Producer Surplus without Apology? Evaluating Investments in RD

THE ECONOMIC RECORD. VOL. 73, NO. 221. JUNE 1997. 146-158

Producer Surplus without Apology? Evaluating Investments in R&D*

WILL MARTIN World Bank

Washington DC. USA

and JULIAN M. ALSTON Department of Agricultural and

Resource Economics, University of California,

Davis and

Member, Ciannini Foundation of Agriculiural Economics

Comparison of producer surplus with definitive meusures based on the profit function reveals potential problems with using changes in producer surplus to measure the benefits of some common types of technicul change. Some illustrative applications indicate that the conventional producer surplus measures may seriously underestimate the change in profit induced by new technology. depending on the characteristics of the underlying technology which dejne the nuture of the supply function, and the nature of the technical change. We provide guidelines for identifiing cases where producer surplus will under-estimate producer research benefits. and suggest alternative measures.

I Introduction In this journal. a quarter century ago, Duncan

and Tisdell ( 197 I ) first highlighted the fact that the specific form of a technical change resulting from research and development (R&D) can have major implications for the resulting welfare benefits. This insight has subsequently been widely acknowledged and used, and a number of studies of research benefits have extended Duncan and Tisdell’s results to a broader set of market and

An early version of this paper was presented at the Australian Agricultural Economics Society’s annual conference, Wellington. New Zealand. February 7-1 I , 1994. The authors are grateful for helpful comments from the conference participants and, on more recent versions, from John Freebaim and Richard Perrin, as well as two anonymous referees.

policy c0nditions.I From this literature. it seems clear that the choice of the specification of technical change often has a major bearing on the measured welfare consequences. Any conclusions are specific to the p;Uticular specification of technical change, and the results of any analysis based on only one specification of technical change cannot be generalized. This point is relevant both for theoretical studies (e.g.. de Gorter, Nielson, and Rausser 1992; Chambers and Lopez 1993). that have arbitrarily chosen certain specifications

Early contributions were made by Scobie (1976). Jarrett and Lindner (1977). Sarhangi, Logan, Duncan and Hagan ( I 977). Lindner and Jarrett ( 1978, 1980) and Rose (1980). among others. Norton and Davis (1981) reviewed that literature. More recent studies-including Edwards and Freebairn (1981. 1984). Davis, Oram, and Ryan (19871, Gardner (1989). Murphy, Furtan, and Schmitz (1993). and Voon and Edwards (1992). among others-are reviewed and summarized by Alston. Norton, and Pardey ( I 995).

I46

1997. The Economic Society of Australia. ISSN 0013-0249.

I997 PRODUCER SURPLUS WITHOUT APOLOGY? 147

of the type of technical change (often for analytical convenience), as well as for many empirical studies, estimating rates of return to research, that invariably have chosen a particular type of research-induced supply shift.

While Duncan and Tisdell were careful to specify technical change in terms of its effects on costs, much of the subsequent litenture has specified different technological chan es in terms of

tended to use either parallel or proportional shifts in supply curves, representing unspecified types of reduction in the underlying cost structures of production. Given the theoretical restrictions that the parameters of a model of a production technology must satisfy. such ad hoc representations are clearly incomplete.3

Recent pressures on public research budgets and increased demands for accountability through ex post evaluation of the benefits. along with the increasing requirements for rigorous assessment of benefits before resources are allocated, mean that accurate measures of research benefits are becoming more important. In much of the literature on evaluating the benefits from technical change. producer surplus measures are used as though they provide definitive estimates of the direct benefits from technical change. It is important to remember, however, that the rationale for using producer surplus is typically to provide a measure of quasi-rents or returns in excess of variable costs (Just, Hueth. and Schmitz 1982. p. 54). This can be done more directly by specifying a modern. dual profit function indicating the maximum achievable excess of returns over variable costs (Comes 1992, p. 117). In this paper, we explore the implications of using this approach, rather than producer surplus, and the resulting insights into the sources of benefits from research.

The procedures we outline for evaluating the benefits from research are potentially applicable

* As discussed by Alston, Norton, and Pardey ( 1 995). parallel supply shifts have been used with linear supply curves while proportional supply shifts have been used with linear or constant elasticity functions; the choice has been dictated almost entirely by analytical convenience, given a prior choice of functional form for

For example, given the theoretical requirement that supply curves be homogeneous of degree zero in prices, it is not possible to change the own-price elasticity of supply without adjusting one or more cross-price elasticities.

shifts in single supply curves. P Studies have

su ply.

in a wide range of circumstances where investments are made, by either the public or private sector, to lower the variable costs of production. We choose to focus on the case of evaluating investments in R&D because this is the one where producer surplus measures have been most widely used, and the issues are important.

