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Product Reliability, R&D, and Manufacturing Cost Shocks.

Introduction

Consider an industry where product failure imposes considerable costs on both the buyer and the seller. A seller who warrantees a product by promising to repair or replace a defective unit incurs the associated costs. The buyer has the costs associated with returning the product. Suppose the firm can do R&D which increases the proportion of products that are judged by customers to be "reliable" in that they are not returned. The key research question of this paper concerns the effects of such R&D for firms with market power under various assumptions about marginal manufacturing costs. Suppose, for example, there is a shock to input prices that increases the marginal manufacturing cost. The firm might respond by increasing its R&D. The resulting improvement in product reliability would then offset some of the increase in the manufacturing costs. On the other hand, the increase in the marginal manufacturing cost may cause quantity produced to fall sufficiently, so that the optimal response is to do less R&D. The primary result is that a manufacturing cost shock prompts the firm to do less R&D in the cases where the replacement cost is low or the marginal manufacturing cost is high. Conversely, if the replacement cost is high and the marginal manufacturing cost is low, then the firm increases R&D, mitigating some of the increase in the manufacturing cost.

The paper also compares the outcomes for reliability, profits, consumer surplus, and social surplus for the optimal R&D case as compared to the case of doing no R&D, paying particular attention to how exogenous changes in the marginal manufacturing cost affect this comparison. The first section considers a monopoly seller in a single market. In that case it will be seen that exogenous increases in the marginal manufacturing cost increase the positive effect of an optimal R&D strategy compared to no R&D for all outcomes except for reliability itself. The second section considers a two-country, two-firm, intra-industry trade model. In the monopoly model R&D increases firm profits. In the duopoly model optimal R&D increases reliability, consumer surplus, and social surplus more than it does in the equivalent monopoly model, but it does not necessarily increase profits. The duopoly R&D game has a prisoner's dilemma type solution. Profits would be higher for both firms if neither does R&D, but profits are even worse if your competitor does R&D and you do not. Equivalently, if you are the only firm to do R&D your profits increase. The effect of an exogenous increase in the marginal manufacturing cost for a symmetric duopoly is analogous to that of a monopoly, except for the effect on profits just mentioned. The duopoly model also allows for the case of country-specific marginal manufacturing cost shocks.

Monopoly models with reliability or quality as the output of R&D are found in Daughety and Reinganum (1995) and Saha (2007). El Ouardighi et al. (2014) is a recent addition to the duopoly literature initiated by d'Aspremont and Jacquemin (1988, 1990) where R&D expenditures cause a reduction in a marginal cost parameter. Ma (2015) presents a monopoly model where R&D increases market size but has no effect on cost. The international flavor of that model is that production may shift from North, where the R&D is done, to South.

Haaland and Kind have a set of papers with an international duopoly model similar to that of the present paper in that R&D increases product quality. In Haaland and Kind (2006) R&D increases customer demand but has no effect on costs, while in Haaland and Kind (2008) R&D does not affect demand but reduces manufacturing cost. In Jinji and Toshimitsu (2006) R&D affects demand but not costs, while in DeCourcy (2005) R&D affects costs but not demand. Both effects of R&D spending are found in Gretz et al. (2009) and Highfill and McAsey (2010). As far as we are aware, the research question of the present paper has not been addressed in the literature: does a firm that experiences a shock in its manufacturing cost increase or decrease its R&D expenditures in response? The general quadratic structure of the R&D expenditure function follows that of Jinji and Toshimitsu (2006), DeCourcy (2005), Haaland and Kind (2006, 2008), and Gretz et al. (2009): increasing R&D expenditure increases reliability but with diminishing marginal returns.

