THE 1990 R&D TAX CREDIT: A UNIFORM TAX ON INPUTS AND A SUBSIDY FOR R&D HARRY WATSON *

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1 THE 990 R&D TAX CREDIT THE 990 R&D TAX CREDIT: A UNIFORM TAX ON INPUTS AND A SUBSIDY FOR R&D HARRY WATSON * Abstract - Starting in 990, the base of the R&D tax credit was linked to a moving average of a firm s sales. The complicated nature of the base has made it difficult to identify the incentive effects of the credit. This analysis shows that the current system is equivalent to a uniform tax on all inputs in combination with a subsidy on R&D expenditures. Given current tax code parameters, the level of input taxation is insignificant, so that the credit might reasonably be viewed as just a subsidy on R&D expenditures. The analysis also shows that the current system offers a greater marginal incentive for R&D than the system that was in place prior to 990. INTRODUCTION It is often argued that the externalities associated with R&D make its social rate of return greater than the return to those undertaking R&D, a view that is * Department of Economics, George Washington University, Washington, D.C supported by empirical research (Griliches, 99). Concern about this divergence between social and private rates of return induced the United States to begin subsidizing R&D investments in 98 with the use of a tax credit. The R&D tax credit that was adopted in 98 had a relatively simple structure, so that its incentive effects, such as its impact on the after-tax cost of R&D, were well understood. The structure of the credit remained essentially unchanged until 990, when Congress altered the base of the credit, making it significantly more complex. Because of the increase in complexity, the incentive effects of the current R&D credit are not well understood. Even basic measures of the effects of the current credit, such as a user cost of R&D, are not known. This is unfortunate because while there is some empirical evidence that indicates that R&D spending by corporations in the United States has been sensitive to tax incentives (Bailey and Lawrence, 992; Hall, 993), our ability to analyze one of the more important of these incentives is limited. 93

2 NATIONAL TAX JOURNAL VOL. XLIX NO. This analysis provides some qualitative insights about the incentives produced by the current R&D credit. Within the framework of a neoclassical model of the firm, a user cost expression for R&D is derived, and it is shown that the credit is equivalent to a combination of a uniform tax on all inputs and a subsidy on qualifying R&D expenditures. The input tax component of the credit, which was noted by Gravelle (993), may be of limited importance. Its uniform nature means that it will not create a bias in input choices of firms and, given current tax code parameters, its level is quite low. The limited impact of this input tax suggests that viewing the current complicated credit system as simply a subsidy on qualifying R&D expenditures may be a good approximation. It is also possible to show that the current credit will provide a greater marginal incentive for R&D than the credit system that was in place prior to 990. Before analyzing the incentives produced by the current R&D credit, it is necessary to briefly describe the evolution of the R&D credit. Between 98 and 990, the R&D credit was defined as a percentage, c, of the difference between qualified expenditures in the current year and a base level. The base level was the maximum of 50 percent of R&D expenditures in the previous year or the average level of R&D expenditures over the prior three years. For most research-active firms, the base would have been the moving average of the prior three-year period, so that if c was the credit rate and E t was spending on R&D in year t, the credit was c {E t (/3) [E t + E t 2 + E t 3 ]}. The moving average nature of this credit scheme means that increasing current spending on R&D increases the average level which, in turn, reduces the size of the credit in the future. This feedback effect produces an effective credit rate equal to 2 c { (/3) [/( + r) + /( + r) 2 + /( + r) 3 ]} where r is the firm s discount rate (Eisner, Albert, and Sullivan, 984). The average discounted sum appearing in this expression reflects the fact that an additional dollar of spending today reduces the value of the credit in the succeeding three years. In 990, Congress altered the R&D credit to reward firms that are able to increase their R&D to sales ratio relative to the ratio that existed in the period between 985 and Specifically, the credit is equal to a percentage of the difference between the current level of R&D spending and the level of spending produced by multiplying the historical R&D to sales ratio (985 89) by an average of the firm s sales over the preceding four years. Thus, if c is the credit rate, s is the historical R&D to sales ratio, E t is spending on R&D in year t, and S t is sales in year t, the credit is equal to cb, where 3 B = E t (s/4){s t + S t 2 + S t 3 + S t 4 }. If a firm s historical R&D ratio exceeds a certain level (now 0.6), it can use this lower value in calculating its credit. 3 94

3 THE 990 R&D TAX CREDIT Assessing the effects of this credit system is considerably more complicated than the old system, because the impact of the credit is no longer limited to the firm s current and past R&D spending decisions. Now, the credit links current R&D spending with all the input decisions that might influence the firm s sales, which has some interesting implications. For example, the value of the credit is reduced if sales are increased, so that the credit may contribute to the cost of inputs other than R&D. The cost of using an additional unit of an input should reflect the adverse impact on the credit that will result from the increased output (sales) generated by the additional input. This suggests that the credit may create what might be described as an implicit tax on the use of all inputs. The complex nature of the current system also makes it difficult to determine whether it provides greater incentives for R&D spending than the old system. While the current R&D credit has received some attention in the literature, the issues raised above have not been fully explored. In particular, there is no formal model of a firm s R&D decision making that provides a framework for analyzing the effects of the current credit. To fill this gap in the R&D literature, we use a neoclassical model of the firm that creates R&D knowledge, which is then used as an input in the firm s production process. The model s solution generates explicit expressions for input costs, including a user cost for R&D. These expressions indicate how the current credit affects the cost of inputs used in producing output as well as inputs used in augmenting the firm s R&D knowledge. The next section describes the basic model that will be used. Subsequent sections deal with the issues raised above and with a generalized model of R&D growth. All derivations have been relegated to the Appendix. Model Apart from tax issues and the treatment of R&D, the model follows the typical neoclassical description of firm behavior. R&D can be incorporated in this framework by treating the stock of R&D knowledge as physical capital. Investment that augments the firm s stock of knowledge occurs when the firm purchases an R&D input, which is subject to the R&D tax credit. On the other hand, this stock of knowledge may depreciate over time as a result of obsolescence, 2 loss of trade secrets, and the like. It will be assumed that this depreciation is equal to a fixed fraction of 2 the stock of R&D knowledge. (A more general approach to modeling the production of R&D knowledge is presented later.) The firm then combines R&D knowledge with capital and labor to produce output. This description of the firm s choices is formalized in the following assumptions.. Output is determined by R&D knowledge, R, labor, L, and capital, K, according to the production function F(R,K,L). The price of output is. 2. The growth of the stock of R&D is equal to z αr, where z is the input subject to the R&D tax credit and α is the depreciation rate of R&D knowledge. Thus, z is equivalent to gross investment in R&D knowledge and αr is depreciation of that knowledge. Each unit of z costs p z. 3. The growth of the capital stock, K, is equal to I δk, where δ the economic depreciation rate and I is gross investment. Each unit of I costs q. 95

4 NATIONAL TAX JOURNAL VOL. XLIX NO. With the exception of the tax treatment of the R&D input, z, the assumptions concerning the tax code are fairly standard. In particular, the following are assumed. 4. Labor cost, wl, and the cost of the R&D input, p z z, are deductible and, following the tax code (Section 74), the deduction for p z z is reduced by the amount of the R&D credit. The statutory value of the R&D tax credit is given by cb, where c is the credit rate and B, as defined in equation 3, is equal to the difference between current expenditures on z and the product of the historical R&D to sales ratio and average sales over the prior four years. 4 If τ is the corporate tax rate, then the after-tax cost of the R&D input, z, is given by p z z τ( p z z cb) cb = ( τ)(p z z cb). 5. For tax purposes, the discounted value of depreciation deductions per dollar of new investment in K is given by d. This deduction lowers the cost of gross investment to q τqd = q( τd). With these assumptions, the after-tax cash flow of the firm can be written as 4 ( τ)[f(r,k,l) wl - p z z + cb] q( τd)i. Subject to the constraints on the growth of R&D and capital, the firm will choose L, K, and R to maximize the discounted value of this cash flow. The optimal choices of these inputs require that their marginal products be equated with their costs. Letting F L, F K, and F R denote the marginal product of labor, capital, and the stock of R&D knowledge, these optimality conditions are 5 F L = 6 F K = q(r + δ)( τd)/( τ) T 7 T where w F R = ( c) p z (r + α) T 8 T = sc 4 Σ 4 i= ( + r) i and r is the firm s discount rate. 5 To interpret these results, it is important to remember that the R&D tax credit can be thought of as a combination of a subsidy for R&D expenditures and an implicit tax on all inputs. The credit is equal to c percent of current expenditures on z (the subsidy) less c percent of the product of the historical R&D to sales ratio, s, and average sales over the prior four years (the tax ). The T term above captures the tax effect of the R&D credit. To see this, note that if an increase in the use of an input raises sales by one dollar, this will raise the moving average level of sales over the succeeding four years and this, in turn, will lead to an erosion in the value of the credit in the future. While the firm has one dollar more in revenue, its 96

5 THE 990 R&D TAX CREDIT credit is eroded by an amount equal to the c times the R&D to sales ratio times the four-year average discounted value of a dollar, or 9 sc 4 Σ4 i= ( + r) i which is T. Put differently, the after-tax value of an increase in sales of $ is only $T. If the tax portion of the R&D credit were eliminated, we would be left with a subsidy of c for qualified R&D expenditures. This can be accomplished by setting the R&D to sales ratio equal to zero (so T = ). Consequently, when s = 0, only the subsidy portion of the credit will remain. With these observations in mind, equations 5 7 are easily explained. From equation 5, the tax effect of the subsidy raises the cost of labor from w to w/t. Similarly, the term multiplying /T on the right-hand side of equation 6 is the standard Hall Jorgensen user cost of capital expression, so that this user cost is increased by a factor of /T. Finally, keeping in mind that α is the deprecation rate for R&D knowledge, p z (r + α) can be interpreted as a user cost of R&D in equation 7. This user cost is reduced by a factor of c, which reflects the subsidy component of the R&D credit, and it is increased by a factor of /T, which reflects the input tax component of the R&D credit. If only the subsidy portion of the R&D tax credit were kept that is, if s were set to zero so that T = then the only impact of the tax credit on the optimality conditions given by equations 5 7 would be a reduction of c percent in the user cost of R&D. As will be shown below, viewing the R&D tax credit as just a subsidy may be reasonable because the impact of the term /T is likely to be quite small. Observations The Impact of the R&D Tax Credit on Input Costs While it is possible to identify part of the R&D tax credit as a tax on inputs, this tax may not have significant consequences. In particular, this tax should not create any substitution effects across inputs. This can be seen by noting that the T term affects all input costs (the right-hand sides of the equations above) in the same way, so that T will not appear in any ratio of input costs. Furthermore, given current tax code parameters, the magnitude of the tax effect generated by /T is likely to be quite small. To see what the tax effect of /T might be, it is only necessary to note that /T is increasing in s and decreasing in r. Because a firm s R&D to sales ratio in the period between 985 and 989, its s value, is capped at 0.6 for the purpose of calculating the R&D credit, the maximum value of /T occurs when r = 0 and s = 0.6. With these values of r and s and with c set at its current value of 20 percent, /T =.033. This means that, at most, the tax effect of the R&D credit raises input costs by 3.3 percent. It is important to note that 3.3 percent is the maximum tax effect of the R&D credit. For most discount rates and s values, the tax created by /T will be substantially smaller. This is reflected in Table, where the tax rate implied by the factor /T is given for different discount rates, r, and different historical R&D to sales ratios, s. As indicated in 97

6 NATIONAL TAX JOURNAL VOL. XLIX NO. TABLE IMPLICIT TAX RATE ON INPUTS CREATED BY R&D TAX CREDIT, 00*[/T ] Discount Rate, r R&D to Sales Ratio, s r = 2% r = 4% r = 6% s = 4% 0.8% 0.7% 0.7% s = 0%.9%.8%.7% s = 6% 3.2% 3.0% 2.9% the table, any firm that has a historical R&D to sales ratio that is no greater than ten percent will face an implicit tax rate of less than two percent for a wide range of discount rates. There are two factors that may make this tax effect even less important. First, in a period in which a firm s sales are rising and R&D expenditures are flat, the firm may not be eligible for the credit. During this period, the tax effect described above would not be operative (that is, T would equal one). Second, the model used here assumes the firm carries no inventories of its products, so that increasing the use of any input yields an immediate increase in sales and, concomitantly, an erosion in the value of the credit. If input use were increased in order to add to inventories of output, however, sales would not immediately increase and the erosion in the value of the R&D credit would be postponed. Therefore, the discounted value of the postponed erosion would be less, and this suggests that the corresponding implicit tax on input use would be less. net, the subsidy effect swamps the tax effect. This should be clear from the discussion above concerning the magnitude of /T. Because /T is bounded between and.033, ( c)/t is bounded between 0.8 and when c = 20 percent. This means that regardless of the firm s discount rate, r, and its historical R&D to sales ratio, s, the R&D credit will lower the user cost of R&D by at least 7.4 percent and by at most 20 percent. Some illustrative values of the effective R&D subsidy, ( ( c)/t), are given in Table 2 for a range of discount rates and historical R&D to sales ratios. As the entries suggest, the effective subsidy rate falls as the historical R&D to sales ratio increases and rises when the firm s discount rate increases. In all cases, however, the effective subsidy rate differs little from the 20 percent rate that would be found if the input tax component of the credit were ignored. These results suggest that approximating the overall effect of the R&D credit on the user cost of R&D by just assuming that the credit is equivalent to a simple subsidy of c = 20 percent would not be inaccurate. This means that researchers who need to create user cost or marginal effective tax rate expressions could probably ignore the complicated nature of the R&D tax credit in their calculations. The user cost of R&D The impact of the credit on the user cost of R&D the right-hand side of equation 7 is captured by the term ( c)/t. The subsidy portion of the credit lowers the user cost by c, while the tax effect increases it by a factor of /T. On TABLE 2 EFFECTIVE SUBSIDY RATES FOR R&D, ( ( c)/t)*00 Discount Rate, r R&D to Sales Ratio, s r = 2% r = 4 r = 6% s = 4% 9.3% 9.4% 9.4% s = 0% 8.4% 8.5% 8.6% s = 6% 7.5% 7.6% 7.7% 98

7 THE 990 R&D TAX CREDIT Comparison with the old credit In its previous incarnation, the credit was equal to a percentage of the excess of current expenditures on z over the average of the preceding three years. 6 Using the approach taken above, it is easy to derive the optimality condition for the firm s choice of the R&D asset. The optimality condition in this case is given by 0 F R = p z τ c( [ (r + α) τ 3 3 Σ i= )] ( + r) i where c is the credit rate. Is the current credit system more generous than that which was in place prior to 990? To answer this, it is only necessary to compare the user cost expression for R&D in equation 7 with the user cost of R&D in equation 0. Because the credit rate is the same (20 percent) in both systems, it is easy to show that when a firm s discount rate is small, the current system will be more generous than the old system regardless of the firm s historical R&D to sales ratio and the value of the corporate tax rate, τ. When the discount rate increases, it is not clear which system will yield a smaller user cost for R&D. However, for a corporate tax rate of 34 percent, the current system will be more generous in the sense of producing a lower user cost of R&D for any reasonable values of the firm s discount rate and any value of s. This is reflected in Table 3, which gives the ratio of the user cost under the current system to the user cost under the old system, for various values of s and r. Because these ratios are all TABLE 3 RATIO OF USER COST OF R&D WITH CURRENT SYSTEM RELATIVE TO OLD SYSTEM Discount Rate, r R&D to Sales Ratio, s r = 2% r = 4% r = 6% s = 4% s = 0% s = 6% less than unity, the current system is somewhat more generous for this range of parameter values. 