires labour ` and knowledge H: xit = F (Hit , `it ) = Hit `αit . Producers can hire labour to produce output, or to produce knowledge H. Knowledge increases by own research or by spillovers: X hjt + hit ) Hit = Hi,t−1 (1 + γ j6=i
where Hi is the initial or inherited level of knowledge; hit the amount of labour devoted to research by firm i at time t, and γ the fraction of knowledge spillovers,0 ≤ γ ≤ 1. Knowledge is non-rival, and so usable by producer as well as imitator. Note that here, in contrast to the model of the last section, we have treated the durability of a firm’s own knowledge asymmetrically to the spillovers to other firms, so as to consider separately the impact of durability and spillovers on incentives to invest (which was not an issue in a model in which firms reacted passively an inheritance from the past and did not plan for the future). • Dynamics : The evolution of knowledge is Hit = Hi,t (1 + ((1 − γ)hit + γ
N X
hjt ));
j=1
i = 1, · · · , N.
Aggregate output evolves as Xt =
N X
xit =
i=1
N X
Hit `αit .
i=1
4.1. Competition and Equilibrium • As in the previous models, consumers’ utility maximisation implies the inverse demand, or price function P (X) = X −θ . 2
This maintains comparability with the static model where every firm is of size 1.
12
• Firms inherit knowledge Hit−1 andPchoose hit , `it , xit at each t. Profits in period −θ t are vit = Pt xit − wt (hit + `it ) = ( N j=1 xjt ) xit − wt (hit + `it ) with wt the wage rate that clears the labour market at t. The present value of profits is Vit =
∞ X
β s vi,t+s .
s=0
• Each firm chooses hit , `it , xit ≥ 0 every period, to maximize discounted profits Vit , and takes the actions of other firms, of consumers, and the wage rate as given. • The equilibrium concept is subgame-perfect Nash equilibrium. Let H−1,i = H > 0 be the initial level of knowledge. An equilibrium is {(hit , `it , xit ; i = 1, · · · , N), Pt , wt }∞ t=0 such that 1. hit ≥ 0, `it ≥ 0, 2. xit = Hit `αit ,
PN
i=1 (hit
+ `it ) = N ;
3. Hit = Hi,t−1 (1 + ((1 − γ)hit + γ P −θ 4. Xt = N i=1 xit ; Pt = Xt .
PN
i=1
hit ))
5. The forward sequence {(hit+s , `it+s , xit+s )i = 1, · · · , N }∞ s=0 is a Nash equilibrium of the game defined by payoffs Vit and state (Hit , · · · , HNt ). A symmetric equilibrium is an equilibrium with xit = xt , `it = `t , hit = ht ;
i = 1, · · · , N.
at each t. A stationary symmetric equilibrium has `it = `, hit = h = 1 − ` for all i at each t. Note that at any symmetric equilibrium, knowledge grows whenever ht > 0: Hit = Ht = Ht−1 (1 + γN ht ) with γN = Nγ + (1 − γ). At a stationary symmetric equilibrium, Ht+1 = (1 + γN h)Ht Xt+1 = (1 + γN h)Xt . 13
The stationary growth rate of output is just g = γN h. Firms choose hit , `it ≥ 0 to maximize Vit . The first order conditions for an optimum must hold at any interior maximum: `it > 0 ⇒
∂Vit = 0; ∂`it
hit > 0 ⇒
∂Vit = 0. ∂hit
Further , ∂Vit ∂Vit |`it =0 ≤ 0; hit = 0 ⇒ |h =0 ≤ 0. ∂`it ∂hit it To represent the first-order conditions, we need : `it = 0 ⇒
1.
∂xit Pt ∂xit
2.
∂xit ∂`it
= α x`itit
3.
∂xit ∂Hit
=
4.
∂xit+s ∂hit
5.