We first set out the approach of using a profit function to evaluate the welfare effects of technical changes. Then we consider the specification of the types of technical change of interest in terms of producer profit functions that represent the production technology and producer behaviour under the hypothesis of competitive profit maxi- mization. We compare and contrast the algebraic measures of producer benefits from the profit function with the corresponding single-market producer surplus measures. We show that the differences between changes in producer surplus and changes in profit can be interpreted in many cases as being equivalent to changes in avoidable fixed cost.. which are reflected in profits but not in producer surplus. Numerical examples are provided to illustrate the practical implementation of the profit function approach, and to provide an indi- cation of the order of magnitude of the differences that may arise between it and the conventional producer surplus approach. In contrast with the well-known case of using consumer surplus to measure welfare effects of price changes, considered by Willig (1976). our results show that using producer surplus to measure the benefits from technical change may lead to serious under- estimation when typical values are used for parameters.

I1 The Specification of Technical Change For our purposes, it is convenient to represent

production technology using a producer profit function. Changes in this profit function directly provide estimates of the benefits to producers from technical ~ h a n g e . ~ In the equations below, P@, v, T) is producer profit for a given vector of

4 This specification is quite gened with respect to analytical length of run and analytical focus. An input or output that is fixed in the short run may be varied in a long-run analysis. A revenue function of the type used by Dixit and Norman (1980. p. 30) is an important case. Long-run supply functions derived from such revenue functions may slope upwards because of induced changes in factor prices. rather than the presence of fixed factors emphasized in the partial-equilibrium literature.

148 ECONOMIC RECORD JUNE

domestic prices, p. endowment of resources or fixed factors, v, and state of technology represented by T. The state of technology is determined outside the control of the profit-maximizing producers in the industry, by factors such as public investment in R&D or by knowledge spillovers, as emphasized in the modem literature on endog- enous growth? The profit function is one component of the overall evaluation of the national benefits from technical change, but for a small open economy without distortions it is a complete measure.6

We consider two approaches to specifying dis- embodied exogenous technical change: (a) the direct incorporation of technical change variables in the function (e.g., Binswanger 1974; Kohli 1991). and (b) the use of a distinction between actual and effective quantities and prices, and output- or input-augmenting technical change (e.g.. Dixon. Parmenter. Sutton and Vincent 1982)? In general form the resulting specifications of producer profit (and hence technology) may be represented as:

T = g(p, v, T I a) (la)

T = g@W. v I a) (Ib) where a is a vector of parameters of the profit function and all other variables are as previously defined. The technology may be represented by any of a wide range of dual functional forms that specify the profit function directly, or may be specified in primal form allowing profit to be evaluated indirectly.8

The approach is more general than this. It could be used, for instance. to evaluate the benefits to suppliers of labour from a technical change that reduced the cost of supplying labour or increitsed its efficiency. This interpretation would involve the use of an expenditure function, e@. v, u. t ) instead of the profit function, where u is the exogenously fixed utility level. For con- creteness and clarity we focus on the profit function interpretation.

A more complete measure allowing for changes in world prices, and for the presence of distortions, can be obtained using the balance ofrrode function (Woodland 1982. Lloyd and Schweinberger 1988. Anderson and N e q 1992) as applied to the measurement of benetits from technical change by Martin and Alston (1994).

A third approach is to use a varying-parameter specification in which the coefficients of a static model are themselves functions of technical change (e.g.. Fulginiti and Pemn 1993). That approach is not considered here. * Kohli (1991) discusses a range of possible functional forms.

These two approaches to the specification of technical change provide great flexibility in incorporating different forms of technical change. As we show below. such approaches can capture the types of technical change used in commodity market models of research benefits, while allowing the restrictions required by economic theory to be satisfied. The profit function approach. unlike the producer surplus approach, can also be generalized readily to allow for multiple sources and more complex forms of technical change. For simplicity and for consistency with the literature. we choose to illustrate these issues using a normalized quadratic profit function, which implies the linear supply functions that have been widely used in modelling research benefits, and can incorporate the commonly used types of technical change; our approach can readily be generalized to other functional forms.

