241B Lecture Application: Returns to Scale in Electricity Markets

Transcription

1 241B Lecture Application: Returns to Scale in Electricity Markets We work from Nerlove (1963): a classic study of returns to scale in a regulated industry. The Electricity Supply Industry In 1963, the US electricity industry was characterized by: Privately owned local monopolies who supply power on demand. Rates (electricity prices) set by regulators. Factor prices (e.g. wage rate) are given to the rm, either because of perfect competition in the market for factor inputs or through long term contracts with labor unions. Deregulation has occurred since Multiple rms are allowed to supply power in the same market and rates are no longer strictly regulated. The Data Nerlove assembled a cross-section data set for The set contains 145 rms in 44 states. The variables are: total costs, factor prices (the wage rate, the price of fuel, the rental rate of capital) and output. Rental rate of capital: Although rms own capital (e.g. power plants), standard investment theory (Jorgenson 1963) tells us that (as long as there are no costs in changing the capital stock), the rm should behave as though it rents capital on a period-to-period basis from itself at a rental price (the user cost of capital) (r + ) p I r is the real interest rate, is the depreciation rate and p I is the price of capital. With this logic, capital can be treated as a variable factor of production, just like labor and fuel inputs. Data construction:

2 Output, fuel and labor costs (which together with capital costs make up total costs) from the Federal Power Commission. Wage rate - statewide average wage for utility workers. Capital costs - ideally calculated as reproduction cost of capital times user cost of capital. Due to data limitation, constructed from interest and depreciation charges available from the rms books. Why do we need econometrics? Why not simply plot average cost (for each rm) against output to determine if the average cost curve is downward sloping? Because the electricity industry faced substantial regional di erences in factor prices, so each rm can have a di erent average cost curve. Nerlove pioneered the method of accounting for di ering factor prices by estimating a parameterized cost function. A second factor that shifts rm level average cost curves is production e ciency. Consider an industry with two rms. The average cost curve for rm A lies above the average cost curve for rm B, because rm B is more e cient. Further, rm B is likely to produce more electricity, because it is more e cient. Connecting the two observed output levels would lead one to wrongly infer increasing returns to scale, when the upward slope of the average cost curves implies decreasing returns to scale. (See Figure 1 for this lecture) Cobb-Douglas Technology To derive a parameterized cost function, we start with a Cobb-Douglas production function Q i = A i x 1 i1 x 2 i2 x 3 i3 ; where Q i is rm i s output, x i1 is labor input (for rm i), x i2 is capital input, x i3 is fuel input and A i captures unobservable di erences in production e ciency (sometimes referred to as rm heterogeneity). The sum = is the degree of returns to scale. Thus it is assumed a priori that the degree of returns to scale is constant across rms. As the electric utilities are privately owned, it may be reasonable to assume that they minimize costs. The cost function associated with the Cobb-Douglas production function is also a Cobb-Douglas function as total costs for rm i are T C i = (A i x 1 i1 x 2 i2 x 3 i3 ) 1= Q 1= i p 1= i1 p 2= i2 p 3= i3 :

