Welfare measures in dynamic firm R&D games.
Keywords Differential games * Innovation competition * Product reliability
JEL Classifications D92 * O31 * F12
That firm level Research and Development (R&D) decisions are important in a dynamic economy borders on a truism, something agreed upon by economists and policymakers alike. The present paper examines the welfare effects of a dynamic R&D game at the firm level in a two-country, two-firm, intra-industry trade context. As economists, we use social surplus to evaluate the benefits of such a game. One innovation of the present paper is that the social surplus functions allow for firm ownership to be divided between citizens of both countries. Economists do not typically think of the trade balance as a measure of economic welfare, but in the public arena it seems as if the trade balance is the only thing that matters, as if the trade balance were per se a measure of economic welfare. The primary result of the paper is that the dynamic time path of social surplus and the trade balance do not track well together. This paper suggests that economists thinking about dynamic R&D games will have to defend imports as having a positive effect on social surplus regardless of trade balance effects.
A literature with models of dynamic intra-industry trade with R&D is beginning to be developed. Papers in the foreign direct investment literature typically focus on export versus investment type questions (e.g., Petit et al. (2012), Sanna-Randaccio (2002)). Papers in the trade literature focus on questions like whether firms should form R&D cooperative ventures (e.g., Cellini and Lambertini (2009), Petit and Tolwinski (1999)) or the effect of spillovers (e.g., Femminis and Martini (2011)). The present paper is an extension of Highfill and McAsey (2013) to include social surplus. It has some similarities to Highfill and McAsey (2010a, b) except here the R&D paths for both firms are endogenous while in the earlier papers, only the R&D path for one firm is endogenous. Static papers in a similar vein that focus on various policy questions are Haaland and Kind (2006, 2008) and Gretz et al. (2009, 2012). Finally, there is related empirical literature on firm-level R&D (e.g. Lokshin et al. (2008)).
Assume a two-country, two-firm, intra-industry trade model. Countries are denoted i= A, B and firms j=1,2. Firm 1 is located in country A in the sense that its R&D and production are conducted there; firm 2 is located in country B. Sales of a firm's output in its own country are denoted [Q.sub.A1](t) and [Q.sub.B2] respectively; imports and exports are from country A's point of view, that is, [Q.sub.M](t) denotes sales of firm 2 in country A and [Q.sub.X](t) denotes sales of firm 1 in country B.
The basic setup is that for each firm product quality [R.sub.j](t) is determined by its R&D expenditures [E.sub.j](t); the production functions (for quality improvement) are increasing but subject to diminishing marginal returns:
d[R.sub.j](t) / dt = [z.sub.j](1 - [R.sub.j](t))[quare root of (E.sub.j](t))] (1)
where [z.sub.j] > 0 is a constant and 0[less than or equal to][R.sub.j](t)[less than or equal to]1. Suppose on the benefit side that improvements in product quality reduce the firm's manufacturing costs as well as increase demand for the firm's product. While these things can be thought of completely abstractly, we find it intuitively useful to think of product quality as product reliability. Suppose the decision on whether or not a product is reliable is the customer's, and it is binary. If the product is returned for replacement or repair the product is not reliable; otherwise it is reliable. The proportion of a firm's items that are reliable is [R.sub.j](t).
