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Firm metrics with continuous R&D, quality improvement, and Cournot quantities.

Abstract The paper considers a dynamic two-firm model of intra-industry trade in which the firms compete for the same market on the basis of product reliability. By assumption, the home firm always has the reliability cost advantage but it may or may not have the manufacturing cost advantage. The results suggest that reliability improvement always helps customers in that they pay a lower full quality price. Comparing the home firm with the foreign firm, metrics such as price, sales, profit margins, and variable profits depend on the relative costs, with the low cost firm performing better. Finally, although this is not the common outcome, the paper suggests that it is possible for the reliability cost advantages gained by R&D expenditures to overcome manufacturing cost disadvantages.

Keywords Dynamic R&D * Quality improvement * Intra-industry trade

JEL F12 * O38 * H25

Introduction

Consider a two-firm model of intra-industry trade in which the firms compete in a world market on the basis of product reliability. The home firm chooses its dynamic R&D path knowing that some proportion of its quality improvements will be backward engineered by its foreign competitor. The primary research questions of the paper concerns the effect of the endogenous quality improvement on customers and on the difference between firms of various firm metrics: price, sales, profit margins, and variable profits, that is, profits before R&D expenditures. The effect on customers is measured by the full quality price, that is, the purchase price plus a term that reflects the expected customer cost of product failure.

The primary results of the paper are that, compared to a base case of no R&D and, thus, no reliability improvement, customers always pay a lower full quality price. Along an optimal path, the reliability may improve quickly enough in some cases at the beginning of the planning period so that full quality price actually falls initially and then grows because the demand parameters are growing over time. As for the firm metrics, the total market sales and the home firm's output are always higher along an optimal path than for the base path. Compared to the base case, the foreign firm sometimes sells more in the optimal case and sometimes less.

On the other hand, comparing the sales and variable profits of the two firms, the one with the cost advantage sells more. Whether this firm is the home firm depends on the relative manufacturing and reliability costs. By assumption, the home firm always has the reliability cost advantage (see (11)) but it may or may not have the manufacturing cost advantage. Relative sales, profit margins, and variable profits then depend on the sum of the manufacturing and reliability costs. There are three cases to be considered. First, if the home firm has the manufacturing cost advantage then it sells more with higher profit margins and variable profits than the foreign firm does. Second, if the home firm has a manufacturing cost disadvantage that is large enough to outweigh its reliability cost advantage, then the home firm sells less with lower profit margins and variable profits than the foreign firm does. Third, in the most interesting case, the home firm begins with a net cost disadvantage (combining manufacturing and reliability costs) but improves its quality fast enough to emerge at the end of the planning period with a net cost advantage. In this case, the home firm's metrics are less than the foreign firm's at the beginning of the planning period but greater by the end of the planning period. Although this case happens in only a little over 10% of the cases in the Monte Carlo simulation, it suggests that it is possible for the reliability cost advantages gained by R&D expenditures to overcome manufacturing cost disadvantages.

Although there is a growing body of work on dynamic R&D games between duopoly firms, as far as we are aware, none of them do the comparisons between firms of the present paper. Confessore and Mancuso (2002) are interested in the effect of certain spillovers on R&D paths and, thus, assume identical firms. Joshi and Vonortas (2001) assume firms selling a homogeneous good but with differing initial R&D levels and thus production costs. Their primary result is that the asymmetry between firms disappears over time. Dockner et al. (1993), in the context of a race for technological breakthrough, simply assume identical firms. Colombo et al. (2009) investigate the relationship between market size and closed/open loop steady state equilibria but, again, assume firms sell a homogeneous good. Suetens (2008) examines the very interesting question of whether cooperation in R&D prompts collusive prices and finds that it does. Although firms are not identical, the cooperation in R&D prompts firms to (theoretically) charge identical prices.