111 Incorporating Technical Change Variables Directly

The first specification of technical change is well known from the empirical literature on the estimation of flexible functional forms (see Bin- swanger 1974). Using a normalized quadratic profit function to illustrate this approach, we begin with9:

. r r = q + a ' P + % P A P (2) where T is normalized profit, P = [p' v' r'l' is a vector including normalized prices, p. of variable outputs and inputs (where inputs are represented by negative quantities); quantities of fixed inputs and outputs, v; and technology variables, T; a' = [ a , , . . . , a,] is a vector of panmeters; and A is a matrix of parameters, conformable with the P vector (for clarity we represent elements of A by pv and 8, reserving aih for technology variables).

By Hotelling's lemma, differentiating the profit function with respect to the netput prices yields the output supply and (negative) input demand functions. For the quadratic function represented in equation 2. this will result in the technology variables entering the output supply and input demand functions as linear shift variables. The

Shumway. Jegasothy and Alexander (1987) discuss the normalized quadratic function in detail. The normalized quadratic profit function presented here includes the widely used Generalized McFadden estimating form as a special case (Diewert and Wales 1987).

1997 PRODUCER SURPLUS WITHOUT APOLOGY? I49

output supply or input demand function for any non-numenire good islo:

n m

The specification represented by equation 3 allows only for parallel shifts in the output supply or input demand equations, since the effect of a change in any T h on qi does not depend on either output or price. For a single technical change (i.e., a change in a particular q, parameter). the change in the ith supply function will involve only one term (i-e., ai,,ATh). In terms of its effect on the supply curve for a single output, this technical change is represented by the move from S, to S , in Figure I , which is reproduced from Duncan and Tisdell's figure I (1971, p. 127)." This form of technical change is often contrasted with a proportional supply shift in the quantity direction, such as that represented by the move from S, to S , , which Lindner and Jarrett (1978) termed a pivotal shift in order to distinguish it from a proportional shift in the price direction.

Consider only a single technical change. represented by an increase in a particular panmeter, Th. A Taylor-series expansion around the initial value of the profit function yields an exact expression for the change in profit resulting from an increase in Th. as follows:

The change in profit includes the effects on all of the supply functions. but. for comparability with producer surplus, we consider a technical change that causes a parallel shift in only one supply function, that for commodity i. In this case only one coefficient on T h , a;,,. is nonzero, so that one element of the summation in equation 4 is

lo The quantity (output or input) of the numeraire good can be determined from the outputs of the non- numeraire goods and the profit function-it is not inde- pendently determined.

I ' The same approach could be used to evaluate the effects of investing in additional factor endowments by considering the effects of a change in vk

FIGURE I Pivotal and Parallel Supply Ships

t Price

S 2

1 0

0 Quantity

nonzero. l 2 Further, this first-order term captures all of the effects of the technical change; there are no higher-order effects to be considered. This technical change also has a very simple geometric interpretation since the change in the aih term is the magnitude of the horizontal shift in the supply curve for good i.

Interpreting the effects of this technical change geometrically, we see that in this case of a parallel shift in the supply curve for good i only, the gain in net revenue can be expressed very simply as the price of good i times the shift in the supply curve, or as pi aih ATh = pi Aq;. In Figure 2 (panel a or panel b), this gain is represented by the rec- tangle q@qI (which is equivalent to the parallelogram abdc). This result is an interesting one with powerful intuitive appeal: it implies that a technical change that increases the output of good i by Aqi units, without affecting the output of any

I* Given the definition of the functional form. in this case there are no adding-up effects that require corresponding shifts in other commodity supplies or factor demands. In some other situations it might not be possible to define a theoretically consistent technical change that shifts only one supply function.

13 The quadratic term involving 7 h in the profit function (i.e., since T~ is being mated as an argument of the function, like one of the prices or fixed factors) drops out from the summation in equations 3 and 4 since ah), = 0.

I50 ECONOMIC RECORD JUNE

Rice 1

FIGURE 2

Gains in ProJit from a Parallel Supply Shij?

(a) (b)

S Price t s

other good or the required quantity of any input, will increase profits by pi Aqi. The resulting measure is equivalent to the 'approximation' to gross annual research benefits (GARB) used by Griliches (1958), when demand is perfectly elastic, as is assumed here.

