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A closed loop analysis of competitive innovation
Economics Letters North-Holland
A CLOSED
LOOP ANALYSIS
Moshe JUSTMAN Ben Gurion
Received
University
13 March
339
16 (1984) 339-344
and Abraham
OF COMPETITIVE
INNOVATION
MEHREZ
of the Negeo, Beer Sheba 84 105, Israel
1984
Using dynamic programming, the symmetric, closed loop equilibrium of an oligopolistic market with competition in R&D is derived. Numerical analysis indicates that increasing the number of market firms improves economic performance. This differs from the result obtained by Spence (1982).
1. Introduction Since Schumpeter advanced the hypothesis that monopolies are the engine of progress, extensive formal analysis has been undertaken to support or refute this view. The research reported in this letter is in this vein. The limited tractability of the problem led most early efforts, notably Arrow (1962) and a long line of papers by Kamien and Schwartz (1978, e.g.), to consider single-firm models with exogenous rivalry. This has obvious drawbacks; it does not address symmetric market structures; and the exogeneity of rivalry is, essentially, an inconsistency of the model. In a more recent effort, Spence (1982) constructs a Nash equilibrium model of competition in research and development that overcomes these drawbacks. This is purchased, however, at some cost: the model is deterministic, the equilibrium is open loop, and for the most part analysis is restricted to the case of a zero discount rate. Our present work overcomes these limitations by applying the dynamic programming approach used by Aldrich and Morton (1975) in the single-firm case to a symmetric multi-firm market. This allows us to construct a stochastic model of R&D competition in an oligopolistic 0165-1765/84/$3.00
0 1984, Elsevier Science Publishers
B.V. (North-Holland)
M. Justman,
340
A. Mehrer / Competmue
rnnovation ana!vsis
market and solve it for the optimal (expected profit maximizing) closed loop strategies. Then, following Spence, comparative statistics analysis of the market equilibrium can be carried out by sampling points in the solution space.
2. Definition
of the model
There are n firms in the market, i = 1,. . , n, all engaged in the same R&D project. As each firm completes the project successfully (a discrete event) is ceases its research activities and it enters the manufacturing market and begins to earn a return on its research effort, this return being greater the fewer the firms in the manufacturing market. To fix ideas we assume a constant elasticity demand function, x = ape”, constant unit manufacturing costs V, and NashhCournot equilibrium in the product market. Consequently, if at time tn, firms have innovated successfully then the temporal return to each of these firms is
R(r) = (a/E)(l
- l/Q$-‘PE.
(1)
Thus the return at time t depends on t only through n, and we can write R = R(n,s). The return is thus uniform across firms that have completed innovation successfully. We further assume that all firms employ the same constant discount rate, r. With regard to the innovative process, following Aldrich and Morton (1975) let m,(r) be the rate of dollar spending of firm i at time t, let z,(t) be the cumulative effort devoted to the project by time t, and let F( z,) be the probability that the firm has completed the project successfully by the time its cumulative effort is z,. Assume F(z) = 1 - e-“‘, an exponential distribution function of effort. Let dz,/dt = g(m,(t)) be the function that relates the rate of change of cumulative effort to the rate of dollar spending, and assume that g(0) = 0 and its derivative is positive, decreasing, and asymptotically zero.
