LABOR ECONOMICS

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(1)LABOR ECONOMICS Sponsored by a Grant TÁMOP-4.1.2-08/2/A/KMR-2009-0041 Course Material Developed by Department of Economics, Faculty of Social Sciences, Eötvös Loránd University Budapest (ELTE) Department of Economics, Eötvös Loránd University Budapest Institute of Economics, Hungarian Academy of Sciences Balassi Kiadó, Budapest. Author: János Köllő Supervised by: János Köllő January 2011.

(2) LABOR ECONOMICS Week 9 Labor demand – Topics János Köllő • •. Two factors: capital and labor More than two factors. Appendix 1: Demand for labor in the short run Appendix 2: Scale effect with homogeneous production function. •. The slides draw from P. Cahuc–A. Zylberberg „Labor Economics” (MIT Press 2004), pp. 171–193. •. We restrict ourselves to the case of homogeneous production functions. Results for the general case are mentioned without proof.. •. We study the two factors case in detail. For the single factor case see Appendix 1. The multifactor case is discussed briefly.. 2.

(3) Two factors: capital and labor Preparations: the production function Bivariate, homogeneous of degree : Returns to scale diminish if <1, constant if. F ( K , L). =1, increasing if. F ( K , L). 0,. >1.. ( K , L). Returns to factors of production are diminishing: FL>0, FK>0, FLL<0, FKK<0. Preparations: cost function The minimum cost at given output Y is written as The cost function describes minimum cost as a function of w, r and Y:. wL rK a) C is homogeneous of degree 1 in prices.. C. C ( w, r , Y ). b) C is concave: Cww<0, Crr<0. c) Satisfies Shepard’s lemma, i.e. optimal factor demands are given by the partial derivatives of the cost function:. L. C w ( w, r , Y ) és. K. C r ( w, r , Y ). 3.

(4) Preparations: market power Perfect competition is an extreme case. Market power is measured with the non-positive elasticity of sales price wrt sales [drawn from the inverse demand function P=P(Y)] p Y. Under perfect competition the firm is price-taker Under imperfect competition the price varies with Y Market power is measured with defined as:. 1 1. p Y. p Y p Y. 0 0. P (Y ) Y P (Y ). 1. Under perfect competition the firm has no market power Under imperfect competition the firm has market power. 1 1. Preparations: the profit function When prices change, the optimal size of the firm changes, too. The profit function is written as:. ( w, r , Y ). P (Y )Y. C ( w, r , Y ). The FOC from differentiation by Y is given by: Note that in the optimum : price=marginal cost*markup P(Y ) CY ( w, r , Y ) ahol 1 /(1 Yp ) Y. ( w, r , Y ). P (Y )Y. P(Y ) CY ( w, r , Y ) 0. By multiplying out P(Y) and recalling that the output elasticity of the inverse demand function is ηYp. P'(Y)Y/P(Y) you immediately get to the above formula. 4.

(5) After these preparation we look at the following questions Conditional demand (Optimal choice at given level of output in response to changing factor prices) • •. Conditional demand for labor and capital Cross effects. Scale effects (Optimal output in response to changing factor prices) Unconditional demand • The substitution and scale effects together Conditional (compensated) demand for labor diminishes with the wage and increases with the user cost of capital From Shepard’s lemma:. L. L w. C w ( w, r , Y ). C ww. 0. Since compensated demand depends only on relative factor prices, the demand for labor increases if the user cost of capital goes up:. L r. C wr. 0. Similarly, the conditional demand for capital decreases with the price of capital and increases with the wage.. 5.

(6) Cross effects Cross elasticity: positive, non-symmetric Elasticity of substitution: positive and symmetric* L r. r L , L r. K w. w K K w. L r. ,. K w. *) This is not necessarily true for more than two factors of production. See later.. w r (K / L ) K L (w / r ). 0. Cross effects derived from the cost function The elasticity of substitution can be derived from the cost function (Uzawa 1962, Cahuc-Zylberberg 2004, 237–238). w r (K / L ) K L (w / r ). C wr C C wCr. L. It follows that sK (1 s L ) r sK is the share of capital in total cost (sK = rK/C = 1–sL = 1–wL/C). log L L r r Cwr Y Y0 Y Y0 log r r L L Considering the above formula it is straightforward to see that : r C Cwr C Cwr C L 1 Cwr r sK L rK L K C w C r L. r. 6.