This simple result from the profit function approach may differ from the result provided by the conventional producer surplus approach, depending upon the elasticity of supply. First, consider the case of inelasric supply (panel a in Figure 2). where the linear supply curve has a positive intercept on the quantity axis (i.e., a positive quantity of good i is produced at a zero price). In this case, the producer surplus approach yields the same result as direct evaluation of the profit function. Because an inelastic linear supply function has implausible implications when the function is extrapolated back to the axes, some users of the producer surplus approach have been concerned about the interpretation of producer surplus measures. Some ad hoc solutions have been offered, but these amount essentially to rejecting the combination of the linear functional

form and elasticities less than one.I4 The issue of functional form is pervasive. Different functional forms might avoid this particular problem but introduce others. For instance, constant elasticity supply functions pass through the origin and imply that positive output will be supplied at prices near zero.1J

An elastic linear supply function (panel b in Figure 2) is more plausible at output levels near zero than either an inelastic linear supply function or a constant elasticity supply function, since it

l4 For insme, Rose (1980) and Lindner and Jamt t (1978. 1980) discussed kinking linear supply functions at the initial equilibrium quantity or price in order to avoid having a negative intercept on the price axis.

I5 An alternative type of supply function has been proposed by L y n m and Jones (1987). which nests the linear and constant elasticity models as special cases. It is a 'consrant-elasticity' form but displaced from the origin so that it cuts the price axis at a positive price: q = f3(P - ar. This ty+ of model. embedded in a complete supply system by Bumiaux and van der Mens- brugghe (1991), was used to measure research benefits by Alston and Martin (1993).

1997 PRODUCER SURPLUS WITHOUT APOLOGY? 151

implies a positive shut-down price (i.e., a price at which zero quantity would be produced). The producer surplus approach yields an estimate of the gains equal to area abefin this case, compared with the profit function measure equal to area ubdc.

The difference between the measure obtained from direct use of the profit function and the producer surplus measure arises because, in the producer surplus measure, the supply curve is truncated at the price axis. The profit function. however, is defined over positive prices. and when the parameters imply elastic linear supply functions as in Figure 2(b), the part of the supply function in the north-west quadrant, where positive prices are associated with negative quantities, is relevant as well. I t can be interpreted in several ways.

One possible interpretation is in terms of netputs: when the price becomes sufficiently low. below point f on curve S in Figure 2(b), the firm or industry changes from a net seller to a net buyer of the good; the good which was an output (positive netput) becomes an input (negative netput). This netputs interpretation would be more widely intuitively appealing and literally applicable in the case of intermediate goods produced within a vertically integrated firm. or in the case of supplied factors, discussed in footnote 5. Under this interpretation. the m a under the curve is a measure of costs of producing this commodity for both positive and negative quantities supplied to the market over the range of positive prices.

Another potential interpretation. that is relevant in a partial equilibrium context, is in terms of avoidable fixed costs-costs that do not vary with output but that can be avoided by ceasing to produce altogether. If negative quantities are not possible, so that the supply curve is truncated at the origin and f represents a choke price. the segment of S below f does not represent a supply curve. Nevertheless. the area under the curve associated with prices below f (triangle Ofc) might still be relevant as a measure of costs-equivalent to costs that can be avoided by shutting down when price falls below J avoidable fixed costs. This latter interpretation is convenient in that it is consistent with a widely held view that producer surplus differs from profit by the amount of any avoidable fixed costs, for a single-output firm. Even if avoidable fixed costs exist, so long as the firm stays in production, the change in producer surplus for a change in price is equal to the change in profit. However, when the supply curve shifts from S to S', the area corresponding to avoidable fixed costs falls from Ofc to Oed in Figure 2@),

and the change in producer surplus understates the change in profit by this amount.

A further possible interpretation of the profit function specification is simply that it provides a locally valid representation of profits. On this view, there is no need to attempt to provide a literal interpretation of what would happen if price fell to the point where output is zero, or if price itself went to zero; the only interest is in whether the profit function adequately represents local changes in outputs, inputs, and profits. Taking this view of the profit function, the expression for the change in profits obtained in equation 4 would be taken as definitive, whether or not it could be rationalized in terms of areas in the positive quadrant on a.supply and demand graph.

( i ) Welfure Meusures under Cost-Reducing Technical Change

The profit function approach can be adapted to deal with technical changes that are best represented as a reduction in the marginal cost of producing a good-a popular specification of technical change. To do this for any good arbitrarily defined as good I . it is necessary first to specify a short-run profit function (dR) in which the output of good I is initially quasi-fixed16 at 4. while the (normalized) supply price of the good, p: . becomes an endog- enous function of the technology, the (normalized) prices of other outputs and variable inputs, and the quantities of other fixed factors. Differentiation of the profit function with respect to this element t h y yields the negative of the virtual supply price, p I , needed to give rise to any specified level of output.17 This equation may be written, following equation 4. as:

n

16 This tixed-output case corresponds to a situation where output is conmlled by a quota.

l7 This virtual price indicates the value of foregone variable outputs. plus any change in the cost of variable inputs resulting from a one-unit increase in the output of good 1. It is also equal to the price that would have resulted in a profit-maximizing producer choosing to produce the specified quantity of good 1. N e w and Roberts (1980) i n d u c e d the concept of the virtual price, as the price that would induce consumers to choose an exogenous quantity.