Petit and Tolwinski (1999) and Petit et al. (2000) examine respectively the social optimality of research joint ventures and the choice between exporting (as in the present paper) and foreign direct investment. Both papers assume firms selling a homogeneous good and, thus, the firm metrics are the same for both firms. Two of these metrics, however, can be compared to certain results of the present paper. Their cumulative R&D spending paths are similar to our product reliability paths, Fig. 1. Their price paths are strictly declining over time, presumably because the demand parameters are static, where prices in the present paper generally rise with the increase in the demand parameters, Fig. 3, left panel. Petit et al. (2009) again investigate exporting versus foreign direct investment paying special attention to locational spillovers. As in the previous papers the cumulative R&D spending paths resemble our reliability paths; they do not report price paths perhaps because they again assume identical firms and a homogeneous good. Finally, Highfill and McAsey (2010) have a paper with many formal similarities to the present paper. It examines the effect on the home firm of the "competitiveness" of a foreign competitor. Though with its focus on the home firm, it does not compute such metrics as price, sales, profit margin, and variable profits for both the home and foreign firm, let alone compare them.

[FIGURE 1 OMITTED]

The key formal assumptions of this paper can be found in various places in this literature. Linear demand curves are found in Confessore and Mancuso (2002), Joshi and Vonortas (2001), Colombo et al. (2009) (except for the treatment of a transportation cost term), Highfill and McAsey (2010), and Suetens (2008). The assumption that firms sell to a single market can be found in Herguera and Lutz (2003), Herguera et al. (2000), and DeCourcy (2005). It is shown in Highfill and McAsey (2010) that the single market assumption is equivalent to a world without transportation or other trade costs. A quadratic R&D function is found in Highfill and McAsey (2010) and Dockner et al. (1993). A two-stage game at each moment with quality (or R&D) decided in the first stage and quantity in the second can be found in Confessore and Mancuso (2002), Joshi and Vonortas (2001), Petit and Tolwinski (1999), Petit et al. (2000), Petit et al. (2009) (although Petit and her coauthors typically have a third stage in the game as well) and Highfill and McAsey (2010). Suetens (2008) has a similarly structured game, except that prices are chosen in the second stage. The game in Colombo et al. (2009) is such that quantity and R&D are chosen simultaneously.

The Model and an Example

Suppose the firms play a two-stage game with order of play as follows. Product reliability is chosen first and then each firm chooses its own quantity taking the other firm's quantity (and both reliabilities) as given. Solutions are computed using generalized backward induction. The quantity decisions are thus considered first, and these depend in turn on manufacturing costs and the demand side assumptions of the model. Cournot quantities are used to write a variable profits function which is then used in the dynamic reliability/R&D problem.

Customer Behavior, Cournot Quantities, and Prices

Suppose customers' reservation prices, v(t), for a perfect product at time t are distributed uniformly on the interval ([W.sub.0][e.sup.rt], [V.sub.0][e.sup.rt]). Imperfect products impose costs, [K.sub.0][e.sup.rt], on the customer that are not reimbursed by the firm. Customers whose reservation price, v(t), satisfies

v(t) [greater than or equal to] [P.sub.1](t) + (1 - [X.sub.1](t))[K.sub.0][e.sup.rt] = [P.sub.2](t) + (1 - [X.sub.2](t))[K.sub.0][e.sup.rt] (1)

purchase the product. Where, for the ith product: [P.sub.i](t) is the purchase price; l-[X.sub.i](t) is the probability of product failure; [X.sub.i](t) is product reliability; (l-[X.sub.i](t)[K.sub.0][e.sup.rt] is the expected customer cost of product failure; and FQP(t) = [P.sub.i](t) + (1-[X.sub.i](t))[K.sub.0][e.sub.rt] is the full quality price; the home firm is i=1; the foreign firm i=2. Notice from (1) that, although in general the firms' costs, reliabilities, and prices need not be the same, the full quality prices must be the same for both firms to have positive sales.

Market quantity, Q(t) [equivalent to] [Q.sub.1](t) + [Q.sub.2](t), is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [N.sub.0][e.sup.st] is the potential market size at time t. The indirect demand functions are:

[P.sub.i](t) = [e.sup.rt] ([V.sub.0] - [K.sub.0](1 - [X.sub.i](t)) - [[[V.sub.0] - [W.sub.0]]/[[N.sub.0][e.sup.st]]]([Q.sub.1](t) + [Q.sub.2](t))). (2)

Suppose firm i has a per unit manufacturing cost of [mc.sub.i0][e.sup.rt]. The (expected) cost of replacing or repairing the defective units is [mc.sub.i0][e.sup.rt](1 - [X.sub.i](t)) [Q.sub.i](t), where (1 - [X.sub.i](t)) [Q.sub.i](t) is the expected number of defective units. Manufacturing and repair costs are:

[e.sup.rt] ([mc.sub.i0] + [mc.sub.i0](1 - [X.sub.i](t)))[Q.sub.i](t). (3)

Variable profits for firm i are thus [VP.sub.i](t) [equivalent to] [margin.sub.i](t) [Q.sub.i](t) where [margin.sub.i](t) = [P.sub.i](t) - [e.sup.rt]([mc.sub.i0] + [mc.sub.i0](1 - [X.sub.i](t))), i.e., price minus per unit manufacturing and repair costs.