Assume a constant per unit manufacturing cost. Suppose that units of the product that fail are returned by the customer and replaced (or if repaired that the repair cost is the same as the replacement cost). For firm 1, for example, manufacturing costs have three components: (1) the cost of the original units is [mc.sub.10][e.sup.rt][Q.sub.1](t), where [mc.sub.10] is a constant, costs grow exponentially, and sales are the sum of domestic sales and exports: [Q.sub.1](t)=[Q.sub.A1](t)+[Q.sub.X](t); (2) the (expected) cost of replacing or repairing the defective units is [mc.sub.10][e.sup.rt](1-[R.sub.1](t))[Q.sub.1](t), where (1-[R.sub.1](t))[Q.sub.1](t) is the expected number of defective units; and (3) these costs are reduced by [sigma][e.sup.rt][R.sub.2](t) which captures the inter-firm spillovers, i.e., the reduction in costs for firm 1 from improvements in its competitor's quality. The spillover parameter 0 is assumed to be small. Firm 2 costs are analogous, defining its sales as [Q.sub.2](t)=[Q.sub.M](t)+[Q.sub.B2](t) and noting that its exports are country A's imports. So the manufacturing costs are:
[c.sub.l] (t) = [e.sub.rt][mc.sub.l0] (1 + (1-[R.sub.l](t)) - [sigma][R.sub.2](t))[Q.sub.l](t)
[c.sub.2] (t) = [e.sub.rt][mc.sub.20] (1 + (1-[R.sub.2](t)) - [sigma][R.sub.1](t))[Q.sub.2](t) (2)
Without loss of generality, define
Cost Ratio = [mc.sub.20] / [mc.sub.10] = [[rho].sub.MC]. (3)
That is, firm 2's costs are greater than firm 1 's if and only if [rho]MC>1. Equivalently, [mc.sub.10]= [mc.sub.0] and [mc.sub.20] = [rho]MC [mc.sub.0].
On the demand side, define the full quality price as the purchase price of a product plus the expected cost for the customer of an unreliable product:
[FQP.sub.A](t) = [P.sub.A1] + (1 - [R.sub.1](t)) [K.sub.A0][e.sup.rt](t) + (1 - [R.sub.2](t))[K.sup.A0][e.sup.rt] (4)
[FQP.sub.B](t) = [P.sub.X] + (1 - [R.sub.1](t)) [K.sub.B0][e.sup.rt](t) + (1 - [R.sub.2](t))[K.sup.B0][e.sup.rt]. (5)
For both firms to have positive sales in a given country, the full quality prices must be equal (although in general the sales prices and the reliabilities will not be the same). Market demands are linear in both quantity and reliability:
[Q.sub.A](t) = [Q.sub.A1](t) + [Q.sub.M](t) = [V.sub.A0][e.sup.st] - [e.sub.(s-r)t]FQ[P.sub.A] (6)
[Q.sub.B](t) = [Q.sub.X](t) + [Q.sub.B2](t) = [V.sub.B0][e.sup.st] - [e.sub.(s-r)t]FQ[P.sub.B] (7)
where s is a parameter, perhaps related to the population growth rate. We will discuss s and r more below Eqs. (9)--(12).
Suppose that the ratio of the intercepts between countries is the same as the ratio of the customer costs of product failure:
Demand Side Ratio = [V.sub.B0] / [V.sub.A0] = [K.sub.B0] / [K.sub.A0] [[rho].sub.DS]. (8)
Intuitively, when Pas >1 country B might have a higher income so that it values the product more and has a higher opportunity cost of customers' time when the product fails--or it may simply prefer the product in terms of its tastes and preferences.
With these assumptions, the demand curves are:
[P.sub.A1](t) = [e.sup.rt]([V.sub.0] - [K.sub.0](1 - [R.sub.1](t))-[e.sup.-st]([Q.sub.A1](t) + [Q.sub.M](t))) (9)
[P.sub.M](t) = [e.sup.rt]([V.sub.0] - [K.sub.0](1 - [R.sub.2](t))-[e.sup.-st]([Q.sub.A1](t) + [Q.sub.M](t))) (10) [P.sub.X](t) = [e.sup.rt]([[rho].sub.DS][V.sub.0] - [K.sub.0](1 - [R.sub.1](t)))-[e.sup.-st]([Q.sub.B2](t) + [Q.sub.X](t))) (11)
[P.sub.B2](t) = [e.sup.rt]([[rho].sub.DS][V.sub.0] - [K.sub.0](1 - [R.sub.2](t)))-[e.sup.-st]([Q.sub.B2](t) + [Q.sub.X](t))) (12)
Notice the formulation permits demand to grow over time. The intercepts of the demand functions grow at a rate of r, perhaps related to the inflation rate. The parameter s allows the slope of the demand functions to stay constant over time (when s = r) or to get a bit steeper or flatter.