3 Taking logs, we have the log-linear form ln (T C i ) = i + 1 ln Q i + 1 ln p i1 + 2 ln p i2 + 3 ln h with i = ln (A i x 1 i1 x 2 i2 x 3 i3 ) 1=i. The coe cients are elasticities, for example the coe cient on ln p i1 is the elasticity of total cost with respect to the wage rate (i.e. the percentage change in total cost brought about by a 1 percent change in the wage rate). The degree of returns to scale, which appears as the reciprocal of the output elasticity of total cost, is independent of the level of output. If we let = E ( i ) and de ne U i = i, then E (U i ) = 0. The quantity U i captures the inverse of rm i s productive e ciency, relative to the industry average. Thus rms with positive values of U i are relatively ine cient and so have higher costs of production. With this notation, the cost function can be written in standard form ln (T C i ) = ln Q i + 3 ln p i1 + 4 ln p i2 + 5 ln + U i ; where 1 =, 2 = 1, 3 = 1, 4 = 2, and 5 = 3. We now see how a parametrized cost function accounts for variable factor costs: Cobb-Douglas production technology leads to a cost function that depends explicitly on the factor prices allowing the researcher to control for varying factor prices. Further, the error term is interpretable as an economic quantity. Is Cobb-Douglas Correct The Cobb-Douglas function satis es certain properties we require, such as declining marginal productivity for factors. Yet many other production functions also satisfy these requirements. In practice, although not in the application considered by Nerlove, the Cobb-Douglas function has been found to be surprisingly accurate at describing data. Are the OLS Assumptions Satis ed? A key assumption is strict exogeneity. It seems reasonable to assume, as in most cross-section studies, that X i is independent of U j. Thus, if X i is independent of U i, then E [UjX] = 0. One feature of the industry is that factor prices are given to the rm, without regard to the rm s e ciency. Hence it is reasonable that the price regressors are independent of U i. But what about output? Because the rm s output is supplied on demand, the level of output depends on the price of electricity. As the price is set by a regulator, if the regulator sets price without

4 regard to the rm s e ciency, then output too is independent of U i and strict exogeneity is satis ed. This, indeed, is what Nerlove assumes. If, however, the price is set to cover the rm s average costs, then output and e ciency are correlated. (Highly e cient rms would have lower prices set by the regulator and higher quantities. As the error measures ine ciency, we would see a negative correlation between output and the error.) This form of endogeneity would also hold if the industry were competitive. The assumption that the error terms are not correlated would be violated if there were technology spillovers between rms in a given area. Such spillovers are not assumed relevant for the study at hand. There is no good reason to assume that conditional homoskedasticity is satis ed. Indeed, Nerlove devotes much of his paper to dealing with conditional heterogeneity. Restricted Least Squares The regression equation is overidenti ed because there are ve regression coef- cients to capture four parameters from the total cost function. The implication is that there is a restriction among the coe cients. Indeed, the fact that = implies that = 1. The restriction also has economic meaning; a generic property of cost functions is that they are linearly homogenous in factor prices (so multiplying total costs and all factor prices by a common factor leaves the cost function intact). Indeed, the cost function is linearly homogeneous in factor prices if, and only if, = 1. To impose the restriction take any one of the factor prices, say, and replace 5 with 1 ( ) to obtain T Ci pi1 pi2 ln = ln Q i + 3 ln + 4 ln + U i : Estimating the rewritten equation with OLS yields the restricted least squares estimators. The parameters of the cost function can be uniquely recovered from the four parameters of the rewritten model. To obtain an estimator of 5, simply use 1 (B 3 + B 4 ). Testing Homogeneity of the Cost Function We have asserted that cost functions should be linearly homogeneous, but it is always sensible to test such a restriction before imposing the restriction. The

5 unrestricted estimates from Nerlove s data are ln T C i = 3:5 (1:8) + 0:72 (0:017) ln Q i + 0:44 (0:29) ln p i1 0:22 (0:34) R 2 = 0:926 mean of dependent variable = 1:72 SER = 0:392 SSR = 21:552 n = 145: ln p i2 + 0:43 ln (0:10) Because 2 = 1, the degree of returns to scale implied by the estimate is about 1.4 (= 1=0:72). The estimate of 4 = 2 = has the wrong sign. As Nerlove notes, there are reasons to believe that the rental price of capital (p i2 ) is poorly measured. This may explain why b 4 is so imprecisely measured that one cannot reject the hypothesis that 4 = 0 (the t-ratio is: 0:22=0:34 = 0:65). To test linear homogeneity in factor prices we could use an F test for R = r. Alternatively, we can construct a test from the SSR for both the unrestricted and restricted regressions. The restricted estimates from Nerlove s data are T Ci pi1 pi2 ln = 4:7 + 0:72 ln Q i + 0:59 ln 0:07 ln (0:88) (0:017) (0:20) (0:19) R 2 = 0:932 mean of dependent variable = 1:48 SER = 0:390 SSR = 21:640 n = 145: The F test of the homogeneity restriction is performed as follows. Step 1: The test statistic is calculated as (21:640 21:552) =1 21:552= (145 5) = 0:57: Step 2: Find the critical value. The number of restrictions (equations) in the null hypothesis is 1 and K (the number of coe cients) in the unrestricted model is 5. So the degrees of freedom are 1 and 140 (= 145 5). From the table of F distributions, the critical value is about 3.9. Step 3: As the estimated value is far less than the critical value, we can comfortably fail to reject the null hypothesis of linear price homogeneity, a comforting conclusion for those of us who take microeconomics seriously. A Cautionary Note on R 2