Solving the first order conditions, d [VP.sub.i](t)/d[Q.sub.i](t) = 0, recalling (2), yields

[Q.sub.i](t) = [1/3] [[N.sub.0]/[[V.sub.0] -[W.sub.0]]] [e.sup.st] ([V.sub.0] - 2[mc.sub.i0] + [mc.sub.j0] - 2([K.sub.0] + [mc.sub.i0]) (1 - [X.sub.i](t)) + ([K.sub.0] + [mc.sub.j0]) (1 - [X.sub.j](t))) (4)

where j denotes firm 2 in [Q.sub.1](t) and conversely. Using (2) again,

[P.sub.i](t) = [e.sup.rt] ([1/3] ([V.sub.0] + [mc.sub.10] + [mc.sub.20]) - [K.sub.0](1 - [X.sub.i](t)) + [1/3] [summation over (k)](([K.sub.0] + [mc.sub.k0]) (1 - [X.sub.x](t)))). (5)

Notice first that if products were 100% reliable, the quantities and prices would be the same as in the typical Cournot game. To aid the intuition of the effect of reliability improvements, define the marginal value of reliability as [k.sub.0]+[mc.sub.i0]. An improvement in reliability helps customers by reducing the cost of product failure as captured by [K.sub.0] and helps firms because they don't have to pay the replacement cost of [mc.sub.i0]. Thus, setting aside the slope term [N.sub.0]/([V.sub.0]-[W.sub.0]) in (4), for [Q.sub.i](t) an improvement in its own reliability (other things being equal) increases quantity by two-thirds the amount of the improvement in reliability, while an improvement in its competitor's reliability (other things being equal) reduces quantity by one-third the amount of the improvement in reliability.

Next, consider prices (5). The direct effect of an improvement in reliability is that the firm can simply raise its price by [K.sub.0] times the change in reliability. That is, the improvement in reliability reduces the expected customer cost of failure and the firm captures the whole of that reduction by an increase in price. The indirect effect of an improvement in reliability is to increase market quantity which increases price by one-third of the value of the improvement (Recall the effect of the [N.sub.0]/([V.sub.0]-[W.sub.0]) is to change quantities into dollar values, i.e., it is the slope of the demand function). Notice that an improvement in the competitor's reliability has the same indirect effect as an improvement in the own firm's reliability, but, of course, there is no direct effect. The net effect on price of the firm's change in its own reliability is positive when [mc.sub.i0][less than or equal to]2[K.sub.0], which is always the case for the Monte Carlo simulations reported later.

Notice that the growth rate r in the demand and cost parameters need not be the same as the growth rate s of the potential market. The former might be related, for example, to the inflation rate while the latter might be related to the population growth rate. In the base case of no R&D the change in quantities over time is driven by the growth rate of the potential market, while the change in price over time is driven by the growth rate in the demand and cost parameters.

Although the computations will not be shown, Eqs. 1-5 imply:

Q(t) = [1/3][[N.sub.0]/[[V.sub.0] - [W.sub.0]][e.sup.st] (([V.sub.0] - [mc.sub.10]) + ([V.sub.0] - [mc.sub.20]) - [summation over (i)] (([K.sub.0] + [mc.sub.i0]) (1 - [X.sub.i](t)))) (6)

[margin.sub.i](t) = [[[V.sub.0] - [W.sub.0]]/[N.sub.0]][e.sup.(r-s))t] [Q.sub.i](t) (7)

[VP.sub.i](t) = [[[V.sub.0] - [W.sub.0]]/[N.sub.0]][e.sup.(r-s)t] [Q.sub.i][(t).sup.2] (8)

FQP(t) = [P.sub.i](t) + [K.sub.0](1 - [X.sub.i](t))[e.sup.rt] = [e.sup.rt] ([V.sub.0] - [[[V.sub.0] - [W.sub.0]]/[[N.sub.0][e.sup.st]]]Q(t)).

