Lecture 7 - Static Labor Demand
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Lecture 7 - Static Labor Demand References: Daniel McFadden "Duality of Production, Cost and Profit Functions" (avail- able on course web site). A very comprehensive presentation of the basic results of production theory. David Card and Thomas Lemieux. "Can Falling Supply Explain the Rising Returns to College for Young Men? A Cohort-Based Analysis." QJE 116 (2001): 705-746 Gianmarco Ottaviano and Giovanni Peri "Rethinking the Effects of Immigra- tion on Wages”. Journal of the European Economic Association, 2011. available at http://www.econ.ucdavis.edu/faculty/gperi/Publications.htm Peter Kuhn and Ian Wooten. "Immigration, International Trade, and the Wages of Native Workers. In John Abowd and Richard Freeman, editors. Im- migration Trade and Labor. University of Chicago Press for NBER, 1991. Some additional background reading: Lawrence Katz and Kevin Murphy. "Changes in Relative Wages, 1963-1987: Supply and Demand Factors," QJE 107 (1992): 35-78. Claudia Goldin and Lawrence Katz. The Race Between Education and Technology. Harvard University Press, 2008. Richard Anderson and John Moroney. "Substitution and Complementarity in CES Models". Southern Economic Journal 60 (April 1994): 886-895. David Card. "Is the New Immigration Really So Bad?" Economic Journal 115 (2005): F300-F323. David Card and Ethan Lewis. "The Diffusion of Mexican Immigrants During the 1990s: Explanations and Impacts." In G. Borjas (editor), Mexican Immi- gration to the United States, Univ. of Chicago Pressm 2007. George Johnson and Frank Stafford. "The Labor Market Implications of International Trade". In Ashenfelter and Card, Handbook of Labor Economics volume 3. (available online). The building block of labor demand is a production function f(x1, x2, ...xn) where xj is the input of "factor" j (say hours of work by some skill group). We usually assume that f has constant returns to scale (CRS). This means that in a competitive industry the scale of individual firms is undefined - firms per se are unimportant. What we can measure and analyze is industry-wide demand. As you may recall in the two-input case with CRS the shape of isoquants is summarized by the elasticity of substitution. Technically this is defined as d log( x1 ) = x2 d log( f1 ) f2 1 f1 where f(x1, x2) = y (i.e., the derivative is along an isoquant). Since is the f2 slope of the isoquant, this is the proportional change of the relative use of the two factors x1 and x2 per percent change in the slope of the isoquant (which, under cost-minimization, would be their relative factor prices). It can be shown that f f = 1 2 . f f12 A simple version of the proof follows. To begin note that with CRS, we have that f1 and f2 are both HD0. So x1f11 + x2f12 = 0 x1f21 + x2f22 = 0. Also, f = x1f1 + x2f2. Now define s as the slope of the isoquant at a point: f (x , x ) s = 1 1 2 f2(x1, x2) log s = log f1(x1, x2) log f2(x1, x2) f11 f21 f12 f22 d log s = [ ]dx1 + [ ]dx2 f1 f2 f1 f2 x2 f21 f21 f12 x1 f12 = [ ]dx1 + [ + ]dx2 x1 f1 f2 f1 x2 f2 f2f21x2 f21f1x1 f2f12x2 + f12f1x1 = [ ]dx1 + [ ]dx2 x1f1f2 x1f1f2 f12 dx1 f12 dx2 = [f1x1 + f2x2] + [f1x1 + f2x2] f1f2 x1 f1f2 x2 f12f dx1 dx2 f12f x1 1 x1 = [ ] = d log[ ] = d log[ ]. f1f2 x1 x2 f1f2 x2 x2 Thus d log[ x1 ] x2 = . d log s The classic examples are Cobb-Douglas ( = 1) and the "CES" 1/ρ f(x1, x2) = ( x + (1 )x ) 1 2 which has = 1/(1 + ). It is a lot easier in most cases to work with the cost function C(w1, w2, y). With CRS this has the form C(w1, w2, y) = (w1, w2)y 2 where () is the "unit cost" function. We will show that: C C = 1 2 . CC12 To do so, start with Sheppard’sLemma x1 = C1(w1, w2, y) x2 = C2(w1, w2, y) Now Cj is HD0 in input prices, so we can write: w1 w1 x1 = C1( , 1, y) = g( , y) w2 w2 w1 w1 x2 = C2( , 1, y) = h( , y) w2 w2 Also @x1 1 w1 @x2 1 w1 C11 = = g1( , y),C21 = = h1( , y). @w1 w2 w2 @w1 w2 w2 Now fixing y: x w w log 1 = log g( 1 , y) log h( 1 , y) x2 w2 w2 so w1 d log x1 w2 x2 w g h = = 1 1 1 d w1 w2 g h w2 w w C C w = 1 2 11 21 2 w g k 2 w C w C = 1 11 + 1 21 C1 C2 Now w1C11 + w2C12 = 0 since C1 is HD0. Substituting we get w C w C C (w C + w C ) C C = 1 21 + 2 12 = 12 1 1 2 2 = 12 C2 C1 