(7) The compensated price effects thus can be written as:. wL. rL L r. (1 s L ). We have seen that Substitution yields :. L w. 0 r L L r. L r L r. r Cwr and L. L r. (1 s L ). C Cwr that is Cwr C wCr. 0 C wCr C. (rCwCr / L C ). Define the share of capital as s K. 1 sL. According to Shepard' s lemma : Cw. L and Cr. rL K LC Thanks to symmetry, the derivation of It immediately follows that. rK / C. L r. sK L w. K (1 sL ). proceeds similarly. The effect of an exogeneous change of output Under homogeneous production function If the production function is homogeneous, a rise in Y (without a change in relative factor prices) increases the demand for both capital and labor. From homogeneity of degree and Shepard’s lemma it follows that:. L (w / r, Y ). L ( w r ,1)Y 1/ és K ( w r , Y ). K ( w r ,1)Y 1/. Remark: in the general case the demand for at least one factor will rise.. 7.

(8) Unconditional demand When factor prices change, optimal output (Y*) will change, too. The firm’s problem is to solve: How Y*, C and (as a consequence) L* will change in response to a change in w?. ( w, r ). max. ( w, r , Y ). Y. p Y ). w ( w, r ) [ P(Y )(1. CY ( w, r , Y )]. Y w. Cw ( w, r , Y ). Under optimality the term in brackets is zero (see Profit function). On the other hand, Shepard’s lemma states that Cw(w,r,Y*)= L*. So we arrive at Hotelling’s lemma: Unconditional demands are decreasing in own prices*: w ( w, r ). L w. L. és 0. ww. r ( w, r ). and. K r. K rr. 0. *) From the concavity of the cost function it follows that the profit function is convex, so the second derivatives are positive. Unconditional demand – The effects of a change in the wage The wage has a direct and an indirect effect: Multiply throughout with w/L*, and the second term with Y*/Y*. L w. L w w L. Cww CwY. w L. C ww. Y w. w L. C wY. Y Y w Y. L w. w L. C ww. Y C wY L. Y w. 8.

(9) What is this? The first term is the conditional demand elasticity at output level Y=Y*. w L. w. Cww. L w. L. L /L w/ w. L w. The wage has a direct and an indirect effect:. L w. Cww CwY. Y w. Multiply with w/L*, and the second term with Y*/Y*. L w w L. w L. C ww. w L. C wY. Y Y w Y. L w. w L. C ww. Y C wY L. Y w. What is this? The first component of the second term is the output elasticity of demand at output level Y=Y* holding relative factor prices constant:. Y C wY. Y. L. L. L Y. L /L Y /Y. L Y. The wage has a direct and an indirect effect: Multiply with w/L*, and the second term with Y*/Y*. L w. Cww CwY. Y w. From step 1 and step 2 we finally have:. L w w L. w L. C ww. w L. C wY L w. Y Y w Y L w. L w. w L. C ww. Y C wY L. Y w. L Y Y w 9.

(10) The total effect of a change in the wage is thus L w. L w. Own-wage elasticity. L Y Y w Scale effect (–). Compensated elasticity of substitution (–). The negativity of the scale effect is easy to prove if the production function is homogeneous. See two slides later! Remark: in the general case it can be proven that the two components of the second term are differently signed (Cahuc–Zylberberg p. 184. and footnote 5 to Chapter 4). Employment effect of a change in the user cost of capital After similar steps we have:. L r. L r. L Y Y r. Compensated elasticity of substitution (+). Cross price elasticity (?). Scale effect (–). L r. 0. K and L are gross complements. L r. 0. K and L are gross substitutes. 10.