Research & Development, Reliability, and the Home Firm's Profit Maximization Problem

Improvements in reliability require expenditures on research and development, [E.sub.1](t)[greater than or equal to]0, but are subject to diminishing marginal returns, as captured for convenience by the square root function. Specifically,

d[X.sub.1]/dt = k(1 - [X.sub.1](t))[square root of ([E.sub.1](t))], [X.sub.1](0) [equivalent to] [X.sub.10] (10)

where k>0, 0 [less than or equal to] [X.sub.1](t) [less than or equal to] 1, and [X.sub.10]>0. Formally, [X.sub.1] is the state variable, and [E.sub.1] is the control. Notice that the closer the reliability is to one the less productive a given level of R&D expenditure will be. Because [E.sub.1] is typically orders of magnitude larger than [X.sub.1], the parameter k is a small fraction. For the sake of simplicity we refer to [E.sub.1] as the expenditure on R&D but it is really the component of expenditure which varies with reliability.

Suppose firm one, the home firm, is the leader in R&D in the sense that firm two's reliability is determined by the following rule (for 0 [less than or equal to] [xi] [less than or equal to] 1):

[X.sub.2](t) = (1 - [xi])[X.sub.10] + [xi][X.sub.1](t). (11)

The home firm knows that its competitor's reliability path will be a weighted average of the initial reliability and its own. The theory is agnostic about how the foreign firm achieves its reliability improvement. Finally, recall the definition of the "base case" of no R&D or reliability improvement:

[X.sub.1](t) = [X.sub.2](t) = [X.sub.10]. (12)

The optimal control problem for firm one can now be stated; for notational convenience, the firm subscripts are suppressed. The firm maximizes the (discounted) difference between variable profits ((8) using (4)) and R&D expenditure E(t), subject to (10-11). Choose E(t) to maximize the integral: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] subject to dX/dt = k(1 - X(t))[square root of(E(t))], X(0) = [X.sub.0] assuming 0 [less than or equal to] X(t) [less than or equal to] 1 where [sigma] = s + r - [rho], and [rho] is the discount rate.

The constants [[alpha].sub.0] and [[beta].sub.0] are found from (8) using (4) and (11):

[[alpha].sub.0] [equivalent to] [square root of ([[N.sub.0]/[9([V.sub.0] - [W.sub.0])]])] ([V.sub.0] - [K.sub.0] - 4[mc.sub.10] + 2[mc.sub.20] - ([K.sub.0] + [mc.sub.20]) (1 - [xi])[X.sub.0]

[[beta].sub.0] [equivalent to] [square root of ([[N.sub.0]/[9([V.sub.0] - [W.sub.0])]])] (2([K.sub.0] + [mc.sub.10]) - [xi]([K.sub.0] + [mc.sub.20])).

The Hamiltonian is:

H = [e.sup.[sigma]t][[[[alpha].sub.0] + [[beta].sub.0]X(t)].sup.2] - [e.sup.-[rho]t] E(t) + p(t)k(1 - X(t))[square root of (E(t))].

The first order condition is [partial derivative]H/[partial derivative]E = 0 which gives 0 = -[e.sup.-[rho]t] + [p(t)k(1 - X(t))/2[square root of(E(t))]].

This equation can be solved for E(t) = [(1/2p(t)k(1 - X(t))).sup.2][e.sup.2[rho]t]. The costate equation is given by dp/dt = -[partial derivative]H/[partial derivative]X which becomes:

[dp/dt] = -2[e.sup.[sigma]t][[beta].sub.0][[[alpha].sub.0] + [[beta].sub.0]X(t)] + p(t)k[square root of (E(t))]

= -2[e.sup.[sigma]t][[beta].sub.0][[[alpha].sub.0] + [[beta].sub.0]X(t)] + p(t)k([1/2]p(t)k(1 - X(t)))[e.sup.[rho]t]

= -2[e.sup.[sigma]t][[beta].sub.0][[[alpha].sub.0] + [[beta].sub.0]X(t)] + [1/2]p[(t).sup.2][k.sup.2](1 - X(t))[e.sup.[rho]t].