Define "Variable Profits," that is, instantaneous profits before R&D expenditures:
V[P.sub.1](t) = ([P.sub.A1](t) - [c.sub.1](t))[Q.sub.A1](t) + ([P.sub.X](t) - [c.sub.1](t))[Q.sub.X](t)
V[P.sub.2](t) = ([P.sub.M](t) - [c.sub.2](t))[Q.sub.M](t) + ([P.sub.B2](t) - [c.sub.2](t))[Q.sub.B2](t) (13)
Substituting (2) and (9)-(12) into (13), variable profits are written as functions of both firms' quantities and reliabilities (and the parameters). The games the firms play can now be described. The firms choose their R&D expenditures [E.sub.j] (f), which determine the product reliabilities. Then, quantities are chosen by Cournot competition. Using backward induction, the quantities (as functions of the reliabilities) are described using the quantity first order conditions: ??V[P.sub.1](t)/??[Q.sub.A1](t) = 0 ??V[P.sub.1](t)/??[Q.sub.X](t)=0, ??V[P.sub.2](t)/??[Q.sub.B2](t) = 0, and ??V[P.sub.2](t)/??[Q.sub.M](t) = 0.
Substituting the quantities into the variable profit functions means that variable profits are now written as a function of the reliabilities.
The formal optimization problems can now be stated. The objective for each firm is to maximize profits: variable profits minus the expenditure on R&D. Choose [E.sub.j](t) to maximize the integral:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)
[dR.sub.j](t) = [z.sub.j](1-[R.sub.j](t))[square root of ([E.sb.j])](t) (15)
[R.sub.j](0) = [R.sub.j0] 0[less than or equal to][R.sub.j][less than or equal to]1, j= 1,2 (16)
where [rho] is the discount rate, and defining 0[less than or equal to][[empty set].sub.j][less than or equal to]1 as the firm's share of R&D expenditures; that is, each country pays the proportion 1-[[empty set].sub.j] of the R&D expenditures for the firm located there. In other words, 1-[[empty set].sub.j] is the subsidy rate. An equilibrium is the solution of (14)-(16) for j=1,2. (15) (16)
Finally, in order to investigate the effect of the firm's behavior on welfare, we need economy-wide measures. We will use social surplus and the trade balance as these measures. Consumer surplus in each country is given by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (18)
These are straightforward calculations, except perhaps for the exponentials; for details see the Appendix. In both consumer surpluses, the second equality uses the quantity first order conditions.
Instantaneous social surplus is defined as follows:
[SS.sub.A](t) = [CS.sub.A](t) + [[theta].sub.A]([VP(t).sub.1] - [[phi].sub.1][E.sub.1](t)) + (1-[theta].sub.B])([VP.sub.2](t)-[[phi].sub.2][E.sub.2](t)) - (1-[[phi].sub.1])[E.sub.1](t) (19)
[SS.sub.B](t) = [CS.sub.B](t) + (1-[theta].sub.A]([VP.sub.1](t)-[[phi].sub.1])[E.sub.1](t)) + [theta].sub.B]([VP.sub.2](t)-[[phi].sub.2])[E.sub.2](t)) - (1-[[phi].sub.2])[E.sub.2](t). (20)
Social surplus for a given country is consumer surplus, country i's proportion of the profit of the firm located in country i, country i's proportion of the profit of the firm located in the other country, less the funding agency in country i's share of the R&D expenditure of the firm located there. The parameter [[theta].sub.A] is the home ownership parameter in country A. that is, the proportion of the stock of firm 1 owned by citizens of country A, and [[theta].sub.B] is the proportion of the stock of firm 2 owned by citizens of country B.
The trade balance from country A's point of view is
[TB.sub.A](t) = [P.sub.X](t)[Q.sub.X](t) - [P.sub.M](t)[Q.sub.M](t). (21)
It will be convenient to aggregate these over time, so the cumulative (discounted) social surpluses and trade balances are:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (22)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (23)
Characterization of a Dynamic Equilibrium
Numerical methods are used to analyze optimal solutions. The Forward-Backward Sweep method is used along with a Runge--Kutta fourth order differential equation routine to solve the optimality conditions above. See Lenhart and Workman (2007) for a discussion of the method. We confine our attention to parameter sets yielding interior solutions; see Highfill and McAsey (2013) for a discussion.