152 ECONOMIC RECORD JUNE

Price

FIGURE 3

Gains in Projitftom a Parallel Cost Reduction

(a) (C)

If the only effect of the technical change that increases r,, is to reduce the short-run supply price of only output I , only one of the technology- related coefficients, +],, is nonzero. In this case, a Taylor-series expansion around the initial value of the profit function yields an expression for the change in profit:'*

(6) The measure given in equation 6 cornsponds to the area agfe in Figure 3(a). In contrast, the producer surplus area with fixed output is ugdc.

ATsR = xp+I@zh = X: 4:

Because it involves truncating the supply curves at a price of zero, the producer surplus measure is smaller than the profit function measure.19 When the supply elasticity is initially equal to one, the producer surplus area is again smaller than the profit function measure, as is shown in Figure 3(b). When the supply curve is locally elastic, as in Figure 3(c). the profit function-based measure with fixed output and the producer surplus measure are both equal to area aced.

In the more common situation, where the output of good 1 is able to adjust, it follows from the Le Chiitelier principle that the gains will be greater than are indicated by equation 6. The measure from the shon-run profit function (with a fixed quantity of good I ) understates the measure from the long-run function (with the quantity of good I variable), because it ignores the benefits from the increase in output made profitable by the technical change. Following the te+chnical change, the marginal cost of good I , p I , is less than the market price, pI. and so profits can be increased by expanding output. To take this into account we define long-run profits as

(7)

where 4: is the long-run optimal value for q I . The

I 9 = d%J2....pn. v, 4: 1 + p,q:

In common with ad hoc treatments of technical change, this assumes then is no direct effect of technical change on profits through other elements of the profit function. 1980).

l9 The ad hoc approach of kinking the supply curve at point D would allow the additional area (cdfe) to be picked up in the producer surplus measure (e.g., Rose


optimal quantity of output of good 1 must satisfy the first-order condition

Satisfying this first-order condition requires that the outputoof good 1 increases from its initial quantity, q to its optimal quantity, 9;.

A seconA-order approximation to the resulting change in long-run profit is given by:

= @ I - + Y2 (5) A9j

= !4 Ap;Aq,. (9)

This expression must be added to the change in short-run profits identified in equation 6 to obtain an estimate of the change in long-run profits resulting from the technical change. In terms of the diagrams, this implies adding the area abc to each of the areas identified with the short-run changes in profits in Figures 3(b) and (c). and area abg in Figure 3(a).

(ii) Quantitative Direrences between Producer Surplus and Profit Measures

To indicate the magnitude of the likely differences between the measures, we estimated the profit function-based measures and the producer surplus measures of welfare change for both the output-increasing and cost-reducing technical change, with local supply elasticities at the production point of 0.5. 1.0, and 2.0. The results, expressed as percentages of initial gross revenue, are presented in Table I . For a given research- induced supply shift, the quantitative importance of the difference between the profit function measure and the producer surplus measure depends only on the elasticity of supply in the neighbourhood of initial output.

With supply elasticities of 0.5 and 1.0. and a horizontal shift of the supply curve, the profit function measure and the producer surplus measure are identical. However, with an elasticity of 2.0, the producer surplus measure is approxi- mately half the profit function-based measure (for an infinitesimal change, it would be exactly half) because of the truncation of the supply curve at the price axis. For a cost-reducing technical

TABLE I Welfare Gain from an Additive Shifi

in the Supply Curve

Output Incrrasing Cost Reducing Shift Shift (10%). (lO%P

Supply Profit Producer Long-run Producer Elasticity Increase Surplus Profit Surplus

(%) Increase Increase Increase

0.5 10 10 10.25 5 1 .o 10 10 10.5 10 2.0 10 5.025 I I 1 1

a A 10% increase from the initial output level.

cent of initial price. A uniform reduction in marginal cost equal to 10 per

change (vertical supply shift), the producer surplus measure is half the profit function measure when the elasticity is 0.5; with a unitary elasticity, the two measures are almost equal (for an infinitesimal shift they would be equal); with an elasticity of 2.0, the producer surplus measure and the profit function measure are the same.