So we get two first order differential equations:

[dp/dt] = -2[e.sup.[sigma]t][[beta].sub.0][[[alpha].sub.0] + [[beta].sub.0]X(t)] + [1/2]p[(t).sup.2][k.sup.2](1 - X(t))[e.sup.[rho]t], p(T) = 0 (13)

[dX/dt] = [1/2]p(t)[k.sup.2][(1 - X(t)).sup.2][e.sup.[rho]t], X(0) = [X.sub.0].(14)

An Example

In order to get the flavor of the results consider an example where the foreign firm's reliability path is simply the average of the initial value and the home firm's reliability path; that is, in Eq. 11, [xi] = 1/2. The other parameters are in line with the analysis of Highfill and McAsey (2010) and the parameter values are given in the Appendix.

Consider first the reliability paths. As shown in Fig. 1, reliability improvement is faster at the beginning of the planning period. For these parameters, 25% of the total reliability improvement that will occur during the 3 year planning period is accomplished in about 0.4 years, 50% in about 0.9 years, and 75% in about 1.5 years.

Recalling that only the home firm R&D is endogenous in the model, the home firm expenditure on R&D and the co-state variable are shown in Fig. 2.

[FIGURE 2 OMITTED]

As suggested perhaps by Fig. 1, Fig. 2 shows that R&D expenditure starts at a relatively high level and then declines over the planning period. The shadow price--a measure of the effort to increase the rate of reliability improvement--also declines over the planning period.

Now that the reliability paths have been characterized, the goal is explore the effect of reliability improvement on the various firm metrics. Considering prices, Fig. 3, left panel shows that home firm always charges the higher price (because it has the higher reliability). From (5) the difference in prices depends only on the reliability path. Price paths for the base case without R&D are not shown, but for the Monte Carlo results reported later, the price for firm one is greater than the base price for all t in about 90% of the cases. On the other hand, the price for firm two is less than the base price for all t in about 97% of the cases. Considering Fig. 3 right panel, the full quality price path falls Initially because of the reliability improvement and then increases as reliability improvement slows; see Fig. 1. It is always the case that the full quality price is less than it would be in the base case of no R&D although customers' savings come in the form of reduced expected customer cost of failure rather than a lower purchase price. For the Monte Carlo results reported later, about 56% of the observation had a U-shaped full quality price path; for the remainder the path was strictly upward sloping.

[FIGURE 3 OMITTED]

See Fig. 4 for quantities and variable profits.

[FIGURE 4 OMITTED]

Both quantities and variable profits are higher for firm one because it has the reliability advantage and manufacturing costs are the same in this example. (The next section considers the effect of differing manufacturing costs.) Comparing quantities and variable profits, for the Monte Carlo results reported later, total quantity always increases, (see (6)). Firm one always sells more than the base case, but firm two sometimes sells more and sometimes sells less. Recall also that while (Eq. 8) variable profits are the quantities squared, the exponential growth rate for variable profits is the sum of that for prices and quantities.

Another Look at the Differences in Firm Metrics

The analysis of the present section is based on the fact that the difference in firm metrics depends on total sales and the difference in sales between firms. These in turn depend on three factors: (1) the manufacturing cost advantage of firm one, (2) the reliability cost advantage of firm one, and (3) the total cost of product failure. (The first two are positive or negative as firm one or firm two has the cost advantage; the third is always nonnegative.) The ultimate goal of the section is to construct a single graph which shows that time helps customers and favors the home firm over the foreign firm.

We will begin with the definitions. The manufacturing cost advantage of firm one, MCA, is MCA = [mc.sub.20] - [mc.sub.10]; the reliability cost advantage of firm one, RCA, is RCA(t) = (([K.sub.0] + [mc.sub.20])(1 - [X.sub.2](t)) - ([K.sub.0] + [mc.sub.10])(l - [X.sub.1] (t))); and the total cost of product failure, TCPF, is TCPF(t) = [summation over i] (([K.sub.0] + [mc.sub.i0])(1 - [X.sub.i](t))).