Single Market Monopoly Model

Consider a product that sometimes "fails" in the sense that customers are not satisfied, and suppose the seller pledges to repair or replace any returned unit, perhaps as a signal of the quality of the product or to demonstrate the firm's commitment to caring about the customer experience. A product that does not "fail" is "reliable." Reliability, denoted R, is the proportion of all products that do not fail, or equivalently, the probability that any given unit will not fail. Customers know this probability, and consider the costs associated with product failure in their decision making, time and postage costs perhaps, or the inconvenience. In the model the cost to the customer who happens to buy a unit that fails is measured by the parameter, K. For the arbitrary customer, the expected cost of product failure is (1 - R)K. Customers are risk neutral in the sense that their buying decisions are based on the purchase price and the expected customer cost of product failure.

Similarly, the firm knows the expected number of units it must replace or repair, (1 -R) Q, where Q is output. Suppose the firm has a constant per unit manufacturing cost of mc. As a way to think about the cost of replacing or repairing a unit, suppose that it is simply a multiple z of the marginal manufacturing cost

c = mc + zmc (1 - R). (1)

For each unit originally manufactured at a marginal cost of mc, the proportion that will need to be repaired or replaced is (1 - R), and the cost per unit of doing so is z mc. In some cases the parameter z may be greater than 1, if, for example, there are substantial costs in inventorying the replacement units or if there are substantial handling costs associated with retrieving them from inventory and shipping them to the customer. On the other hand, some products may be repairable at a cost that is less than the cost of making the original unit in which case z would be less than one. We do not make any particular assumption about the relationship between the cost of replacing a unit for the firm and the cost of returning it for the customer.

Consumer Behavior

Suppose customers whose reservation price v satisfies the following condition will purchase the product:

v [greater than or equal to] P + (1 - R)K, (2)

where P is the product's purchase price. The term P + (1 - R)K, the "full quality price," includes both the purchase price and the expected cost of product failure. While a customer does not know whether the specific unit they purchase will fail, they include the expected cost to them of a failure in their decision making. Customers whose reservation price is greater than or equal to the full quality price will purchase it. Assume the distribution of reservation prices is uniform, f = 1/[gamma], on the range 0 [less than or equal to] V - [gamma] [less than or equal to] v [less than or equal to] V; where Vis the maximum reservation price. (This formulation allows the lower bound of the reservation price range to be strictly positive.) Under these assumptions the quantity demanded is:

Q = N[[integral].sup.V.sub.P+(1,-R)K] f dv = N[[integral].sup.V.sub.P+(1,-R)K] 1/[gamma] dv = N/[gamma] (V- P - (1 -R)K). (3)

Market demand is the proportion of potential customers who buy the product times the potential market size N Without loss of generality, choose the unit of measurement of the potential market size so that N/[gamma]= 1 The indirect demand function is:

P = V - (1 - R)K-Q, (4)

and consumer surplus is:

CS = N[[integral].sup.V.sub.P+(1,-R)K] [[v-(P + (1 - R)K)]/[gamma]]dv = [1/2] [Q.sup.2] (5)

Please see the online supplemental appendix, paragraph 1, for the derivation of (5).

Firm Behavior

Improvement in reliability requires R&D. The firm's R&D expenditure is denoted RDE [greater than or equal to] 0. Increasing R&D expenditure improves the reliability of the firm's product, but is subject to diminishing marginal returns. Specifically, we assume the functional form:

RDE = k[(R - [R.sub.0]).sup.2], (6)

where k > 0, and defining 0 [less than or equal to] [R.sub.0] [less than or equal to] 1 as the "default reliability" that would occur in the absence of any R&D expenditure. The interpretation of the parameter k is that it governs the size of the effect of R&D spending on reliability. This is perhaps seen more clearly by writing reliability as a function of RDE:

R(RDE) = [R.sub.0] + [k.sup.-[1/2]][RDE.sup.[1/2]]. (7)

In general, a given amount of R&D spending produces less reliability as k increases. It is shown in the online supplemental appendix, paragraph 2, that an interior solution requires k > 1/4 [(K + mcz).sup.2], that is, k must be larger than (one fourth of) the square of the sum of the per unit costs of product failure for the customer and the firm. For future reference, note:

R'(RDE) = [1/2] [k.sup.-[1/2]] [RDE.sup.-[1/2]]. (8)

Finally, although for the sake of simplicity we refer to RDE as R&D expenditure, more completely it could be thought of as the component of R&D expenditure which varies with reliability. There would normally be many fixed-cost R&D expenditures as well.