Our goal is to examine how changes in the cost ratio (3) and the demand side ratio (8) affect the cumulative measures of social surplus (22) and the trade balance (23). To do so, we will define a base set of parameters and examine its state and control paths. Then we will consider the effect of changes in the cost and demand side ratios. The parameters used in the base case are found in Table 1.
Table 1 Parameter values for base case [V.sub.0] [K.sub.0] [mc.sub.0] [[sigma]] [S=r=[rho]] 200 100 100 0.1 0.025 [V.sub.0] [[phi].sub.j] [[theta].sub.j] [R.sub.j0] [z.sub.j] T 200 0.5 0.5 0.9 0.03 3
The base cost ratio is [[rho].sub.MC]=0.6 and the base demand side ratio is [[rho].sub.DS]=0.9. Figure 1 shows the time paths of the state variables while Fig. 2 shows the time paths of the controls.
Briefly, because [[rho].sub.MC]=0.6<1, firm 2 has a manufacturing cost advantage, [mc.sub.10>[mc.sub.20]; firm 1 has a manufacturing cost disadvantage. Firm 2 responds by doing more R&D and producing a more reliable product than firm 1 does. The co-state variables are similar to those shown in Highfill and McAsey (2013).
The primary outcomes we are interested in are social surplus and the trade balance. These are shown in Figs. 3 and 4 as functions of time; both figures show country A.
Social surplus increases over time, but the rate of increase decreases. Its path is reminiscent of the reliability paths. The trade balance of country A is always in deficit during the planning period because it has the manufacturing cost disadvantage and produces a less reliable product. Notice it is S-shaped, with the deficit first growing rather quickly, then somewhat slower, and then more quickly. The policy implication is that the trade balance over time does not track with social surplus.
Let us briefly consider the effects of the demand side assumptions before we turn our attention to sensitivity analysis in the next section. The demand functions (9)-(12) are written to allow for the intercept to grow over time and the slope to either increase or decrease. Setting r=s=0 is equivalent to assuming a demand function which does not change over time.
Figure 5 shows the effect on social surplus. Briefly, growing demand (i.e., r=s=0.025) means that both firms will do more R&D and produce a better product as compared to constant demand (i.e., r=s=0). The effect on social surplus is generally positive. There is a very small interval of time when the effect on social surplus is negative; this is because by assumption at time zero the reliabilities are the same for the two paths even though the R&D expenditures differ. When only expenditures differ, the path with the higher expenditure has the lower social surplus. Finally, we have shown the social surplus path in the case of constant demand and no R&D spending. The firms' R&D game does increase social welfare, but not right away. A bit of patience is required to see the benefits of R&D spending.
Welfare Measures and Sensitivity Analysis
We now want to turn our attention to our primary task of describing how changes in the cost ratio (3) and the demand side ratio (8) affect the cumulative measures of social surplus (22) and the trade balance (23) calculated over the entire planning period. This is done in Figs. 6 and 7, which are sensitivity analyses in the sense that they describe the result of changes in parameter sets on the equilibrium outcomes. In Fig. 6, the demand side ratio is on the horizontal axis. At [[rho].sub.DS]=1, the market demands in both countries are the same. When [[rho].sub.DS]> 1 , country B has the larger market ([V.sub.BO]>[V.sub.AO]). Reading from left to right then, in Fig. 6, country A is exporting to an increasingly larger market. The vertical axis gives the cumulative trade balance (23) and the cumulative social surplus (22) for country A. Thus, from country A's point of view, reading from left to right, the export market for its domestic firm is getting larger. As country A's exports increase, its trade balance improves and its social surplus increases. By either measure, it benefits country A for its trading partner to grow. Notice that the cumulative trade balance is never positive in this example, but it gets much closer to positive near the right edge of Fig. 6.