These three items are all defined for a given set of parameters, that is, ([W.sub.0], [V.sub.0], [N.sub.0], [K.sub.0], [mc.sub.10], [mc.sub.20], k, s, r, [rho], [xi]) are being held constant. The interpretation of the manufacturing cost advantage is straightforward and constant over time. For the reliability related costs, recall that [K.sub.0] captures the cost of product failure for customers and [mc.sub.i0] the cost for firms. Thus, the cost for society is ([K.sub.0] + [mc.sub.i0])(1 - [X.sub.i](t)), which is positive unless the firm's product is 100% reliable. Although the reliability cost advantage for firm one can be positive or negative, the passage of time (almost) always improves the reliability cost from firm one's point of view (The Monte Carlo analysis suggests that only about 1% of interior solutions are exceptions). If firm one initially has a reliability cost advantage, it will only get larger over time. If it does not initially have a reliability cost advantage, its disadvantage will get smaller over time and possibly disappear altogether. These two relative costs determine the difference in the metrics between firms. On the other hand, total sales and the full quality price depend on the total cost of product failure which is always nonnegative, and the passage of time always reduces it.

To see how these factors determine the metrics, considerthe differences in sales from Eq. 4 [DELTA]Q(t) = [Q.sub.1]t) - [Q.sub.2](t) = [N.sub.0]/([V.sub.0] - [W.sub.0])[e.sup.st](MCA + RCA(t)). Over time, the difference in quantities between firms depends on both the effect that the reliability improvement has on RCA(t) and an exponential growth term. In order to focus on just the changes caused by reliability improvements, we will discount the difference in quantities.

[DELTA]DiscQ(t) = [DELTA]Q(t) [e.sup.-st] = [[N.sub.0]/[[V.sub.0] - [W.sub.0]]] (MCA + RCA(t)).(15)

The other metrics will be discounted in a similar way. From (7) [DELTA]margin(t) = [margin.sub.1] (t) - [margin.sub.2](t) = [[V.sub.0] - [W.sub.0]/[N.sub.0]][e.sup.(r-s)t][DELTA]Q(T) = [e.sup.rt](MCA + RCA(t)) so that

[DELTA]Discmargin(t) = [DELTA]margin(t) [e.sup.-rt] = (MCA + RCA(t)).(16)

The total sales and frill quality price are (from (6) and (9) respectively):

DiscQ(t) [equivalent to] ([Q.sub.1](t) + [Q.sub.2](t))[e.sup.-st] = [1/3][[N.sub.0]/[[V.sub.0] - [W.sub.0]]](2[V.sub.0] - ([mc.sub.10] + [mc.sub.20]) - TCPF(t)) (17)

DiscFQP(t) [equivalent to] FQP(t)[e.sup.-rt] = [1/3]([V.sub.0] + ([mc.sub.10] + [mc.sub.20]) + TCPF(t)) (18)

The difference in (discounted) variable profits requires a little more work. Using (8)

[DELTA]DiscVP(t) = [e.sup.-(r+s)t]([VP.sub.1](t) - [VP.sub.2](t)) = [e.sup.-2st][[[V.sub.0] - [W.sub.0]]/[N.sub.0]]([Q.sub.1][(t).sup.2] - [Q.sub.2][(t).sup.2])

= [e.sup.-2st][[[V.sub.0] - [W.sub.0]]/[N.sub.0]]([DELTA]Q(t)*Q(t))

=[1/3][[N.sub.0]/[[V.sub.0] - [W.sub.0]]](MCA + RCA(t))(2[V.sub.0] - ([mc.sub.10] + [mc.sub.20]) - TCPF(t)).(19)

As will be seen shortly in Fig. 5, the goal is to map various level curves in (TCPF (t), RCA(t)) space for some fixed t. From (15) and (16), noting the (discounted) difference in sales or profit margins depends only on RCA(t), which is the vertical axis in Fig. 5. Level curves for sales or profit margins are simply horizontal lines. Whether firm one sells more than firm two (and earns a higher profit margin) depends on how the RCA(f)compares to the MCA. In the example shown, neither firm has a manufacturing cost advantage. For the only quantity level curve shown, namely the horizontal axis, firm one sells exactly the same amount as firm two. To simplify the picture the other horizontal lines are not shown, but in general firm one sells more (and has a higher profit margin) when RCA(t)>MCA and it sells less (and has a lower profit margin) when RCA(t)<MCA. Comparing level curves, moving up the vertical axis implies that the difference between firm one's and firm two's sales or profit margin is getting larger.