3. Solution
of the model
Adapting Aldrich and Morton for the multifirm case, let n,(t) denote the number of firms still engaged in R&D at time t, and let n, = n,(t) be
M. Justman,
A. Mehrer
/
Competitive
rnnouation
341
analyw
the number of firms that have completed their research program at that time, so that n,+ n, = n. Let V,( n,,, n,, t) be the discounted expected value of the project at time t, for a firm before completion, and let Given that an Y(n,, n/7 t) be the same for a firm after completion. optimal path is followed over the remaining infinite horizon, V, must satisfy
v/( n,,
fl/,
t) = max
-m,At+
hg(m,)AtV,(n,
+ 1, n,-
1, r)
IV,20
+hC~(m,)Atv,(n,+l,n~-l,t)+
i
J+’
-bCg(m,)At-rat I+’
I-bg(m,)At
,
v(n,,nf,t+at)
(2)
1
where the summation X ,+,g( m,) is over all firms still engaged in research, except firm i. The explanation of (2) is the following: If the firm spends at rate m,, its spending in an interval of length At is rn,At. This will lead to completion by firm i in the current period with an asymptotic probability of bg(m,)At, while completion of some other firm in the current period will occur with asymptotic probability bC,(m,)At. The last three terms in the square brackets are thus expected profits, with future profits discounted to a first order approximation at the rate of r. Similarly, for successful firms,
Y(n,r,nf, t) = R(n,)At + bCs(m,)AfK(n,+ 1 -bEg(m,rAt /
i
V,(n,,
1, nf- 1,
t)
nf, t+At),
(3)
where C,g(m,) denotes a summation over all firms currently engaged in research. Eqs. (2) and (3) define a system of equations that can be solved recursively, beginning with P’,(n, 0), noting that for given n,%and n,, v,, v/ and m, are stationary. At each stage then, once v, has been calculated,
342
M. Jus~mun,
A. Mehrez
/
C’mpetitioe
innormtion
cmu!ysrs
5 and m, can be derived from (2) and the first order conditions characterizing maximization of (2) over m, for given m,. The symmetric solution is then derived by setting m, equal to m,, and so on. Then L$(O, n) is the value of the research program for a firm ex ante, against which any initial fixed costs involved in carrying out the research program would need to be weighed. Given the solution described above, economic performance of such a market can be assessed by calculating the expected discounted net benefits it generates. These would equal the difference between the total surplus generated in the product market and the costs of development. Again, calculation is recursive. When all firms have completed their research programs successfully, the present value of net benefits is
B(n, 0) = T(n)/r, where T(n) is total surplus generated by the market T(n) = a(l/(E supplying it. [In our case l/nE)E-‘~‘-E.] For n,
when n firms are 1) + l/nE)(l -
XB(n,+l,n/-l)]/[r+n,hg(m(n,,n,))], where m(n,, n,) is the solution above, in the symmetric case.
4. Numerical
of the optimization
problem
described
results
Solution of the model for various combinations of demand elasticities, discount rates, and market structures revealed several consistent patterns. The results were consistent with the general consensus of empirical evidence that R&D effort is greatest in oligopolistic industries with few (but more than one) competing firms. However, economic welfare as measured by the expected net present value of the sum of consumer surplus and producer profits not of R&D outlays, increased monotonically with competition. This latter result differs from the findings of Spence (1982) for an open loop deterministic model of cost reducing innovation.
M. Jus~mun,
Table
A. Mehrer
/
Comprtrfiw
wvmutron
uno~sh
343
1
Elasticity
Number
Expected
Expected
of firms
NP V of net
NPVof
benefits
outlays
1.1
10
935.475
3.920
1.1
3
843.099
5.118
629.387
4.238
1.1 2.0
10
88.643
1.922
2.0
3
74.719
2.236
2.0
1
49.272
1.641
3.0
10
43.263
1.270
3.0
3
35.576
1.421
3.0 10.0
10
10.0
3
10.0
22.928
1.010
9.041
0.401
7.000
0.415
4.266
0.278
R&D
Table 1 presents the numerical values of the expected NPV of net benefits, and R&D outlays for several combinations of demand elasticities and market structures. They illustrate the points noted above. For these calculations we set g(m) = m”.*‘, h = 0.25, u = 10, 1)= 1. and r = 0.1.
5. Concluding
remarks
In this letter we reported the initial results of our application of Aldrich and Morton’s (1975) dynamic programming approach to a multifirm model of competition in R&D in the spirit of Spence (1982). This allows us to derive a symmetric closed loop equilibrium within a stochastic setting. It offers a theoretical framework with numerous possibilities for further development along various lines, including spillovers, cost reduction, different competitive regimes, subsidies, and competitive policy.
References Aldrich,
C. and
projects’, Arrow,
T.E.
K.J., 1962,
Nelson. Princeton.
Morton.
Management Economic
ed., The NJ).
rate
1975, Optimal
Science and
welfare
funding
paths
for a class
of ‘risky
R and
D
21, 491-500. and
direction
the allocation of inventive
of resources activity
for invention,
(Princeton
University
in: R.R. Press,
344
M. Jus~mun,
A. Mehrer / Conzpetrtrve innowrron
UII(I!KSI.S
Kamien. M.1. and N.L. Schwartz, 1978, Potential rivalry monopoly profits and the pace of inventive activity, Review of Economic Studies 45, 547-557. Spence. A.M., 1982, Cost reduction. competition, and industry performance, Dwussion paper no. 897 (Harvard Institute of Economic Research, Harvard University. Cambridge, MA).
