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41,794.3

Source: Thomas Pei-Fan Chen, "Economic Growth and Structural Change in Taiwan—1952-1972, A Production Function Approach," unpublished Ph.D. thesis, Dept. of Economics, Graduate Center, City University of New York, June 1976, Table II. *New Taiwan dollars.

Source: Thomas Pei-Fan Chen, "Economic Growth and Structural Change in Taiwan—1952-1972, A Production Function Approach," unpublished Ph.D. thesis, Dept. of Economics, Graduate Center, City University of New York, June 1976, Table II. *New Taiwan dollars.

method (see Appendix 7A, Section 7A.5 for the computer printout):

ln Yi = -3.3384 + 1.4988 ln X2i + 0.4899 ln X3i (2.4495) (0.5398) (0.1020)

From Eq. (7.9.4) we see that in the Taiwanese agricultural sector for the period 1958-1972 the output elasticities of labor and capital were 1.4988 and 0.4899, respectively. In other words, over the period of study, holding the capital input constant, a 1 percent increase in the labor input led on the average to about a 1.5 percent increase in the output. Similarly, holding the labor input constant, a 1 percent increase in the capital input led on the average to about a 0.5 percent increase in the output. Adding the two output elasticities, we obtain 1.9887, which gives the value of the returns to scale parameter. As is evident, over the period of the study, the Taiwanese agricultural sector was characterized by increasing returns to scale.18

18We abstain from the question of the appropriateness of the model from the theoretical viewpoint as well as the question of whether one can measure returns to scale from time series data.

226 PART ONE: SINGLE-EQUATION REGRESSION MODELS

From a purely statistical viewpoint, the estimated regression line fits the data quite well. The R2 value of 0.8890 means that about 89 percent of the variation in the (log of) output is explained by the (logs of) labor and capital. In Chapter 8, we shall see how the estimated standard errors can be used to test hypotheses about the "true" values of the parameters of the Cobb-Douglas production function for the Taiwanese economy.

We now consider a class of multiple regression models, the polynomial regression models, that have found extensive use in econometric research relating to cost and production functions. In introducing these models, we further extend the range of models to which the classical linear regression model can easily be applied.

To fix the ideas, consider Figure 7.1, which relates the short-run marginal cost (MC) of production (Y) of a commodity to the level of its output (X). The visually-drawn MC curve in the figure, the textbook U-shaped curve, shows that the relationship between MC and output is nonlinear. If we were to quantify this relationship from the given scatterpoints, how would we go about it? In other words, what type of econometric model would capture first the declining and then the increasing nature of marginal cost?

Geometrically, the MC curve depicted in Figure 7.1 represents a parabola. Mathematically, the parabola is represented by the following equation:

which is called a quadratic function, or more generally, a second-degree polynomial in the variable X—the highest power of X represents the degree of the polynomial (if X3 were added to the preceding function, it would be a third-degree polynomial, and so on).

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