Under the manufacturing cost assumptions summarized by (1) total manufacturing costs are c times output Q:

cQ = mcQ + zmc(1-R)Q, (9)

and firm profits are:

[PI] = (P-c) Q-RDE = VP-RDE, (10)

defining "variable profits," VP, as the profits before R&D expenditures (i.e., profit margin times quantity, where the margin is the purchase price minus the unit manufacturing cost):

VP(Q) = (P-c)Q = (V-(1-R)K-Q-(mc + zmc(1-R)))Q, (11)

where the right-hand equality uses (4) and (9).

Suppose the firm uses a two-stage process to arrive at a solution: in the first stage the R&D expenditure level is chosen and in the second stage quantity is chosen. Formally, backward induction will be used. The quantity first order condition from (10) is:

[partial derivative][PI]/[partial derivative]Q = VP'(Q) = (V-(1-R)K-Q-(mc + zmc(1-R)))-Q = 0, (12)

(thinking only of interior solutions). Solve (12) for Q,

Q = [1/2](V - (1 - R)K - (mc + zmc(1-R))) (13)

and then use (7) to substitute for R. Therefore Q can be written as a function of RDE:

Q(RDE) = [1/2] (v - (1 - ([R.sub.0] + [k.sup.1/2][RDE.sup.1/2]))K - (mc - zmc(1 - ([R.sub.0] + [k.sup.1/2][RDE.sup.1/2])))). (14)

Notice that the form of this equation is Q = X + Y [square root of RDE], where X and Y are parameters.

For any Q satisfying the first order condition (12), it is shown in the online supplemental appendix, paragraph 3, that

VP = [Q.sup.2]. (15)

To find the optimal level of R&D expenditure, substitute (14) into (10) using (15):

[PI](RDE) = VP(RDE)-RDE = [(Q(RDE)).sup.2]-RDE, (16)

where,

VP(RDE) = [1/4] [(v - (1 - ([R.sub.0] + [k.sup.[1/2]][RDE.sup.[1/2]]))K - (mc - zmc(1 - ([R.sub.0] + [k.sup.[1/2]][RDE.sup.[1/2]])))).sup.2]. (17)

The first order condition with respect to RDE is:

[PI](RDE) = dVP(RDE)/Drde -1 = 0, (18),

where

dVP(RDE)/dRDE = [1/4] ([(K + zmc).sup.2]/k (K + zmc)(V - (1 - [R.sub.0])K-mc (1 + z-z[R.sub.0]))/[k.sup.[1/2]][RDE.sup.[1/2]]). (19)

The intuition of (18) will be developed further below (see Equation (24)) but for now notice at the optimum that the marginal value of an R&D dollar (19) is equal to 1 (the marginal cost of an R&D dollar). It will also aid intuition shortly to notice that the second term of dVP/dRDE in (19) is related to the level of output in the case that no R&D is done so that R - [R.sub.0]. Denote the "no-R&D" outcome with a subscript "0". That is, the no-R&D output is [Q.sub.0], the no-R&D variable profits are V[P.sub.0] = [Q.sup.2.sub.0] and the noR&D profits are the same as variable profits [[PI].sub.0] = V[P.sub.0]. Using this notation the second term of (19) can be written

[mathematical expression not reproducible] (20)

so that the first order condition with respect to RDE (18) can be written as:

[1/2](K + zmc)[[PI].sup.[1/2].sub.0][RDE.sup.[1/2]] = 1 - 1/4 [(K + zmc).sup.2]/k. (21)

The optimal level of R&D expenditure is

[mathematical expression not reproducible] (22)

or alternatively,

RDE = M [Q.sup.2.sub.0] = M [VP.sub.0] = M[[PI].sub.0] (23)

where M = (4k[(K + z mc).sup.2])/[(4k - [(K + z mc).sup.2]).sup.2]. An interpretation of the optimal level of R&D expenditure is that it is the "no-R&D" level of profits (variable profits or quantity squared) times a "multiplier" M, which increases as the sum of the per unit costs of unreliable products increases (i.e., as K + zmc increases) or as the effectiveness of an R&D dollar increases (i.e., as k in (7) decreases). In the numerical analysis of the next section, this multiplier ranges from about 2% to about 25%, so the expenditure on R&D never approaches the level of profits.

The next section focuses on the effect of optimal R&D on model outcomes, and then investigates the impact of a shock to the marginal manufacturing cost. Equation (22) suggests that the main story will involve the contrary effects of an increase in mc on R&D. On the one hand, an increase in the mc makes reliability more valuable so that more R&D will be done. That is, the "multiplier" in Eq. (22) increases. On the other hand, an increase in mc has a negative effect on quantity, including the no-R&D quantity in Eq. (22).

Single Market Monopoly Outcomes

Benefits of R&D: R&D versus No R&D

Figure 1 shows the percentage increases in some key model outcomes when the optimal R&D is done as compared to doing no R&D. These are reliability, R, profits, II, consumer surplus, CS, and social surplus, SS (the sum of profits and consumer surplus). For the parameter values V = 200, K = 100, [R.sub.0] = 0.9, k = 150, 000, and z = 1, when the marginal manufacturing cost is $100, the percentage change in product reliability is just over 3%, profits about 7%, social surplus about 9%, and consumer surplus not quite 15%. (The range of marginal costs that yield an interior solution is about 0 [less than or equal to] mc [less than or equal to] 150.)

The effect of R&D expenditures, as compared to the case of no R&D, on product reliability is slightly concave, increases in mc being associated with first a very mild increase in reliability followed by a decrease. (The comparative statics of mc will be discussed further shortly.) As for the other outcomes in Fig. 1, an exogenous increase in the marginal manufacturing cost will have the largest percentage change effect on consumer surplus, then social surplus, and finally profits, and all curves are increasing. Although not shown, the percentage change in g has a similar convex shape to the percentage change in consumer surplus (recalling CS = [1/2] [Q.sup.2]) and profits (recalling II = [Q.sup.2] - RDE). Intuitively, a high mc is associated with low sales Q, Thus, bad outcomes in terms of profits and consumer surplus but jumping from no R&D to the optimal level of R&D increasingly mitigates those bad outcomes. (The picture is not so straightforward for the full quality price or the purchase price. The full quality price, P + (1 -R)K, is always lower with R&D as compared to no R&D, but the percentage change is not monotone: first decreasing for low values of mc and then increasing for large values. The percentage change in the purchase price is positive and declining for relatively low values of mc and negative but quite small for large values.) A monopoly firm benefits from doing R&D, but consumers benefit more in percentage terms, especially when manufacturing costs are high. Interestingly, this qualitative result holds even though the percentage change in the reliability curve is mostly downward sloping. The effect of exogenous changes in the marginal manufacturing cost on R&D are examined next.

RDE Optimality Condition

Equation (18) gives the formal statement of the R&D optimality condition. In order to develop the intuition of the relationship between marginal manufacturing cost and R&D expenditure it may be helpful to take a second look at it. Recalling (10), the first term shows the effect of an increase in R&D expenditure on variable profits. Abusing the notation slightly, recalling (15) and (14), denote the "marginal value of R&D expenditure" by MVRDE:

MVRDE = dVP/dRDE = 2Q [dQ/dR] [dR/dRDE] = 2Q [1/2] (K + zmc) [dR/dRDE]. (24)

So, we can approximate MVRDE

MVRDE [approximately equal to] Q (K + zmc) (R - [R.sub.a])/([DELTA]RDE) = (KQ + zmcQ)((1 - [R.sub.a])/[DELTA]RDE - (1 - R)/[DELTA]RDE). (25)

Intuitively, let [R.sub.a] denote the reliability before the marginal increase in R&D. The value of a dollar spent on R&D is the reduction in the expected costs of product failure for customers, KQ(1 - [R.sub.a]) - KQ(1 - R), and the reduction in the expected firm manufacturing costs for the replacement units, zmcQ(1 - [R.sub.a]) - zmcQ(1 - R). To see that the MVRDE is downward-sloping, substitute from (20) into (19) and take the derivative.

dMVRDE/dRDE = d (dVP(RDE)/dRDE)/dRDE = - 1/4 (K + zmc) [[PI].sup.[1/2].sub.0] [k.sup.[1/2]] [RDE.sup.-3/2] < 0. (26)

Figure 2 shows the value of the marginal R&D dollar for two levels of the marginal manufacturing cost, mc. From (18), the optimal level of RDE is where MVRDE = 1, where the marginal value of an R&D dollar is equal to that dollar. In this case, an exogenous increase in marginal manufacturing costs will prompt the firm to optimally increase its R&D. As mentioned above, that qualitative result depends on the level of mc.

Regardless of shifts in the MVRDE curve, it is always downward sloping. That is, the value of an R&D dollar is always reduced by doing more R&D.

Comparative Statics of the Marginal Manufacturing Cost

To take a more general look at the comparative statics of mc we will investigate the derivative of MVRDE, Eq. (24), with respect to mc, using (14):

dMVRDE/dmc = dR/dRDE (zQ + (K + zmc)dQ/dmc) = dR/dRDE (zQ - [1/2] (K + zmc) (1 + z((1 - [R.sub.0]) + [k.sup.-[1/2]][RDE.sup.[1/2]])). (27)

When marginal manufacturing cost increases, there are two effects on the marginal value of an R&D dollar. On the one hand, looking at the term,

(K + zmc) dQ/dmc = [1/2](K + zmc)(1 + z((1 - [R.sub.0]) + [k.sup.-[1/2]][RDE.sup.[1/2]]) < 0,

an increase in marginal cost is associated with a lower quantity produced, and thus a lower total cost of product failure. On the other hand, the effect of an increase in mc on MVRDE, holding quantity constant, is positive: z Q > 0. How large this is depends both on quantity produced Q and the parameter z, which measures the relative cost of replacement units. Thus, for some combinations of (mc,z) an increase in marginal manufacturing cost will cause R&D expenditures to increase, and for others R&D expenditures to decrease. When quantities are high because marginal costs are low, an increase in marginal cost is associated with an increase in R&D expenditure. Conversely, when quantities are low because marginal costs are high, an increase in marginal cost is associated with a decrease in R&D expenditure.

Figure 3 maps optimal RDE, Eq. (22), as a function of mc for various values of the replacement cost parameter z. When replacement units are free to manufacture, z = 0, there is always a negative relationship between mc and RDE. A manufacturing cost increase prompts the firm to do less R&D. When replacement costs are the same as the original manufacturing costs, z = 1 (the default assumption), the relationship is upward sloping, where R&D mitigates cost shocks to some degree for marginal costs in the range of about 0 < mc < 40. It is downward thereafter. For z = 1.5, when replacement costs are one and a half times the manufacturing cost, the upward sloping range is about 0 < mc < 55. Clearly, the effect of cost shocks on R&D depends on both the marginal cost parameter and the replacement cost parameter.

Briefly thinking of RDE as a function of just these two parameters, RDE(mc,z), Fig. 4 shows the firm's response to an exogenous increase in the marginal manufacturing cost in the parameter space(mc, z). In summary, for (mc, z) pairs where the replacement cost is high and the marginal cost is low, the firm responds to a manufacturing cost shock which raises mc by doing more R&D. Otherwise, it does less.

Two-Country, Two-Firm, Intra-Industry Trade Model

Assume a two-country, two-firm, intra-industry trade model. A two-stage game is played where quality is chosen first and then Coumot quantities are chosen. Solutions are found by backward induction. Countries are denoted i = A, B and firms j= 1, 2. Firm 1 is located in country A; sales of firm 1 in country A are denoted [Q.sub.A1]. Firm 2 is located in country B; sales of firm 2 in country B are denoted [Q.sub.B2]. Without loss of generality, imports and exports are from country A's point of view, that is, [Q.sub.M] denotes imports in country A produced by firm 2 and [Q.sub.x] denotes exports of firm 1 into country B. The model follows the general lines of the monopoly model, and similar notation is used, with the addition of subscripts for country and firm. The demand side assumptions are:

[v.sub.A] [greater than or equal to] [P.sub.A1] + (1-[R.sub.1])[K.sub.A] = [P.sub.M] + (1 - [R.sub.2])[K.sub.A]; (28)

[v.sub.B] [greater than or equal to] [P.sub.X] + (1 - [R.sub.1])[K.sub.B] = [P.sub.B2] + (1 - [R.sub.2])[K.sub.B]; (29)

so that the market demand functions are:

[P.sub.A1] = [V.sub.A] - [K.sub.A](1 - [R.sub.1]) - ([Q.sub.A1] + [Q.sub.M]); (30)

[P.sub.A1] = [V.sub.A] - [K.sub.A](1 - [R.sub.2]) - ([Q.sub.A1] + [Q.sub.M]); (31)

[P.sub.X] = [V.sub.B] - [K.sub.B](1 - [R.sub.2]) - ([Q.sub.B2] + [Q.sub.X]); (32)

[P.sub.B2] = [V.sub.B] - [K.sub.A](1 - [R.sub.2]) - ([Q.sub.B2] + [Q.sub.X]); (33)

The consumers surpluses are [CS.sub.A] = ([1/2])[([Q.sub.A1], + [Q.sub.M]).sup.2] and [CS.sub.B] = ([1/2])[([Q.sub.X] + [Q.sub.B2]).sup.2].

The basic production-side assumptions are the same as in Eqs. (6) through (9) with the addition of a subscript denoting firm 1 or firm 2. Variable profits in the duopoly model are:

[VP.sub.1] = ([P.sub.A1] - [c.sub.1])[Q.sub.A1] + ([P.sub.X-c1])[Q.sub.X] (34) [VP.sub.2] = ([P.sub.M] - [c.sub.2])[Q.sub.M] + ([P.sub.B2] - [c.sub.2])[Q.sub.B2],

and profits are:

[[PI].sub.j] = [VP.sub.j] - [RDE.sub.j] j = 1,2, (35)

recalling the research and development functions (6).

Using backward induction, firm 1 chooses ([Q.sub.A1], [Q.sub.X]) to maximize [VP.sub.1] and firm 2 chooses ([Q.sub.M], [Q.sub.B2]) to maximize [VP.sub.2]. The quantities are then written as functions of ([RDE.sub.1], [RDE.sub.2]), and the results substituted into the profit functions (35). Finally, firm 1 chooses [RDE.sub.1] to maximize [[PI].sub.1] and firm 2 chooses [RDE.sub.2] to maximize [[PI].sub.2]. While analytical solutions are available, they are so cumbersome that they are not informative.

Therefore, the analysis will proceed by numerical methods, assuming the parameter values from above: [V.sub.A] = 200, [K.sub.A] = 100. [V.sub.B] = 200, [K.sub.B] = 100. [R.sub.01] = [R.sub.02] = [R.sub.0] = 0.9, k = 150,000, and [z.sub.1] = [z.sub.1] = z = 1. The symmetric trade model is considered first, and then an asymmetric model which allows for different marginal manufacturing costs between countries is considered.

Trade Duopoly Outcomes

Symmetric Trade Duopoly

Figure 5 shows the percentage increase in profits, reliability, social surplus, and consumer surplus, comparing the case of optimal R&D versus no R&D. Comparing these results to those of the single country monopoly in Fig. 1, we see immediately that the R&D game for the duopolists actually reduces profits as compared to not doing any R&D at all. The other outcomes are qualitatively similar to the monopoly model; quantitatively, duopoly provides a larger percentage increase in consumer surplus, and in social surplus, even taking the reduction of profits into account. Looking at profits, both firms doing R&D is indeed the Nash equilibrium, as seen for the example in Table 1. If firm 1 does its optimal, [RDE.sup.*.sub.1], and firm 2 does not, firm l's profits go up substantially while firm 2's fall, and conversely. The normal Nash reasoning leads to both firms doing R&D in order to avoid the substantial reduction in profit that would occur if your competitor does R&D and you do not. In general, the difference in profits when both firms do R&D as compared to when neither firm does is not large.

Figure 6 again maps [RDE.sub.j] as a function [mc.sub.j]. The range of marginal costs that yield an interior solution is about the same as for the monopoly case, and the qualitative results are similar to Fig. 3, but notice that in the symmetric trade model R&D expenditures are considerably higher than in the single country monopoly model. For example, when z = 1 and the marginal cost for the monopolist and the symmetric duopoly firms are 30, competition between the firms prompts them to choose an R&D expenditure level which is more than three times that of the monopolist. The resulting improvement in reliability comparing duopoly with monopoly is also substantial, about 0.964 as compared to 0.935. As suggested by comparing Figs. 1 and 5, the increased R&D expenditures in the more competitive market structure help consumers and reduce profits for firms, with an overall improvement in social surplus.

Asymmetric Trade Duopoly

This last subsection considers the case when the firms and countries are identical except that one firm experiences a manufacturing cost shock. Specifically, suppose the marginal manufacturing cost for firm 2 changes exogenously while [mc.sub.1] remains at 40. The results for country A and firm 1 whose mc, has not changed are shown in Fig. 7. (Notice that the range of [mc.sub.2] that generates interior solutions is considerably smaller than in the previous models. Briefly, as [mc.sub.2] gets larger as compared to the fixed [mc.sub.1] - 40 firm 1 increases its reliability, eventually hitting the upper bound of [R.sub.1] = 1.) The percentage change in consumer surplus and social surplus are always increasing, but not always at an increasing rate as they were in Fig. 5. A more significant change is that the percentage change in profits is only negative for small values of its competitor's marginal manufacturing cost, and the percentage change in profits is actually positive for firm 1 when [mc.sub.2] is large. Another significant difference is that the percentage change in reliability is increasing in Fig. 7 while it was concave in Fig. 5. The results for country B and firm 2 whose [mc.sub.2] is subject to a cost shock are shown in Fig. 8.

The percentage changes in consumer surplus and social surplus for country B in Fig. 8 are rather similar to those of country A in Fig. 7. But the percentage change in profits for firm 2 in Fig. 8 is the reverse of the pattern for firm 1 in Fig. 7. It is positive for small values of [mc.sub.2] and negative for large values of [mc.sub.2]. The percentage change in reliability curve for firm 2 is concave in Fig. 8, as it was in Figs. 5 and 1 above, but the range over which it is upward sloping is small, about 0 < [mc.sub.2] < 7. Figure 9 shows the reliability of firm 2's product itself for relatively low levels of [mc.sub.2]. Finally, Fig. 10 shows the effect of exogenous changes in [mc.sub.2] on its R&D expenditures for the various values of the replacement cost parameter.

The results are qualitatively similar to those in Fig. 6 although the R&D expenditures are somewhat higher here for firm 2. For example, suppose the marginal cost for firm 2 in the asymmetric model is 30 recalling that the marginal cost for firm 1 is 40. In the symmetric duopoly case with marginal cost at 30 the R&D expenditures were about 608 with reliability of 0.964. Here the firm with that same marginal cost (firm 2) spends about 701 with a resulting reliability of 0.968. Firm 1, with the higher marginal cost, spends about 525 with a resulting reliability of about 0.959. In an asymmetric duopoly the firm with the higher relative marginal manufacturing cost does relatively less research and development.

Conclusion

The goal of the present paper was to do a simple theoretical investigation of the effect of manufacturing cost shocks on a firm's research and development, in a model where R&D improves product reliability and thus decreases both customer cost of product failure and the firm's costs to replace or repair an item than has failed. In the somewhat extreme case that all product failure costs fall on the customer and there are no costs to the firm in replacing (or repairing) an item, our results suggest that cost shocks that increase the marginal manufacturing cost, always prompt the firm to do less R&D. Briefly, R&D decreases the customer's expected cost of failure and thus increases their willingness to pay for an item, so even in the case that the firm bears none of the costs of product failure the firm optimally does some R&D. But an increase in the marginal manufacturing cost reduces the quantity produced, so the firm gets less benefit from that increased willingness to pay per unit. Thus if there are no firm-level costs of product failure the firm always responds to a manufacturing cost shock that increases costs by doing less R&D. This result continues to hold as long as the firm's replacement costs are low.

But when the firm's replacement cost gets sufficiently high the results get more complicated. In this case, the firm's response to a shock depends on whether the marginal manufacturing cost was low or high before the shock. If it was high, the result is the same as when there were no replacement cost. If the marginal manufacturing cost goes up, the firm does less R&D. But if the marginal manufacturing cost is low and the cost shock does not increase it too much, then the firm responds to an increase in the marginal manufacturing cost by doing more R&D. In this case, the quantity produced is high enough that the value of a marginal R&D dollar actually increases with the cost shock. Mapping R&D expenditures to values of the marginal manufacturing cost parameter, the curve is concave, first upward sloping and then downward sloping, for values of the replacement cost parameter above a given threshold (e.g., about 0.5 in Fig. 4). Finally, as the firm's per unit replacement cost parameter increases this curve becomes even more concave, with a greater range where the slope is positive. Intuitively, the greater the replacement cost, the more often the firm responds to a manufacturing cost shock by doing more R&D, mitigating the effect of the manufacturing cost shock to some degree.

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https://doi.org/10.1007/s11293-017-9565-3

Published online: 29 January 2018

Electronic supplementary material The online version of this article (https://doi.org/10.1007/s11293-0179565-3) contains supplementary material, which is available to authorized users.

Jannett Highfill (1) [ID] * Michael McAsey (2)

[mail] Jannett Highfill

highfill@bradley.edu

(1) Department of Economics, Bradley University, Peoria, IL 61625, USA

(2) Department of Mathematics, Bradley University, Peoria, IL. USA

Caption: Fig. 1 R&D vs no R&D, single market monopoly

Caption: Fig. 2 Marginal value of R&D expenditure, monopoly

Caption: Fig. 3 Marginal cost and R&D expenditure, monopoly

Caption: Fig. 4 R&D response to a manufacturing cost shock

Caption: Fig. 5 R&D vs no R&D, symmetric trade duopoly

Caption: Fig. 6 Marginal cost and R&D expenditure, symmetric duopoly

Caption: Fig. 7 R&D vs no R&D. country A, firm 1, asymmetric duopoly

Caption: Fig. 8 R&D vs no R&D. country B, firm 2, asymmetric duopoly

Caption: Fig. 9 Marginal cost and reliability, firm 2, asymmetric duopoly

Caption: Fig. 10 Marginal cost and R&D expenditure, firm 2, asymmetric duopoly
Table 1 Symmetric trade duopoly Nash equilibirium

[mc.sub.1] = 40 =    [RDE.sub.1] = 0        [RDE.sub.1] =
[mc.sub.2]                                  [RDE.sup.*.sub.1]

[RDE.sub.2] = 0      [[PI].sub.1] = 4737,   [[PI].sub.1] = 5357
                     [[PI].sub.2] = 4737    [[PI].sub.2] = 4171

[RDE.sub.2] =        [[PI].sub.1] = 4171,   [[PI].sub.1] = 4719
[RDE.sup.*.sub.2]    [[PI].sub.2] = 5357    [[PI].sub.2] = 4719
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Author:Highfill, Jannett; McAsey, Michael
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Date:Mar 1, 2018
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