Lecture22-2011 - Defining Changes in Productivity...

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Unformatted text preview: Defining Changes in Productivity Measuring Technological Change Measuring Changes in Productiity: XXII Charles B. Moss1 1 University of Florida November 15, 2011 Charles B. Moss Measuring Changes in Productiity: XXII Defining Changes in Productivity Measuring Technological Change 1 Defining Changes in Productivity 2 Measuring Technological Change Total Factor Productivity and Index Number Theory Distance Functions Charles B. Moss Measuring Changes in Productiity: XXII Defining Changes in Productivity Measuring Technological Change Defining Changes in Productivity y1 y1 y1 Yx Yx p2 y2 y2 Charles B. Moss p1 y2 Measuring Changes in Productiity: XXII Defining Changes in Productivity Measuring Technological Change In this figure, we assume that the inputs are constant at x , but the total level of outputs has increased from (y1 , y2 ) to ￿ ￿ (y1 , y2 ). Charles B. Moss Measuring Changes in Productiity: XXII Defining Changes in Productivity Measuring Technological Change Change in Productivity - Reduced Inputs x1 x1 x1 x2 x2 Charles B. Moss x2 Measuring Changes in Productiity: XXII Defining Changes in Productivity Measuring Technological Change In both cases, most would agree that a technical change has taken place. Further, most would agree that the technical change has increased the economic well being of society. We now have more stuff for the same level of inputs. However, there are some issues that need to be raised. Charles B. Moss Measuring Changes in Productiity: XXII Defining Changes in Productivity Measuring Technological Change First, one could raise the question about embodied versus disembodied technical change. This debate regards whether there has been an increase in knowledge, or whether there has been an increase in the quality of inputs. If the increase in output has been associated with an increase in quality of an input, is it technical change? For example, a large portion of the gains to research literature can be traced to Griliches discussion of hybrid corn. In this case, the increase in technology was associated with the improvement in an input – seed. More recently, some of the most recently observed increases in productivity may be traced to genetically modified organisms (GMOs). Under most concepts of productivity, these increases do not represent changes in productivity in agriculture, but can be traced to changes in the input bundle. Charles B. Moss Measuring Changes in Productiity: XXII Defining Changes in Productivity Measuring Technological Change A second area of concern is whether the changes in technology are neutral with regard to the input bundle. Going back to the figure, the increase in technology implies relatively more x1 it is biased toward x1 . x1 x1 x1 x2 x2 Charles B. Moss x2 Measuring Changes in Productiity: XXII Defining Changes in Productivity Measuring Technological Change Continued In the hybrid corn example, additional fertilizer complemented the use of hybrid corn. In the classic discussion, Hicks developed the notion of labor or capital augmenting technological development. Charles B. Moss Measuring Changes in Productiity: XXII Defining Changes in Productivity Measuring Technological Change Total Factor Productivity and Index Number Theory Distance Functions Measuring Technological Change In the single input-single output analysis, one could directly measure technical change: ￿y ￿ y y = f ( x , t ) ⇒ θ ( t ) = ⇒ t = θ −1 (1) x x Several factors should be considered. We know that profit-maximizing behavior changes the point of production even in the univariate case. We know that the decision maker chooses to produce where the marginal value product equals the price of the input. Thus, if either the price of the input or the price of the output has changed, the ratio of outputs to inputs will change. Even in the single variable case, we would wonder about excluded factors, things beyond the decision makers control. Charles B. Moss Measuring Changes in Productiity: XXII Defining Changes in Productivity Measuring Technological Change Total Factor Productivity and Index Number Theory Distance Functions Extending the analysis to the multivariate world, we begin by examining the product-product relationship in the first graph. Note θ (t ) = ￿ ￿ Y ￿ (x ) p 1 y1 + p 2 y 2 = Y (x ) p 1 y1 + p 2 y 2 (2) could be used as one measure of technical change. Similarly, in the input-input relationship: θ (t ) = ￿ ￿ V ￿ (y ) w 1 x1 + w 2 x2 = V (y ) w 1 x1 + w 2 x2 Charles B. Moss Measuring Changes in Productiity: XXII (3) Defining Changes in Productivity Measuring Technological Change Total Factor Productivity and Index Number Theory Distance Functions Note first that each of these formulations is based implicitly on Shephard duality: θ (t ) = = Y ￿ (x ) π (p , w , t = 1r ) = Y (x ) π (p , w , t = 0) ￿ ￿ p1 y1 (p , w , t = 1) + p2 y2 (p , w , t = 1) p2 y1 (p , w , t = 0) + p2 y2 (p , w , t = 0) V ￿ (y ) c (w , y , t = 1) θ (t ) = = V (y ) c (w , y , t = 0) = ￿ ￿ w1 x1 (w , y , t = 1) + w2 x2 (w , y , t = 1) w1 x1 (w , y , t = 0) + w2 x2 (w , y , t = 0) Charles B. Moss Measuring Changes in Productiity: XXII (4) Defining Changes in Productivity Measuring Technological Change Total Factor Productivity and Index Number Theory Distance Functions More formally, ￿ ￿ c (w , y , t ) = min w ￿ x : x ∈ V (y , t ) x >0 ￿ ⇔ V (y , t ) = x : w x ≥ c (w , y , t ) , w > 0 ∗ ￿ (5) ￿ Thus, by gross simplification, we could envision a cost function 1 c (w , y , t ) = α0 + α￿ w + w ￿ Aw + β ￿ + 2 1￿ y By + w ￿ Γy + θ (w , y , t ) 2 x (w , y , t ) = α + AW + Γy + ∇w θ (w , y , t ) with θ (w , y , t ) being a measure of technical change. Charles B. Moss Measuring Changes in Productiity: XXII (6) Defining Changes in Productivity Measuring Technological Change Total Factor Productivity and Index Number Theory Distance Functions This formulation allows us to discuss several key features of technology measurement. First, in the grossest sense, technological change tends to be a measurement of factors that we don’t understand. From the preceding equation, what is the difference between technology and a residual? One approach is to proxy technical change with a simple time trend, t . Alternatively, several studies have used other proxy variables such as spending on agricultural research. This formulation allows the researcher to examine the neutrality of technical change ∂ c (w , y , t ) xi ( w , y , t ) xi ( w , y ) ∂ wi = = ∂ c (w , y , t ) xj ( w , y , t ) xj ( w , y ) ∂ wj Charles B. Moss Measuring Changes in Productiity: XXII (7) Defining Changes in Productivity Measuring Technological Change Total Factor Productivity and Index Number Theory Distance Functions Finally, it is possible to envision adjusting this measure for differences in input quality. For example, if the quality of one variable increases over time, then we could adjust the price of that variable upward to account for the increase in quality. Charles B. Moss Measuring Changes in Productiity: XXII Defining Changes in Productivity Measuring Technological Change Total Factor Productivity and Index Number Theory Distance Functions Total Factor Productivity and Index Number Theory The index number approach can be looked at as an extension of the single input-single product scenario above dy ∂ f (x , t ) dx ∂ f (x , t ) = + dt ∂x dt ∂t Replacing differentiation with log differences y = f (x , t ) ⇒ d ln [y ] ￿ ∂ ln [y ] d ln [xi ] = + T (x , t ) dt ∂ ln [xi ] dt i ￿ d ln [xi ] = ￿i + T (x , t ) dt i ￿ wi xi d ln [xi ] = + T (x , t ) dt p￿y i d ln [y ] ￿ wi xi d ln [xi ] ⇒ T (x , t ) = − dt dt p￿y i Charles B. Moss Measuring Changes in Productiity: XXII (8) (9) Defining Changes in Productivity Measuring Technological Change Total Factor Productivity and Index Number Theory Distance Functions This formulation is sometimes approximated as the Tornqvist-Theil measure T (x , t ) = ln [yt ]−ln [yt −1 ]− N ￿1 i =1 2 [Vit + Vi ,t −1 ] [ln [xit ] − ln [xi ,t −1 ]] (10) Working backward d ln [y ] = d ln [x ] + T (x , t ) ⇒ d ln [y ] − d ln [x ] = T (x , t ) ⇒ d ln ￿y ￿ ￿ x Qy ⇒ d ln Qx Charles B. Moss = T (x , t ) ￿ = T (x , t ) Measuring Changes in Productiity: XXII (11) Defining Changes in Productivity Measuring Technological Change Total Factor Productivity and Index Number Theory Distance Functions In the Tornqvist-Theil index, the indices were implicitly Divisia output and input indices Qy = ￿ i ￿ w i xi p i yi ￿ ￿ , Qx = p j yj w j xj i j (12) j The linkage in this case is the definition of total factor productivity TFP = y y ˙ ˙ ⇒ TFP = = x x ˙ Charles B. Moss ￿ ∂ f (y , x , t ) dyi ∂ yi dt i ∂ f (y , x , t ) dxi ∂ xi dt Measuring Changes in Productiity: XXII (13) Defining Changes in Productivity Measuring Technological Change Total Factor Productivity and Index Number Theory Distance Functions Distance Functions x1 x1 x1 x2 x2 Charles B. Moss x2 Measuring Changes in Productiity: XXII Defining Changes in Productivity Measuring Technological Change Total Factor Productivity and Index Number Theory Distance Functions These concepts actually define a distance function measure of productivity growth. Define a measure θ for a new technology based on the old technology as Dθ (x , y ) = min [θ : F (x θ) ≥ y ] θ Charles B. Moss Measuring Changes in Productiity: XXII (14) Defining Changes in Productivity Measuring Technological Change Total Factor Productivity and Index Number Theory Distance Functions x1 x1 x1 x2 x2 Charles B. Moss x2 Measuring Changes in Productiity: XXII ...
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This note was uploaded on 02/01/2012 for the course AEB 6184 taught by Professor Staff during the Fall '09 term at University of Florida.

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