A CLOSED
LOOP ANALYSIS
Moshe JUSTMAN Ben Gurion
Received
University
13 March
339
16 (1984) 339-344
and Abraham
OF COMPETITIVE
INNOVATION
MEHREZ
of the Negeo, Beer Sheba 84 105, Israel
1984
Using dynamic programming, the symmetric, closed loop equilibrium of an oligopolistic market with competition in R&D is derived. Numerical analysis indicates that increasing the number of market firms improves economic performance. This differs from the result obtained by Spence (1982).
1. Introduction Since Schumpeter advanced the hypothesis that monopolies are the engine of progress, extensive formal analysis has been undertaken to support or refute this view. The research reported in this letter is in this vein. The limited tractability of the problem led most early efforts, notably Arrow (1962) and a long line of papers by Kamien and Schwartz (1978, e.g.), to consider single-firm models with exogenous rivalry. This has obvious drawbacks; it does not address symmetric market structures; and the exogeneity of rivalry is, essentially, an inconsistency of the model. In a more recent effort, Spence (1982) constructs a Nash equilibrium model of competition in research and development that overcomes these drawbacks. This is purchased, however, at some cost: the model is deterministic, the equilibrium is open loop, and for the most part analysis is restricted to the case of a zero discount rate. Our present work overcomes these limitations by applying the dynamic programming approach used by Aldrich and Morton (1975) in the single-firm case to a symmetric multi-firm market. This allows us to construct a stochastic model of R&D competition in an oligopolistic 0165-1765/84/$3.00
0 1984, Elsevier Science Publishers
B.V. (North-Holland)
M. Justman,
340
A. Mehrer / Competmue
rnnovation ana!vsis
market and solve it for the optimal (expected profit maximizing) closed loop strategies. Then, following Spence, comparative statistics analysis of the market equilibrium can be carried out by sampling points in the solution space.
2. Definition
of the model
There are n firms in the market, i = 1,. . , n, all engaged in the same R&D project. As each firm completes the project successfully (a discrete event) is ceases its research activities and it enters the manufacturing market and begins to earn a return on its research effort, this return being greater the fewer the firms in the manufacturing market. To fix ideas we assume a constant elasticity demand function, x = ape”, constant unit manufacturing costs V, and NashhCournot equilibrium in the product market. Consequently, if at time tn, firms have innovated successfully then the temporal return to each of these firms is
R(r) = (a/E)(l
- l/Q$-‘PE.
(1)
Thus the return at time t depends on t only through n, and we can write R = R(n,s). The return is thus uniform across firms that have completed innovation successfully. We further assume that all firms employ the same constant discount rate, r. With regard to the innovative process, following Aldrich and Morton (1975) let m,(r) be the rate of dollar spending of firm i at time t, let z,(t) be the cumulative effort devoted to the project by time t, and let F( z,) be the probability that the firm has completed the project successfully by the time its cumulative effort is z,. Assume F(z) = 1 - e-“‘, an exponential distribution function of effort. Let dz,/dt = g(m,(t)) be the function that relates the rate of change of cumulative effort to the rate of dollar spending, and assume that g(0) = 0 and its derivative is positive, decreasing, and asymptotically zero.