Figure 7 does a similar analysis for changes in the relative manufacturing costs between firms. The cost ratio parameter [[rho].sub.MC] is on the horizontal axis. At [[rho].sub.MC]=1, the manufacturing costs are the same and the reliabilities are the same. The left side of the figure ([[rho].sub.MC]<l) is where firm 2 has the manufacturing cost advantage while the right side ([[rho].sub.MC]>1) is where firm 1 has the manufacturing cost advantage. Reading from left to right, the manufacturing cost advantage of firm 1, located in county A, increases. The vertical axis again shows the cumulative social surplus and trade balance for country A.
Unlike Fig. 6, where the trade balance and social surplus track together, in Fig. 7 social surplus declines but the trade balance increases as firm 1 's manufacturing costs improve (i.e., get smaller) relative to firm 2's. The trade balance is most negative when social surplus is highest. Other things being equal, with regards to social surplus, it is better to import from a low cost foreign firm (i.e., social surplus is higher towards the left of Fig. 7). With regard to the trade balance, a low cost foreign competitor implies a large deficit. In short, these results suggest that imports are good for social surplus but are associated with a trade balance in deficit.
A standard result of intra-industry trade models is that a country both imports and exports the same good. In the model of the present paper, this phenomenon is explained by differences in reliability caused by differing R&D strategies. The major result of the paper is that using trends in the trade balance as proxies for the welfare effects of firm level R&D is quite misleading. A second major result of the paper considers the question of whether a country benefits by being the home of the relatively low manufacturing cost producer. The answer again depends on whether the trade balance or social surplus is the measure of whether a country benefits. In this experiment, it is certainly possible that social surplus is highest when the trade deficit is largest and when the positive effects of imports for customers outweigh the effects on firms.
Appendix: Demand Side Theory
While the demand functions in the body of the paper can simply be assumed, it is also possible to derive them from a uniform distribution of reservation prices. Suppose reservation prices, vi(t), at any time t in country i are distributed uniformly on the interval (([V.sub.i0]-[[gamma].sub.i0])[e.sup.rt],[V.sub.i0][e.sup.rt]). (In general all cost and demand parameters are assumed to grow at an exponential captured by [e.sup.rt], perhaps related to the inflation rate.) The maximum reservation price any customers have for the product is [V.sub.i0][e.sup.rt] and the range of reservation prices is [[gamma].sub.i0][e.sup.rt] where 0<[[gamma].sub.i0]<[V.sub.i0]. (Notice that the range of distribution prices is getting bigger over time but only at the same rate as the maximum reservation price. The assumption essentially allows demand to grow exponentially over time.) The population distribution function is [f.sub.i] = 1/[[gamma].sub.i0][e.sup.rt] Customers whose ni reservation price satisfies the following condition will purchase the product:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)
for country A and B respectively. Customers are risk neutral in the sense that their buying decisions are based on the full quality price, the sum of the price and the expected customer cost of product failure.
Under these assumptions and using (24), the market quantity demanded for country A is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (26)
That is, market demand is the proportion of potential customers that buy the product times the potential market size [N.sub.i](t)=[N.sub.i0][e.sup.st] which grows exponentially at a rate of [e.sup.st] and may be related to the population growth rate. Consumer surplus in this setup is simply
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (27)
Suppose that the economy with the larger population has a larger range of distribution prices, but that the initial ratio of population to range is the same in the two countries, that is, ([N.sub.A0]/[[gamma].sub.A0])=([N.sub.B0]/[[gamma].sub.B0]). Choose the unit of measurement of the potential market size so that this ratio is one
These assumptions normalize the slope of the market demand functions to one, except for the exponential. Both the market demand functions and the demand functions for each firm's product in each country follow. For the numerical results. [N.sub.AO] = [[gamma].sub.AO] = 100.
Published online: 26 July 2013 [c] International Atlantic Economic Society 2013
Int Adv Econ Rcs (2013) 19:439-449
DOI 1O. 1007/s11294-013-9425-0
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J. Highfill (*) * M. McAsey
Bradley University, Peoria, IL, USA
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|Author:||Highfill, Jannett; McAsey, Michael|
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|Date:||Nov 1, 2013|
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