[FIGURE 5 OMITTED]

From (17) and (18) the (discounted) total sales and full quality price depend only on TCPF(t) and, thus, are vertical lines (not shown in Fig. 5). An increase in the total cost of product failure decreases sales and the full quality price. Imagine two vertical lines were drawn on Fig. 5; the right one would correspond to the higher full quality price and lower market demand.

Finally, the (discounted) difference in variable profits curves can be found by rearranging (19) to get RCA(t) = [[V.sub.0]-[W.sub.0]/[N.sub.0]] [3 [DELTA]DiscVP(t)/2 [V.sub.0]-([mc.sub.10]+[mc.sub.20])-TCPF(t)] - MCA. As suggested by (19) the (discounted) variable profits for firm one are positive when it sells more. Moving up a vertical line (i.e., holding TCPF(t) constant and increasing RCA(t)) implies an increase in [DELTA]DiscVP(t). Similarly, moving to the left along a horizontal line (i.e., holding RCA(t) constant and increasing TCPF(t)) also implies an increase in [DELTA]DiscVP(t).

As a final step, the line segment starting on the horizontal axis and moving northwest is the path over time of the optimal solution of the example of the previous section (parameters in the Appendix). The endpoint on the axis is the location of the total cost of product failure at time zero TCPF(0)=135.68. By construction, this point on the axis is also the location of the base case since reliability is unchanging. But for the optimal solution, time always decreases the total cost of product failure and almost always increases the reliability cost advantage for firm one. Thus, as shown, the time path moves northwest. To interpret these results recall also that the time paths of all variables have an exponential term and the term driven by reliability improvement. By constructing the discounted level curves, we have taken out the effects of the exponential. Thus, Fig. 5 implies that as compared to the base case, discounted market sales are always greater, and the difference is getting larger. Similarly, the discounted full quality price is less than the base case, and the difference is increasing. Time helps customers. Time also favors firm one over firm two as well. In the case shown, the firms were equal in sales and variable profits initially, but by the end of the planning period, firm one has a decided advantage.

Monte Carlo Analysis

By construction, in the example, neither firm has an initial manufacturing or reliability cost advantage. The goal of the present section is to use a Monte Carlo simulation to examine the implications of relaxing that assumption, (see the Appendix for details of the simulation.) The numerical algorithm used is often referred to as the Forward-Backward Sweep Method and uses a Runge-Kutta order four differential equation routine to solve the first order Eqs. 13-14 resulting from the optimal control problem. See Lenhart and Workman (2007) for a discussion of the method.

Considering reliability first, recall from (11) the foreign firm's reliability is a weighted average of the home firm's reliability (at any given t) and the initial reliability. Table 1 gives the descriptive statistics for the reliability improvements, defined as (([X.sub.i](T) - [X.sub.0])/[X.sub.0])* 100.
Table 1 Reliability improvement

                                 Mean   Std. Dev.   Min    Max

Firm 1 Reliability Improvement  12.391   11.922    1.295  61.984
Firm 2 Reliability Improvement   6.346    7.053    0.004  37.726


For the 310 parameter sets, yielding internal solutions, the average reliability improvement for firm one was 12.391% with a range from 1.295% to 61.984%. For firm two, the average reliability improvement was 6.346% with a range of virtually 0-37.726%.

The data for some of the comparisons suggested by Fig. 5 is given in Table 2.
Table 2 Firm comparisons

                              Mean   Std. Dev.     Min      Max

Average Difference Price       3.82     4.28        0.01    24.68

Average Difference Quantity    3.57    10.73      -21.08    34.73

Average Difference Variable  248.15   755.86    -1749.65  2807.94
Profits

Average Full Quality Price   163.94    20.26      114.52   224.55


For each parameter set yielding an internal solution, the average difference in price (quantity, variable profits) was computed from the point of view of the home firm one. On average, the home firm charges $3.82 more for its product, sells 3.57 more units, and earns $248.15 more in variable profits than firm two. Recall the full quality price is the same for both firms.