3. Solution
of the model
Adapting Aldrich and Morton for the multifirm case, let n,(t) denote the number of firms still engaged in R&D at time t, and let n, = n,(t) be
M. Justman,
A. Mehrer
/
Competitive
rnnouation
341
analyw
the number of firms that have completed their research program at that time, so that n,+ n, = n. Let V,( n,,, n,, t) be the discounted expected value of the project at time t, for a firm before completion, and let Given that an Y(n,, n/7 t) be the same for a firm after completion. optimal path is followed over the remaining infinite horizon, V, must satisfy
v/( n,,
fl/,
t) = max
-m,At+
hg(m,)AtV,(n,
+ 1, n,-
1, r)
IV,20
+hC~(m,)Atv,(n,+l,n~-l,t)+
i
J+’
-bCg(m,)At-rat I+’
I-bg(m,)At
,
v(n,,nf,t+at)
(2)
1
where the summation X ,+,g( m,) is over all firms still engaged in research, except firm i. The explanation of (2) is the following: If the firm spends at rate m,, its spending in an interval of length At is rn,At. This will lead to completion by firm i in the current period with an asymptotic probability of bg(m,)At, while completion of some other firm in the current period will occur with asymptotic probability bC,(m,)At. The last three terms in the square brackets are thus expected profits, with future profits discounted to a first order approximation at the rate of r. Similarly, for successful firms,
Y(n,r,nf, t) = R(n,)At + bCs(m,)AfK(n,+ 1 -bEg(m,rAt /
i
V,(n,,
1, nf- 1,
t)
nf, t+At),
(3)
where C,g(m,) denotes a summation over all firms currently engaged in research. Eqs. (2) and (3) define a system of equations that can be solved recursively, beginning with P’,(n, 0), noting that for given n,%and n,, v,, v/ and m, are stationary. At each stage then, once v, has been calculated,
342
M. Jus~mun,
A. Mehrez
/
C’mpetitioe
innormtion
cmu!ysrs
5 and m, can be derived from (2) and the first order conditions characterizing maximization of (2) over m, for given m,. The symmetric solution is then derived by setting m, equal to m,, and so on. Then L$(O, n) is the value of the research program for a firm ex ante, against which any initial fixed costs involved in carrying out the research program would need to be weighed. Given the solution described above, economic performance of such a market can be assessed by calculating the expected discounted net benefits it generates. These would equal the difference between the total surplus generated in the product market and the costs of development. Again, calculation is recursive. When all firms have completed their research programs successfully, the present value of net benefits is
B(n, 0) = T(n)/r, where T(n) is total surplus generated by the market T(n) = a(l/(E supplying it. [In our case l/nE)E-‘~‘-E.] For n,
when n firms are 1) + l/nE)(l -
XB(n,+l,n/-l)]/[r+n,hg(m(n,,n,))], where m(n,, n,) is the solution above, in the symmetric case.
4. Numerical
of the optimization
problem
described
results
Solution of the model for various combinations of demand elasticities, discount rates, and market structures revealed several consistent patterns. The results were consistent with the general consensus of empirical evidence that R&D effort is greatest in oligopolistic industries with few (but more than one) competing firms. However, economic welfare as measured by the expected net present value of the sum of consumer surplus and producer profits not of R&D outlays, increased monotonically with competition. This latter result differs from the findings of Spence (1982) for an open loop deterministic model of cost reducing innovation.
M. Jus~mun,
Table
A. Mehrer
/
Comprtrfiw
wvmutron
uno~sh
343
1
Elasticity
Number
Expected
Expected
of firms
NP V of net
NPVof
benefits
outlays
1.1
10
935.475
3.920
1.1
3
843.099
5.118
629.387
4.238
1.1 2.0
10
88.643
1.922
2.0
3
74.719
2.236
2.0
1
49.272
1.641
3.0
10
43.263
1.270
3.0
3
35.576
1.421
3.0 10.0
10
10.0
3
10.0
22.928
1.010
9.041
0.401
7.000
0.415
4.266
0.278
R&D
Table 1 presents the numerical values of the expected NPV of net benefits, and R&D outlays for several combinations of demand elasticities and market structures. They illustrate the points noted above. For these calculations we set g(m) = m”.*‘, h = 0.25, u = 10, 1)= 1. and r = 0.1.
5. Concluding
remarks
In this letter we reported the initial results of our application of Aldrich and Morton’s (1975) dynamic programming approach to a multifirm model of competition in R&D in the spirit of Spence (1982). This allows us to derive a symmetric closed loop equilibrium within a stochastic setting. It offers a theoretical framework with numerous possibilities for further development along various lines, including spillovers, cost reduction, different competitive regimes, subsidies, and competitive policy.
References Aldrich,
C. and
projects’, Arrow,
T.E.
K.J., 1962,
Nelson. Princeton.
Morton.
Management Economic
ed., The NJ).
rate
1975, Optimal
Science and
welfare
funding
paths
for a class
of ‘risky
R and
D
21, 491-500. and
direction
the allocation of inventive
of resources activity
for invention,
(Princeton
University
in: R.R. Press,
344
M. Jus~mun,
A. Mehrer / Conzpetrtrve innowrron
UII(I!KSI.S
Kamien. M.1. and N.L. Schwartz, 1978, Potential rivalry monopoly profits and the pace of inventive activity, Review of Economic Studies 45, 547-557. Spence. A.M., 1982, Cost reduction. competition, and industry performance, Dwussion paper no. 897 (Harvard Institute of Economic Research, Harvard University. Cambridge, MA).