The appropriate choice of measure may depend upon the problem at hand. If a horizontal supply shift is believed appropriate. the profit function measure generates results that seem more plausible than the producer surplus measure in the case where they differ (i.e. when the elasticity of supply is greater than one). As previously noted, the profit function measure implies that a 10 per cent increase in output, with no other adjustments, will yield an increase in profits equal to 10 per cent of gross revenues. In the case of a cost-reducing technical change, when the supply curve is inelastic. the producer surplus measure omits a first-order term on the grounds that it corresponds to an area outside the positive quadrant. Clearly, however, if the facts of the situation involve a cost reduction on all units of production, then the profit function approach, which allows for this, will be superior.

IV Output- or Input-Augmenting Technical Change

Another specification that has been used widely to model technical change is the distinction between actual and effective quantities and prices utilized by Dixon et af. (1982). Under this approach, technical change is thought of as some-


thing that increases the effective quantity of a good associated with a given physical quantity. An important feature of this specification is that there is a corresponding change in the effective price of the good. An increase in the effective quantity of an output provided by each physical unit will raise its effective price to the producer. While care is needed in defining appropriate effective quantity and price variables, this approach does not alter the behavioural parameters of the model, ensuring that they continue to satisfy any theoretical restrictions imposed in their original construction. Further, the specification has the intuitively appealing feature that improvement in technology stimulates output in two ways: first through a productivity effect which increases output for a given level of inputs and, second, through an increase in competitiveness which draws additional resources from other activities.

Technical change of this nature may come about in many ways. such as an improvement in the physical quality of the good. or from improved information or management that allows the good to be utilized more efficiently. Using this approach, the relationship between physical and effective quantities of a particular good (i.e., inp$ or output) can be represented by q; = 9:. si. where qi is the actual quantity of the good, q: is the effective quantity of the good. and r: is the index of output-augmenting or input-augmenting technical change for good i. The corresponding relationshp between actual and effective prices is pi = p y r , . where p: is the effec2ve price of the good, pi is the actual price and si is the augmen- tation factor. When qi is an input, input-saving technical advance is represented by a decline in s:, which reduces the physical quantity of the input rquired for one effective unit and also lowers the effective price relative to the ytual price. When q; is an output, an increase in ri rep resents output-augmenting technical change: an increase raises the physical quantity associated with a given effective quantity and raises the effective price for a given actual price.

Under this specification of technical change, producers are represented as optimizing over effective quantities and prices, rather than actual quantities and prices. This causes changes in the quantities of goods chosen, and hence in the revenues generated. Once the quantities have been chosen. the revenues may be calculated by simple multiplication using either aCtu3 or effective quantities andprices, since the ri terms cancel when p: and q are multiplied together.

The profit function incorporating technical change is defined by replacing the variables in equation 2 with the corresponding effective values of those variables, and eliminating the terms involving the direct technical change variables, as can be seen in equation 2'.

p = a,, f a'P* + %P*' A P* (2')

where P* = [p*' Y*']'. Considering, for simplicity. technical change affecting only the variable output quantities q. and the corresponding prices. p . output supply or input demand is

n m

where the q: and p: variables are as defined above. Substituting the definitions of p* and q* into equation 10 yields a behavioural function in the actual price and quantity variables:

From inspection of equation 1 I . it is clear that a technical advance of this form for commodity i involves two proportional shifts: one resulting from multiplication of some or all of the price variables by 6 (a proportional shift in the price direction), and one resulting from myltiplication of the entire term in parentheses by 7 . (a proportional shift in the quantity directionhh

A second-order Taylor-series expansion will exactly capture the benefits from this form of the technical change given the quadratic form for the profit function. The resulting expression for the welfare effects of technical change is

Assuming r: is initially unity, substituting equation l l into equation 12 yields:

AT = p A A ( + !h fiii r

2o A combination of this type of shift in the supply curve and a parallel shift of the type considered in the previous section could be used to generate a pivotal shift. such as from So to S I in Figure I.


where ~ f ' = (&pi/& is the elasticity of sup ly of good i at the initial price pi and quantity qf

The first term on the right-hand side of equation 13 can be thought of as representing the welfare effect of the direct increase in the actual output of good i, that is the increase in output that is brought about purely by the increase in productivity, without any reduction in the output of other commodities. The second term (a second-order term) measures the net welfare gain that results from increases in output of good i achieved by reducing the output of other goods.