7 Thus, as a result of the change in the definition of the credit base that occurred in 990, the marginal incentive appears to be between 5 and 9 percent greater. The entries in the table indicate that the benefit provided by the current system, relative to the old system, falls as the firm s discount rate increases and as the firm s historical R&D to sales ratio increases. This suggests that while the current system has created a greater incentive to undertake R&D, the increase in this incentive has not been even across firms. Firms that were doing relatively more R&D in the late 980s (those with high R&D to sales ratios) and firms that were viewed as relatively risky (those with higher discount rates) faced the smallest increase in this incentive. Generalizing the R&D Production Process The analysis above assumed that the growth of R&D knowledge is determined by the level of a noncapital input, z, less depreciation in the stock of R&D knowledge, αr. A more realistic approach would not place restrictions on the manner in which R&D knowledge is produced and would recognize the possibility that some types of R&D require capital inputs. In particular, the model used above could be generalized 99

8 NATIONAL TAX JOURNAL VOL. XLIX NO. by assuming that the growth of R&D knowledge can depend upon a noncapital input, z, which is subject to the R&D tax credit, as well as a capital input, K R, where the R subscript is used to distinguish this type of capital from capital that is used in the production of output. 8 In addition, the linear production function that characterized the growth of R&D knowledge, z αr, could be replaced by a more general production process, say, g(z,k R,R). Making the modifications suggested above is useful for two reasons. First, it is possible to draw some conclusions about the way in which the R&D tax credit will affect the firm s choice of R&D inputs. Second, it is possible to show that the characterization of the R&D tax credit as uniform tax on inputs in combination with a subsidy on qualifying R&D expenditures carries over to this more general setting; that is, the earlier results are not simply an artifact of the model that was used. With regard to the firm s choice of R&D inputs, the optimality conditions require that the ratio of the marginal product of capital, g KR, to the marginal product of z, g z, must equal affect the relative input cost. The subsidy effect of the credit the ( c) term in the denominator raises the relative price of capital used in the R&D production process, so that z will substitute for K R in the production of R&D. For the firm s choices of labor and capital used in the production of output, the optimality conditions are still given by equations 5 and 6. For the firm s choice of R&D knowledge, condition 7 becomes somewhat more complicated because the user cost of R&D will usually be endogenous; that is, its value will depend upon the firm s choices of z and K R. However, there is one situation in which a simple user cost expression can be found. Suppose that the inputs z and K R are used in fixed proportions: each unit of K R requires θ units of z in order to produce R&D knowledge. Suppose, as above, that R depreciates at a rate of α. Then, the growth of R&D is given by g = min [z/θ,k R ] αr. In this case, the firm will equate the marginal product of R&D, F R, with 2 (r + α) T [θ ( c)p z + q R (r + δ )( τd )/( τ)]. R R q R (r + δ R )( τd R )/( τ) ( c)p z where q R is the price of a unit of K R, δ R is the economic depreciation rate for K R, and d R is the discounted value of depreciation deductions per unit of K R. The expression above is simply the user cost of capital used in the production of R&D divided by the subsidized price of z, ( c)p z. Because both z and K R contribute to the growth in R, the input tax effect of the R&D credit does not If the opportunities for substitution between capital and noncapital inputs are limited, expression 2 may be a reasonable approximation to the user cost of R&D. This expression has a simple interpretation. The term r + α is the per unit user cost of R&D knowledge. The tax inclusive cost of each unit of R&D knowledge is equal to the product of the input tax effect, /T, and the cost of producing one unit of R&D 00

9 THE 990 R&D TAX CREDIT knowledge. Because it takes θ units of z and one unit of K R to produce one unit of R&D knowledge, the cost of producing one unit of R&D is θ times the subsidized price of z plus one times the user cost of K R. Conclusions This analysis has shown that the R&D tax credit is equivalent to a uniform tax on all inputs in combination with a subsidy on qualifying R&D expenditures. Because the tax component of the credit is uniform across inputs, it will create no substitution effects in the choice of inputs used in the production of output or in the choice of inputs used in the production of R&D. On the other hand, the subsidy component of the credit will induce firms to substitute noncapital for capital inputs in the production of R&D. Given the current values of tax code parameters, the level of implicit input taxation is sufficiently low that it is unlikely to have any significant effect on firm decision making. This suggests that it