∂xj,t+s ∂hit
= Pt (1 − θ xXitt ).
xit ; Hit
=
1+γ
PN
xit+s ; hjt +(1−γ)hit
j=1
= γ 1+γ P N
xjt+s hit +(1−γ)hit
j=1
Substitutions yield: ∂xit P (Xt ) ∂xit ∂Vit = − wt ∂`it ∂xit ∂`it xit xit = αPt (1 − θ ) − wt ; Xt `it ∞ N X X ∂xit+s P (Xi,t+s ) ∂xjt+s ∂Vit s = β [ ] − wt ∂hit ∂xjt+s ∂hit s=0 j=1 P∞ s xit s=0 β Pt+s xit+s [1 − θγ − θ(1 − γ) Xt+s ] = − wt . P 1+γ N j=1 hjt + (1 − γ)hit
As in the static model (equation 2.3) the three terms represent a quantity effect, a price effect via own output, and a price effect via the output of others. At a symmetric solution, ∂Vt θ xt = αPt (1 − ) − wt ∂`t N `t P∞ s θγN ∂Vt s=0 β Pt+s xt+s [1 − N ] = − wt . ∂ht 1 + γN ht 14
4.2. Results: equilibrium growth rates We would like to know the effects of market structure (N ), and spillovers (γ) on growth rates in equilibrium. There may be many such equilibria: for this analysis, we restrict attention to stationary symmetric equilibrium growth paths. At a stationary symmetric solution, we know that (S1) (S2) (S3)
∂Vt θ (x0 (1 + g)t )1−θ = α(1 − ) − wt ; ∂` N N θ` Nθ (1 − γN ) ∂Vt (x0 (1 + g)t )1−θ = − wt . ∂h (1 − β(1 + g)1−θ ) N θ (1 + γN h) 1 + g = (1 + γN h).
∂Vt ∂Vt − |h=0,`=1 ≥ 0 ∂` ∂h ∂Vt ∂Vt − |h,`=1−h = 0 (S5) g > 0 ⇒ h > 0 ⇒ ∂` ∂h (S4) g = 0 ⇒ h = 0 ⇒
Conditions (S1)- (S5) are useful in finding stationary growth rates3 . In particular, (S1) and (S2) help us to understand the model’s comparative statics. At an equilbrium with positive growth, the marginal profitability of applying a unit of labour to producing output must exactly equal the marginal profitability of applying that unit of labour to producing knowledge (or be weakly greater than it at a zero growth equilibrium). Yet both of these are functions of the level of competition, the level of spillovers and of h, the equilibrium investment in knowledge per producer per period. To characterise the equilibria we proceed as follows. First we rearrange the two expressions for the marginal profitability of goods and knowledge so as to characterize equality between them in terms of equality between two expressions, one of which is a function of h while the other contains no terms in h (nor any other endogenous variables). We can then show that the first expression is increasing in h within the region in which the intertemporal optimization converges, while the second is increasing in N. Taking the value of the first expression for h = 0 then allows us to find a 3
The expression in (S2) derives from X X β s Pt+s xt+s = β s (N xt+s )−θ xt+s s
s
= = Finally,
P∞
s s=0 a
P
s 1−θ xt+s θ N
sβ
X x1−θ t β s [(1 + g)1−θ ]s . θ N s
= 1/(1 − a) with a = β(1 + g)1−θ .