Interestingly, the first-order term in equation 13, representing the benefits from an output- augmenting technical change, is equivalent to that given in equation 4 for the case of a horizontal supply shift ai,,Ash in equation 4 is equivalent to piqiA7; in equation 13 when q, = f). This particular result has potentially very important practical implications-the measured benefit from a given percentage technical change is, to a first- order approximation. the same for these two frequently analyzed types of technical change. Note, however, that a 10 per cent output-augmenting technical change will yield an increase in output that is larger than 10 per cent, with the difference depending upon the supply elasticity.

As we demonstrate in Appendix A. the change in profit. indicated by equation 13 is equal to the change in producer surplus irrespective of whether the supply curve is elastic or inelastic. Thus. for this type of technical change. the two measures are equal even though the producer surplus measure is truncated at the price axis but the profit function measure is not. A geometric interpretation allows us to compare the producer benefit measure from the profit function with the producer surplus measure. With the producer surplus approach, the welfare change is measured by the area abdc plus the area bde (i.e., area abec) in Figure 4. In contrast, the measure of welfare change resulting from the profit function approach is given by the parallelogram fdeg plus the triangle cdh.

The equivalence of the two measures in this case is consistent with both the netputs interpretation and the avoidable fixed costs interpretation of the areas in Figure 4 associated with negative output quantities. The nature of the shifts of the curves is such that the area of triangle Oaf. under the initial curve So, is exactly equal to the area of triangle Obi, under the final curve S, (see Appen- dix A). Under the avoidable fixed costs interpretation, this means that the change in technology

FIGURE 4

Output-Augmenting Technical Change

- f g 0

Quantity

was of a type that did not affect avoidable fixed costs, so that changes in producer surplus would coincide with changes in profit.

The profit function measure also allows an intuitively appealing decomposition of the benefits from technical change. The area fdeg measures the value of the direct increase in output, from d to e, resulting from the technical change. The rota- tion of the supply curve from S, to S, was, after all, due to a multiplicative shift in this supply curve with no consequent reduction in any other supply curve. The area cdh measures the net gain in profit that results from the increase in the effective price of good i. Since this component of the increase in output of good i (the increase from c to d in Figure 4) requires a reduction in the output of other goods. it stands to reason that the welfare gain associated with this increase in output should be substantially less than for the component of output increase that is a 'free' good.

The decomposition of welfare change provided by the profit function approach raises serious questions about the use of the producer surplus methodology to distinguish between the welfare effects of ad hoc parallel and pivotal shifts in supply curves. The pure productivity effect represented by the shift from s, to S, is a pivotal shift in the supply curve but, as is clear from use of the profit function, yields a benefit measured by area degf. In contrast, the pure competitiveness


effect represented by the move from So to S , is more than a pivotal shift of the supply curve, but yields only a second-order welfare benefit. On this interpretation, it does not seem adequate to specify a technical change as an od hoc shift in a supply function. Some pivotal shifts may have larger welfare gains than parallel shifts with the same effect on output. What seems to be needed is a return to focusing on the effects of the technical change on the underlying technology or cost structure-as originally suggested by Duncan and Tisdell ( 197 I).

V Conclusion The traditional producer surplus approach may

not provide an accurate estimate of the changes in profit from technical change. The accuracy of the producer surplus approximation depends on the nature of the technical change. even in the context of the simple case of linear supply functions, corresponding to a normalized quadratic profit function. In all cases considered here, the profit function measures provided definitive and intuitively appealing measures of changes in producer net revenues. In several cases, the results obtained from the producer surplus approach were the same. However. in a number of important cases. the producer surplus measures were misleading because of the truncation of conventional supply curves at the axes. Further, in a number of cases, the profit function approach resulted in measures of welfare impacts that were simpler than the producer surplus approach.

A very clear-cut difference between the two approaches occurred with a parallel shift in one supply curve that is locally price elastic. For a technical change that raised output of one good by 10 per cent, with no other change in input or output levels, the change in profit was 10 per cent of the initial value of output for this good-a result that is clearly correct given the formulation of the problem. Using producer surplus measures, theesti- mated change is smaller by an amount depending on the elasticity of supply. With an elasticity of 2.0, the benefits were under-estimated by 50 per cent because of the truncation of the supply curve at the price axis. Conversely, for a cost-reducing technical change, producer surplus gave the same answer as the profit function approach only in the case of an elastic supply curve. For an inelastic supply curve. the truncation of the producer surplus measure at the quantity axis resulted in an underestimate of the change in profit.