(11) Own-wage elasticity under homogeneous production function L w L w. L w. L Y Y w. = - (1 - s L )s < 0. Y w. 1 q. (a). s (c). (b) (a) We have seen that sL. L r. r L L r. wL / C. Shepard' s lemma : Cw. (b) L ( w, r , Y ). L (w, r ,1)Y 1. C Cwr C wC r. r Cwr and L L és Cr. L r. (rCwCr / L C ) .. K . Substitution yields :. ln L (w,r,Y) ln L (w,r,1 ) 1 θ ln Y. L r. (1 sL ). d ln L d ln Y. 1 θ. (c) For the derivation see Appendix 2. In these formulas it is easy to observe that*: L w. (1 sL ). sL. or. L w. sL. a). The demand for labor decreases with the wage.. b). The substitution and scale effects are additive.. c). Demand is more elastic if capital and labor are ‘easy to substitute’ ( is large) HM– 2.. d). The stronger is market power, the weaker the scale effect. If competition is strong ( ) the scale effect is large and the demand for labor is highly elastic HM–1.. 11.

(12) e). The elasticity of demand for labor increases with labor’s share in total cost provided that < /( - ). The validity of HM–4 depends on how the scope for capital-labor substitution relates to the elasticity of product demand.. More than two factors of production (different types of capital and labor, land, raw materials, etc.) The firm’s problem is to solve: The FOC is essentially identical to that discussed in the two-factors case: n. wi X i. min 1 n. X ... X. k . f . F ( X 1 ,..., X n ) Y. i 1. 1. n. F ( X ,..., X ) Y. és. Fi ( X 1 ,..., X n ) F j ( X 1 ,..., X n ). wi wj. i, j 1,..., n. The cost function is first order homogeneous in w and homogeneous of degree 1/ in Y if F(.) is homogeneous of degree . It is concave and satisfies Shepard’s lemma:. C ( w 1 ,..., w n , Y ) Xi. Ci ( w 1 ,..., w n , Y ). The demand for factors of production diminishes with own prices:. X. i. 1. n. Ci ( w ,..., w , Y ). Xi w. i. Cii. 0. i 1,..., n. But the demand for a given factor does not necessarily increase if the price of another factor goes up. If the price of factor j goes up, its employment will fall and the employment of at least one other factor will rise. However, we cannot predict how the demand for factor i will change (without knowing the technology). X i a particular Xj Cij i, j 1,...,n j i w w. Xi wj Xi wj. 0  i and j are p-subsitutes (Hicks–Allen substitutes) 0.  i és j are p-complements (Hicks–Allen complements). 12.

(13) Cross effects Cross elasticity. Ambigously signed, non-symmetric Direct elasticity of substitution. Defined as in the two-factors case:. Xi wj. i j. wj. Cij. i j. wj Xi Xi ( X i / X j ) ( w j / wi ). d ij. j i. ( w j / wi ) ( X i / X j ). Difficult to interpret: a change in the price of j starts a chain of substitutions so the demand for i will change for several reasons. Allen’s partial elasticity of substitution (derivable from the cost function) tells more: i j. i j. Cij C. C j. w X. j. Ci C j. i j. i js j. The formula known from the two-factors case i i j js j continues to hold but is ambigously signed (unlike in the two-factors case). Corollary: changes in the price of capital, materials and land may affect the demand for different types of labor in different ways. Unskilled labor and capital are usually found to be substitutes, for instance, while skilled labor and capital are complements according to several estimates (capital-skill complementarity). It also continues to hold that i i i Y i, j 1,..., n j j Y i Therefore the sign of the uncompensated elasticity remains an empirical question : i j. 0. i and j are gross substitutes. i j. 0. i and j are gross complements. If the production function is homogeneous and the market is not fully competitive then: i j. sj. i j. i, j. 13.

(14) If the term in the bracket is positive, a rise in the price of j will increase the demand for i. If it is negative, the demand for both factors will fall.. Appendix 1: Demand for labor in the short run Demand for labor in the short run Labor is the only factor of production. The production function can be written as: Y=F(L), