Welfare Measures in Dynamic Firm R&D Games

Abstract

The present paper examines the welfare effects of a dynamic Research and Development (R&D) game at the firm level in a two-country, two-firm, intra-industry trade context. Economists do not use the trade balance as a measure of economic welfare, but it is often used in the public arena. 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.

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, v _i (t), at any time t in country i are distributed uniformly on the interval ((V _i0 − γ _i0)e ^rt, V _i0 e ^rt). (In general all cost and demand parameters are assumed to grow at an exponential captured by e ^rt, perhaps related to the inflation rate.) The maximum reservation price any customers have for the product is V _i0 e ^rt and the range of reservation prices is γ _i0 e ^rt where 0 < γ _i0 ≤ V _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 ${f}_i=\frac{1}{\gamma_{i0}{e}^{rt}}$. Customers whose reservation price satisfies the following condition will purchase the product:

$$ {v}_A(t)\ge FQ{P}_A(t)={P}_{A1}(t)+\left(1-{R}_1(t)\right){K}_{A0}{e}^{rt}={P}_M(t)+\left(1-{R}_2(t)\right){K}_{A0}{e}^{rt} $$

(24)

$$ {v}_B(t)\ge FQ{P}_B(t)={P}_X(t)+\left(1-{R}_1(t)\right){K}_{B0}{e}^{rt}={P}_{B2}(t)+\left(1-{R}_2(t)\right){K}_{B0}{e}^{rt} $$

(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

$$ \begin{array}{c}\hfill {Q}_A(t)={Q}_{A1}(t)+{Q}_M(t)={N}_{A0}\;{e}^{st}{\displaystyle {\int}_{P_{A1}(t)+\left(1-{R}_1(t)\right){K}_{A0}{e}^{rt}}^{V_{A0}{e}^{rt}}\frac{1}{\gamma_{A0}{e}^{rt}}}d{v}_A(t)\hfill \\ {}\hfill =\frac{N_{A0}}{\gamma_{A0}}{e}^{\left(s-r\right)t}\left({V}_{A0}{e}^{rt}-{P}_{A1}(t)-\left(1-{R}_1(t)\right){K}_{A0}{e}^{rt}\right).\hfill \end{array} $$

(26)

That is, market demand is the proportion of potential customers that buy the product times the potential market size N _i (t) = N _i0 e ^st which grows exponentially at a rate of e ^st and may be related to the population growth rate. Consumer surplus in this setup is simply

$$ \begin{array}{c}\hfill C{S}_A(t)={N}_{A0}{e}^{st}{\displaystyle {\int}_{P_M(t)+\left(1-{R}_2(t)\right){K}_{A0}{e}^{rt}}^{V_{A0}{e}^{rt}}\frac{v_A(t)-\left({P}_M(t)+\left(1-{R}_2(t)\right){K}_{A0}{e}^{rt}\right)}{\gamma_{A0}{e}^{rt}}}d{v}_A(t)\hfill \\ {}\hfill =\frac{1}{2}{e}^{\left(r-s\right)t}{\left({Q}_{A1}(t)+{Q}_M(t)\right)}^2.\hfill \end{array} $$

(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 _A0/γ _A0) = (N _B0/γ _B0). Choose the unit of measurement of the potential market size so that this ratio is one

$$ \frac{N_{A0}}{\gamma_{A0}}=\frac{N_{B0}}{\gamma_{B0}}=1. $$

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 _A0 = γ _A0 = 100.