6 The unrestricted R 2 of.926 is surprisingly high for cross-section data. Such a large amount of explained variation could simply be due to the scale e ect, that costs increase with rm size. To see this point, simply subtract ln Q i from both sides of the unrestricted equation and reestimate as T Ci ln = 3:5 0:28 ln Q i + 0:44 ln p i1 0:22 ln p i2 + 0:43 ln (1:8) (0:017) (0:29) (0:34) (0:10) Q i R 2 = 0:695 mean of dependent variable = 4:83 SER = 0:392 SSR = 21:552 n = 145: As you may have anticipated, the estimated coe cient on ln Q i is now 0:28 (= 0:72 1) and the other coe cient estimates are unchanged. The R 2 has changed, to re ect the fact that dependent variable is now average cost rather than total cost. It would not be sensible to select one of the two regressions because the R 2 is higher, as the two equations represent the same model. Clearly to compare R 2, the dependent variable must be the same. Testing Constant Returns to Scale From the above test on homogeneity, we have concluded that the restricted model is the model to study. To test constant returns to scale, = 1, we need simply examine the coe cient on ln Q i, which equals 1 if = 1. The null hypothesis of constant returns to scale is H 0 : 2 = 1. The t test of constant returns to scale is performed as follows. Step 1: The test statistic is calculated as 0:72 1 0:017 = 16: Step 2: Obtain the critical value. From the Gaussian limit theory, the critical value is 1.96 (the exact critical value from the t is almost identical, 1.98). Step 3: Because the test statistic exceeds the critical value in magnitude, we (overwhelmingly) reject the null hypothesis of constant returns to scale. Importance of Residual Plotting

7 The regression has a problem that cannot be seen from the estimated coe cients and standard errors. Figure 2 in the attached sheet mirrors the plot of residuals from the tted equation. One can immediately see that the sign of the estimated residuals is not unrelated to output. It appears the correct relation would have ln (T C) as a concave function of ln (Q). Fitting a line to a concave function yields residuals that are positive for low values of Q, negative for middle values of Q and positive again for high values of Q. Further, the variation in the residuals appears to be highest for low values of Q, suggesting conditional heteroskedasticity. To account for these issues, Nerlove divides the data into ve groups of 29 observations each, according to size. He then estimates the restricted equation separately for each of the ve groups. With ve separate equations, the coe - cients (including 2 = 1=) are allowed to di er for each of the groups as are the error variances. Nerlove nds that the returns to scale diminish steadily with size, from more than 2 to slightly less than 1. Subsequent Developments Much e ort was devoted to generalizing the Cobb-Douglas technology (while maintaining cost minimization). One alternative is the Constant Elasticity of Substitution technology, which (in addition to being highly nonlinear) implies constant returns to scale. As returns to scale do not appear to be constant, the translog cost function was developed, which explicitly allows for varying returns to scale. This model was estimated for electricity rms in 1973 and most of the increasing returns to scale were gone, rms had expanded in size to the region in which average cost curves are essentially at. An innovation of more importance is whether or not output is endogenous. In 1962 authors argued that regulators guarantee a fair-rate-of-return above cost, thus high-cost producers get higher prices, which in turn leads to lower output. Hence output is correlated with e ciency and is endogenous. This line of reasoning morphed into more recent work about the interplay between regulator and rm. The rm has more information about costs and tries to hide this information from the regulator. Modeling the outcomes as arising from regulator-utility interactions with asymmetric information yields a system of supply and demand equations (and so overcomes endogeneity of supply as well).