may be reasonable to ignore the input tax component of the R&D credit and to view the R&D credit as simply a subsidy of 20 percent for qualifying R&D expenditures. The current credit system offers a greater marginal incentive a lower user cost for R&D than the system that was in place prior to 990. The increase in this incentive is greater for firms with lower discount rates and lower historical R&D to sales ratios. ENDNOTES The author is grateful to Joseph Çordes and the referees for helpful comments. Qualified expenditures include most items, such as labor costs, materials, and the like, that could be deducted under the corporate income tax, provided they are incurred in the development of a product or process. Expenditures on quality control, marketing research, or on another firm s patents or processes do not qualify. Some capital expenditures also qualify for the credit and can be expensed. 2 Apparently, Congress felt that linking the base of the credit to a measure of firm s activity would increase the marginal incentive provided by the credit. For a discussion of the link between the definition of a tax credit base and its incentive effects, see Gravelle (993). 3 Note that the value of the historical R&D to sales ratio, s, is exogenous from the firm s perspective; its value is determined by the firm s choices in the period between 985 and 989. Firms without a history, such as start-ups, are assigned an s value of The value of s for some firms may be so small that the product of s and average sales falls below 50 percent of current R&D expenditures; in this case, the credit base is defined to be 50 percent of current R&D expenditures. Because this provision is unlikely to affect firms with an active research record, it will be ignored in the analysis that follows. 4 In the interest of tractability, we will ignore the requirement that B must be non-negative for now; that is, we will assume that the credit is fully recaptured when B is negative. While this symmetry assumption makes the analysis tractable, it may not be appropriate for firms that find it optimal to allow R&D spending to fall relative to sales for any length of time. See Altshuler and Auerbach (990) and Scholes and Wolfson (992) for analyses of the effects of tax law asymmetries. 5 The firm s discount rate generally will be a function of the firm s financing choices and tax code provision; however, the results that follow do not require a specific form for r. 6 For firms with no history of R&D activity, such as start-ups, the base would have been 50 percent of expenditures in the previous year. Because this provision did not affect research-active firms, it will be ignored in the analysis that follows. 7 It may appear from the table that if the discount rate rose enough, a firm might find that the old system was at least as generous as the new system. While this is true, the discount rate required for this to occur would be 68 percent if the firm s s value were 6 percent. 8 Because some capital expenditures can be subject to the R&D tax credit, dichotomizing R&D inputs in this way is not entirely correct. However, capital expenditures that are subject to the credit can also be expensed, so that their tax treatment is similar to that of labor and materials. Futhermore, almost all items that are not subject to the credit, such as structures and acquired research (patents and the like), are durable goods. 0

10 NATIONAL TAX JOURNAL VOL. XLIX NO. REFERENCES Altshuler, Rosanne, and Alan Auerbach. The Significance of Tax Law Asymmetries: An Empirical Investigation. Quarterly Journal of Economics 05 No. (February, 990): Bailey, Martin Neil, and Robert Lawrence. Tax Incentives for R&D: What Do the Data Tell Us? Washington, D.C.: Council on Research and Technology, 992. Budelis, John, and Arthur Bryson. Some Optimal Control Results for Differential- Difference Systems. IEEE Transactions on Automatic Control 5 (April, 970): Eisner, Robert, Steven Albert, and Martin Sullivan. The New Incremental Tax Credit for R&D: Incentive or Disincentive? National Tax Journal 37 No. 2 (June, 984): Gravelle, Jane. Incremental Investment Credits. Congressional Research Service Report No S. Washington, D.C.: Congressional Research Service, 993. Griliches, Zvi. The Search for R&D Spillovers. NBER Working Paper No Cambridge, MA: National Bureau of Economics Research, 99. Hall, Bronwyn. R&D Tax Policy during the 980 s: Success or Failure? In Tax Policy and the Economy, Vol. 7, edited by James Poterba, 35. Cambridge, MA: MIT Press, 993. Kamien, Morton, and Nancy Schwartz. Dynamic Optimization. New York: Elsevier, 99. Scholes, Myron, and Mark Wolfson. Taxes and Business Strategy: A Planning Approach. Englewood Cliffs, NJ: Prentice-Hall, 992. and this must be maximized subject