C1C2 C1C2 using the fact that C is HD1 in input prices. Another useful fact: @x1 = C12 @w2 w @x w C C C w C C 2 1 = 2 12 = 12 2 2 1 ) x1 @w2 x1 C1C2 Cx1 w x = 2 2 = s (1) C 2 3 here s2 = input 20s cost share. Also @x1 @x1 w1 + w2 = 0 @w1 @w2 @x1 w1 = s2 = (1 s1) ) @w1 x1 This says that in the 2-input case the output-constant elasticity of demand for an input is the product of (1 s) and . Finally, with more than 2 inputs we define the partial elasticity of substitu- tion CijC ij = . CiCj Note that @xi wj wj CijC wjCiCj xi = Ci(w, y) ij = Cij = = ijsj ) @wj xi xi CiCj xiC which is a multi-factor generalization of (1). Marshall’sRules Let’sconsider a competitive industry with CRS and a cost function C(w, y) = (w)y. Industry output is priced at p = (w), and there is a downward slop- ing demand curve for the industry’s output y = D(p). We are going to show the "classic" connection between the elasticity of demand for an input by the industry and three key parameters: the elasticity of product demand, ij the partial elasticities of substitution, and sj the cost shares. We start with xi = yi(w) log xi = log y + log (w) ) i @ log x @ log y d log p (w) i = + ij ) @wj @ log p @wj i(w) d log (w) (w) = + ij @wj i(w) (w) yCjC = j + ij (w) iyCjC Therefore @ log xi wjj(w)y wjCijCjC wj = + @wj (w)y CiCjC wjCj wjCj = + ij C C = sj + ijsj = ij sj 4 Now let’sconsider the own-price effect: @ log xi = ii si. @ log wi Finally, consider the 2-input case, so ii = (1 si). Then @ log xi = (si + (1 si)) @ log wi which says that in the 2-input case the own-price demand elasticity is a combi- nation of the final product demand elasticity (a "scale" effect) and the elasticity of substitution (a "substitution" effect). To understand the scale effect, note that when p = MC = (w) @ log p (w) = j @wj (w) @ log p wjj(w)y wjxj wj = = = sj. @wj (w)y C Thus when wj rises by 1%, industry selling price rises by sj%, and this chokes off demand by sj%. The "standard model" of the demand side in labor economics is one in which all firms have CRS and pay the same prices for all factors. In this model firms per se do not matter: in fact the number and size of firms is indeterminant. In trade theory and IO there is considerable interest in models with a lot of heterogeneity across firms (e.g., Melitz, 2003). In these models different firms have different levels of productivity. Less productive firms survive because they produce differentiated products which consumers are willing to buy (though less productive firms are smaller). On the labor side, all firms pay the same wages, so the heterogeneity in firms does not matter directly. An important and growing area of work in labor economics focuses on the impact of firms. The starting point for this work is the recognition that "who you work for matters". See the presentation for the "Vancouver School of Economics" in Sept. 2013 (on the class web site) that tries to summarize some of the older ideas and new thrusts in this area. Some Functional Forms Cobb Douglas —often used for modeling labor and capital in contexts where the assumption that = KL = 1 is not too crazy: 1 y = f(K, L) = AL K Note that labor’sshare is which is constant. This used to be approximately correct. However, over the 2000’s, labor’s share has fallen from the "histori- cal" value of about 65% to about 58%. See Fleck, Glaser and Sprague, "The 5 Compensation-Productivity Gap": A Visual Essay" Monthly Labor Review Jan- uary 2011 (Figure 5 is attached). Note that in any CRS 2-input production function f(x1, x2) = x1f(1, x2/x1) = x1g(x2/x1) f(x1, x2)x1 = g(x2/x1). ) In the C-D case we get 1 y/L = A(K/L) which says that with fixed A "labor productivity" (y/L, the average product of labor) is a concave function of K/L. With C-D we get @y = A(K/L)1 @L so marginal product is just a constant times average product. These equations are widely used as a starting point for understanding productivity trends. CES - often used to study high/low skill labor: 1/ρ f(x1, x2) = ( x + (1 )x ) . 1 2 Many textbooks (e.g. Silberberg) have a section on the derivation of the CES from the differential equation d log( x1 ) x2 = , a constant d log( f1 ) f2 a problem that was solved by Arrow et al (Re Stat, 1961).