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value of N above which a zero growth equilibrium exists. Setting the two expressions equal to each other at h > 0 allows us to find sufficient conditions for positive-growth equilibrium. Finally, using the fact that one expressions are increasing in h while the other is increasing in N allows us to conclude that, in a positive-growth equilibrium, the higher is the degree of competition the higher will be the equilibrium growth rate. Define the function φN (h) =
1−h . (1 + γN h)(1 − β(1 + γN h)(1−θ) )
and the quantity uN = α
N −θ ; N − γN θ
t t We note that ∂V ≥ ∂V ⇔ uN ≥ φN (h) at a symmetric stationary solution. An ∂` ∂h equilibrium with positive growth satisfies
uN = φN (h); with (1 + g) = (1 + γN h). Our first result establishes conditions for the existence of a (stationary symmetric) equilibrium with zero growth. The second establishes conditions for a co-ordination problem, namely the existence of multiple such equilibria, including at least one with positive growth. The third establishes that growth rates in the positive growth equilibrium are increasing in N. Proposition 2 (Zero Growth). A symmetric equilibrium with zero growth exists if, and only if N satisfies (G0) (1 − β)uN ≥ 1. Proof:A symmetric equilibrium with zero growth is stationary by construction. At such an equilibrium, g=0⇔h=0 ∂Vt ∂Vt − ]h=0,`=1 ≥ 0 ⇔[ ∂`t ∂ht 1 ⇔ uN ≥ φ(0) ≡ . (1 − β) This yields the required condition (G0). 2
Condition (G0) is necessary and sufficient for the existence of an equilibrium with g = 0. Obviously, equilibrium growth rates are necessarily positive when it fails. In the next result, we show that there may be multiple equilibria (or co-ordination failure): a positive growth equilibrium co-exists with a zero growth equilibrium. 16
Proposition 3 (Coordination Problem). Suppose (G0) holds, and g = 0 is an equilibrium. A positive growth equilibrium exists, in addition, if β (1 + γN )(1−θ) > 1
(G1)
Proof: A positive growth equilibrium satisfies uN = φ(h) for 0 < h ≤ 1. Let h∗ satisfy β(1 + γN h∗ )(1−θ) = 1; condition (G1) implies h∗ < 1. The function φ(.) is continuous on [0, h∗ ); (G0) implies φ(0) < uN , while (G1) implies φ(h∗ ) = ∞. Thus, uN = φ(h) is true for some h < h∗ < 1, and a positive growth equilibrium exists.2 We need to establish that (G0) and (G1) are compatible for some N . The next proposition establishes a sufficient condition. Proposition 4 (Coordination failure). Conditions (G0), (G1) are simultaneously true if N is large enough and (G2) θγ > 1 − α(1 − β). Proof: Condition (G0) is true whenever β < aN ≡ 1 −
1 ; uN
and condition (G1) is true whenever β > bN ≡
1 . (1 + γN )1−θ
Note that an is monotone increasing in n; and that limn→∞ an = 1 − 1−θγ . Suppose α (G2) holds, implying β < limn→∞ an . For any β satisfying (G2), we can find an integer Na such that an > β whenever n ≥ Na . Further, bn is monotonically decreasing in n, and limn→∞ bn = 0. For any β > 0, we can find Nb such that bn < β whenever n ≥ Nb . Conditions (G0), (G1) are simultaneously true for each N ≥ max[Na , Nb ]. 2 Propositions 2-4 show that an economy with large N can have an equilibrium with zero growth, and another equilibrium with positive growth; there may be several positive growth equilibria. This suggests that an increase in N may increase the maximal growth rate compatible with equilibrium. A further proposition evaluates this claim. Proposition 5 (Comparative Statics). Suppose (G0), (G1) hold. Equilibrium growth rates can increase with N . Proof: A positive equilibrium growth rate, gN satisfies uN = φ(h) implying 1 =1 ηN (g) ≡ uN (1 + g)(1 − β(1 + g)1−θ ) + g γN 0 at g = gN . The function ηN (g) is concave in g, and ηN (0) > 1, ηN (0) > 0 whenever 0 (G0) holds. Thus, η (gN ) < 0 at equilibrium, where ηN (gN ) = 1. It follows that gN increases with N if ηN+1 (gN ) > ηN (gN ) For gN close to zero, ηN ' uN (1 − β); uN increases with N , implying the result. 2
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4.3. Implications of the results There has been much debate about the relevance of Schumpeterian phenomena in the growth progress, and about the impact of increased competition on the growth rate. Empirical evidence often fails to corroborate the theoretical prediction of growth models that competition may be harmful to growth (see Aghion and Howitt (1998)). Our model provides a natural way to reconcile these conflicting points of view, since it can generate both equilibria with zero steady-state per capita growth (which we may call A-H or Aghion-Howitt equilibria since they arise from free-rider effects on innovation) and equilibria with a strictly positive steady-state growth rate ( R or Romer equilibria). The first result establishes that A-H equilibria will always exist if there is enough competition. The second establishes that