to dr/dt = g(z,k R,R), dk/dt = I δk, and dk R /dt = I R δ R K R. The base for the R&D credit is given by B = p z z(t) s 4 t F [ R(x),K(x),L(x)]dt. t 4 Because the price of output is unity, the integral above represents the average sales of the firm between time t 4 and t. The presence of the moving average (the integral in the definition of the credit base, B) makes this a rather unusual dynamic optimization problem. Because both the limits of integration for this integral depend upon time, the integral cannot be dealt with in the usual manner; that is, the definition of B above cannot be treated as an isoperimetric constraint or an integral equation constraint. The following strategy will be used in solving this problem. B will be converted to a state variable by differentiating the definition of B given above with respect to time. The resulting differential equation for db/dt includes a lagged time variable; however, using a result due to Budelis and Bryson (970), it is possible to derive optimality conditions that take this time lag into account. (A relatively painless discussion of this methodology can be found in Section 9 of Kamien and Schwartz. 99.) Differentiating the definition of B with respect to time yields db/dt = p z dz/dt (s/4)[f(t) F(t 4)], where F(t) = F[R(t),K(t),L(t)] and F(t 4) = F[R(t 4),K(t 4),L(t 4)]. Note that although z is a control, the time derivative of z appears in the equation above. In fact, z can be treated as a state variable rather than a control, provided it is recognized that the time derivative of z is not constrained in any way. Because dz/dt is not constrained, simply define dz/dt = y, where y is an additional control variable that must be chosen. APPENDIX The model described here is the more general version dealt with in the next to the last section, where it was assumed that the growth of R&D could depend upon a capital input, K R, as well as the input subject to the R&D credit, z. In particular, suppose that the growth in the stock of R&D is determined by g(z,k R,R) and that K R grows according to I R δ R K R, where I R is gross investment in R&D enhancing capital and δ R is the deprecation rate for this type of capital. If q R is the cost of a unit of I R and d R is the discounted value of depreciation deductions per unit of K R, then the after-tax cost of a unit of I R is q R ( τd R ). The firm s discounted after-tax cash flow is given by 0 e rt [( τ)(f wl p z z + cb) q( τd)i q R ( τd R )I R ]dt The Hamiltonian for this problem can be written as e rt [( τ)(f wl p z z + cb) q( τd)i q R ( τd R )I R ] + λ z y + λ R g + λ B [p z y (s/4){f(t) F(t 4)}] + λ K (I δk) + λ KR (I R δ R K R ) where λ i is the costate variable for state variable i. Note that any of the first-order conditions involving L, K, or R will have to take into account the lagged values of these variables in the differential equation for B. Following Budelis and Bryson (970), this is done by including in these first-order conditions the impact on F of a change in L, K, or R times the value of the costate variable for B evaluated four years in the future. Using this approach, the first-order conditions for the controls L, I, I R, and y are, respectively, 02

11 THE 990 R&D TAX CREDIT e rt ( τ)[f L (t) w] (s/4)f L (t)[λ B (t) λ B (t + 4)] = 0 e rt q( τd) + λ K (t) = 0 e rt q R ( τd R ) + λ KR (t) = 0 Using the solution for λ B, the first-order conditions for y, I, and I R, and time derivatives of the first-order conditions for y, I, and I R, the costate variables in the first order condition for L and in the equations involving dλ K /dt, dλ R /dt, and dλ KR /dt can be eliminated, yielding λ z (t) + p z λ B (t) = 0. F L = w/t (The time argument has been included in the functions above to avoid any confusion.) For the costate variables, the firstorder conditions are dλ z /dt = { ( τ)p z e rt + λ R (t) g z } F K = q(r + δ)( τd)/( τ)t F R = ( c)p z Tg [ r g + dg /dt z R g ] z z dλ B /dt = {c( τ)e rt } dλ R /dt = {( τ)e rt F R (t) λ R (t)g R (t) (s/4)f R (t)[λ B (t) λ B (t + 4)]} g KR g z = q R (r + δ R )( τd R )/( τ) ( c)p z dλ K /dt = {( τ)e rt F K (t) δλ K (t) (s/4)f K (t)[λ B (t) λ B (t + 4)]} where dλ KR /dt = {λ R (t) g KR (t) δ R λ KR (t)}. Assuming that the usual transversality conditions hold that is λ B 0 as t the equation for the costate variable λ B can be solved. This yields λ B = ( τ)e rt r. T = sc 4r ( e 4r ). The results in the text for the simple version of the model can be found by setting g R = α, g z =, and g KR = 0. The definition of T given in the text is the discrete time analogue of the definition for T given above. When g = min [z/θ,k R ] αr, the analysis above has to be altered slightly by adding the constraints z/θ = K R (the firm is at a corner of its R&D isoquant) and dr/dt = K R αr. 03