Noting that, on average, firm one had about twice the reliability improvement of firm two, Table 2 shows that for our observations, firm one always charges the higher price. As shown above, whether firm one has the higher variable profits depends on sales. On average firm one did earn higher profits, but as suggested by the Minimum column, there were certainly cases where the opposite result held. Firm one had the higher quantity and variable profits for the entire planning period in 163 cases (53%); firm two had the higher quantity and variable profits for the entire planning period in 109 cases (36%). For the remainder, 38 cases (12%) firm one initially had lower sales and variable profits but overtook firm two during the planning period. In these cases, firm one always had a manufacturing cost disadvantage, but its reliability cost advantage allowed it to eventually overtake firm two in terms of sales and variable profits. The average period of time it took firm one to overtake firm two was 0.89 years, or about 30% of the planning period. The longest such time was 2.43 years or about 81% of the planning period. Thus, our data suggest that when the home firm starts out with a cost disadvantage, and is able to use its advantage in the rate of reliability improvement to overtake firm two, it will do so relatively quickly, in something under a year.

The effect of the reliability improvement for customers is best captured by the full quality price as compared to the base full quality price (which is not itself reported). The average savings for customers from a lower (than base) full quality price was about 3.37%, or about $5.96, with the range of savings being from about 0.36% to 11.77% of the full quality price. Recalling the time path of the full quality price in Fig. 3, right panel, in about 44% of the observations full quality price was rising during the entire planning period. But in the remainder, the reliability improvement actually "bought" customers an interval of falling full quality price. For these cases, on average the full quality price fell for 1.18 years or about 39% of the planning period. At the top end, the full quality price fell for 2.39 years, or about 80% of the planning period.

Conclusion

The paper considers the case of the optimal R&D path of a home firm that knows that its quality improvements will be partially copied by its foreign competitor. In the case that the home firm also has a manufacturing cost advantage, time does nothing but increase the home firm's advantages over the foreign firm, in terms of sales, prices, profit margins, and variable profits (profits before R&D expenditures). The case that the home firm has a manufacturing cost disadvantage may perhaps be even more interesting. In some cases, our simulation suggests that the dynamic reliability improvement of the home firm may allow it to overtake the foreign firm, even though the foreign firm maintains its manufacturing cost advantage during the entire planning period. Although beyond the scope of this paper, it may be the case that extending the planning period might increase the likelihood of the home firm overtaking the foreign firm. One might speculate that given enough time, an advantage in reliability improvement might always outweigh a manufacturing cost disadvantage.

Appendix

The parameters for the main example are [W.sub.0] = 100, [V.sub.0] = 200, [N.sub.0] = 60, [mc.sub.10] = [mc.sub.20] = 72, [K.sub.0] 140, k=0.03, r = s = [rho] = 0.025, [xi] = 0.5, [X.sub.10] = [X.sub.20] = 0.68, and T=3.

For the Monte Carlo simulation, the parameters were chosen independently and randomly from a 60%-interval ([+ or -]30%) around their value in the example. The parameter k, as suggested by Highfill and McAsey (2010), must be orders of magnitude smaller than the other parameters and is the same for all observations.

The selection of 60%-intervals was determined by two factors. First, increasing the width of the randomization interval resulted in significant decreases in the number of data sets with positive prices and quantities. Using the 60% intervals there were 310 observations that generated such solutions out of 700 randomly generated parameter sets. Second, with significantly longer randomization intervals, the number of data sets that failed to produce convergent solutions increased significantly.
Table 3 Descriptive statistics for parameters and selected variables

Parameter               Mean    Std.Dev.    Min      Max

[W.sub.0]               93.74    14.98     70.03   129.13
[V.sub.0]              223.75    26.59    151.25   259.62
[N.sub.0]               60.32    10.92     42.03    77.96
[mc.sub.10]             71.09    12.05     50.57    93.31
[mc.sub.20]             72.82    11.95     50.82    93.38
r,s,[rho]                0.025    0.004     0.018    0.032
[xi]                     0.517    0.285     0.001    0.997
[X.sub.10]=[X.sub.20]    0.706    0.107     0.482    0.881
[K.sub.0]              137.84    23.89     98.10   181.96


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Jennett Highfill * Michael McAsey

J. Highfill (*) M. McAsey

Department of Economics, Bradley University, Peoria, IL 61625, USA

e-mail: highfill@bradley.edu

DOI 10.1007/S11294-010-9269-9

Published online: 16 July 2010

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Date:Aug 1, 2010
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