When technical change is of an output- augmenting kind, the profit function-based measures and the producer surplus measures provide identical results. The profit function approach. however, provides an illuminating decomposition of the benefits into a component due to a pure pro- ducrivify effect with given resources and a corn- peritiveness effect, that is associated with the transfer of resources from other activities, and does not depend on the supply elasticity. The pure productivity effect is first-order in magnitude and equal to the benefits from an equivalent parallel supply shift even though it produces a proportional shift of the supply curve and a much smaller gain in producer surplus. The competitiveness effect, which depends on the supply elasticity, is second- order in magnitude, implying that accurate knowledge of the supply elasticity is less important than might be implied by the producer surplus approach.

Even in the relatively simple cases of technical change that we have considered, we find that the profit function is more reliable than the producer surplus measures, and allows useful, intuitive, interpretations of the results to be drawn. These advantages, and the greater generality of the profit function measures, should make the profit function more often the approach of choice in the increasingly important field of analyzing the benefits from research and development.

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APPENDIX A Projir and Producer Surplus under Output-

Augmenting Technical Change

A.1 Change in Projit

Assume a normalized quadratic profit function with two variable commodities, a commodity of interest, x having an exogenous price p . and a numeraire commodity, y having a price w. Output-augmenting technical change is represented by T. with p* = sp and q* = q/r. Without loss of generality, define w = I , so that we can use p = p/w as the normalized price. Then the normalized profit function is:


Using Hotelling's lemma the supply function for q is defined by

an 4 = - = a,* + IT'P = [al + a, ,TPIT. ( ~ . 2 ) aP Similuly, differentiating the profit function with respect to T:

iJ7l P - = alp + a,,~p? = [ a , + a,,rplp = q ; dr (A.3)

The second derivative of profit with respect to the technology index is:

(A.4)

A second-order Taylor-series approximation to the change in profit due to technical change is:

thr $77

ar d r 2 AT = -At + % -(AT)2

P T

= q -AT + K a, ,&AT)?. (A.5)

Defintg the initial value of lo = I . so that p: = po and 90 = 9@

2 AT = qop,,At + !4 a,Ipo(AT)2

= P& [ xo + !h QI I P&]. (A.6)

A.2 Change in Producer Surplus-Elastic Supply

In the supply function defined in equation A.2. the intercept on the price axis is = -a,/a,,T. Initially, when T = T,, = changes, T = T,, producer surplus before and after the technical change (i.e., for i = 0 or I . respectively), is given by Psi = (p - &)9@ Thus, initially PSo = (a, + a, 1p)-/2a , and,

The change in producer surplus is. therefore. given by PSI - PS, = alp(rl - I ) + %aIlp2(T3 - I). Using

after the technical change, PSI = ( a , + aIITIp! i /2aII.

2 the definition that (zl - I ) = AT and. thus, (T, - I ) = UT + (AT+, this simplifies to

PSI - PSo = a ,pAT + a,,$AT + % a,l$(AT)2

= PAT qo + % a IIpA-r } . (A.7)

This is identical to the measure given in equation A.6.

A.3 Chunge in Producer Surplus-Inelastic Supply

In the supply function defined in equation A.2, the intercept on the quantity axis is 7 = a,r. When T = T" = I . 7 = yo = al. After technology changes. -r = 7,. and 7 = yI = alrl. The producer surplus before and after the technical change (i.e.. for i = 0 or I , respectively), is given by Ps i = p(qi + 7;)/2. Thus. initially PSo = p(2a, + a, ,p)? and after the technical change, PSI = p(2aI7, + aIITIp)R. Let T, = I + AT. Then

PSI = % p ( l + Ar)(2a, + all(l +Ar)p)

[

= pso + (axI + aIIp)pAr + % all$(+&

Hence,

PSI - PS, = a , p A T + a, ,$AT + % a, ,&AT+

= PAT qo + I/i a , , p A z ] . (A.9)

This, also. is identical to the measure given in equation A.6.

A.4 Avoidable Fued Costs under Outputdugmenting Technical Chunge

In Figure 2(b), the area in the negative quantity quad- n n t beneath the supply curve, ofc, m y be interpreted as avoidable fixed costs, or the costs of supplying a netput of exactly m. This area can be mevured as the area of a triangle of height equal to the intercept on the price axis (p above) and width equal to the magnitude of the intercept on the quantity axis (-y above). Thus the cost is C = where p = -a,/a,,z and 7 = alr. Hence, C-=q/aII and does not depend on the value for T under output-augmenting technical change.

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Producer Surplus without Apology? Evaluating Investments in RD

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