there may in addition exist R equilibria for large enough γ and large enough N. To the extent that divergence between countries may arise from multiple equilibria and not just from sensitive dependence on initial conditions, this suggests that successful countries may be those that overcome coordination failures in the production of knowledge.These co-ordination failures arise because of spillovers: research by other firms increases productivity including the productivity of research. There may be many Romer equilibria: the function φN (h) (which is initially decreasing and eventually explodes) may have more than one turning-point. The third result establishes that the highest-growth equilibrium has a growth rate increasing in N . Taken together, therefore, these three results show that competition (proxied by an increase in the number of firms) has two rather different effects in incentives. First, it makes an A-H equilibrium more likely to exist (this is the ”moral hazard” effect at the bad equilibrium). Secondly, it increases the highest growth rate that could be attained: this is the ”returns to scale” effect due to spillovers. Competition may be beneficial in societies that can solve the associated coordination problems, while being problematic in countries that cannot4 . This in turn focusses theoretical and empirical attention on the institutions that may enable countries to overcome co-ordination problems. It should not be thought that adverse competitive impacts on growth arise solely from the possibility of A-H equilibria. Simulations (not reported) suggest that both competition and the degree of spillovers may have non-monotonic effects on growth even in the Romer equilibria. The two influences also interact in interesting ways: when spillovers are positive but small, competition reduces growth rates, while this is no longer true when spillovers are large. This suggests interesting problems for public policy: the optimal degree of competition and the optimal degree of protection of 4
Bessen & Maskin (2000) show that absence of patent protection in software has not been damaging to innovation, and they argue that competition has market-expanding effects that may more than offset the theoretically familiar free-rider effects on innovation. This argument is similar to our assumption that research by one firm not only improves the productivity of others’ production but also the productivity of their research.
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intellectual property rights will interact in possibly complex ways. Our analysis therefore establishes that both positive and negative effects of competition on growth can be observed, along extremal growth equilibria. Multiple growth equilibria arise in many frameworks (e.g. Azariadis and Drazen (1990), or Acemoglu and Zillibotti (1997), among others; indeed, are thought to be the norm in economies with non-convex production sets of the kind required to explain sustained equilibrium growth (Lucas (1987)). In our analysis, the comparative statics of extremal equilibria provides a device to reconcile theoretical predictions with actual performance. Both competition and spillovers act to increase the range of possible outcomes in a world in which investment in knowledge is driven by expectations of the value of knowledge in the future.
5. Concluding remarks 6. Bibliography Acemoglu, D. and F. Zillibotti (1997) “ Was Prometheus unbound by chance? Risk, diversification, and Growth ” . Journal of Political Economy, 105 (4) 709-51 Aghion, Philippe and Peter Howitt (1998), Endogenous Growth Theory, MIT Press Aghion, Philippe, Nicholas Bloom, Rachel Griffith, Peter Howitt & Richard Blundell (2002), ”Competition and Innovation: an inverted-U relationship”, NBER working paper no. 9269. Azariadis, C. and A. Drazen (1990) “ Threshold Externalities and Economic Development” Quarterly Journal of Economics, 105(2), 501-26 Bessen, J. & E. Maskin (2000): ”Sequential Innovation, Patents & Imitation”, working paper, department of economics, MIT. Carlin, W., S. Fries. M. Schaffer & P. Seabright (2001): ”Competition and Enterprise Performance in Transition Economies: evidence from a cross-country survey”, discussion paper no 2840, Centre for Economic Policy Research. Lucas, R.E. (1987) “ On the Mechanics of Economic Development”, Journal of Monetary Economics, 22, 3-42
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Figure 1: Total Output and the Number of Firms 0,25
0,2
0,15
gamma 0.8, alpha 0.1 gamma1, alpha 0.1 gamma1, alpha 0.2 gamma1, alpha 0.5
0,1
0,05
0 1
2
3
4
5
6
7
8
9 10 11 12 13 14 15
Number of Firms
Figure 2: Average rate of real sales growth by number of domestic competitors Average rate of sales growth by number of competitors
8
6
4
2
0 0
1-3
3+
-2
-4
-6
Number of domestic competitors Source: Carlin et.al. 2001
1
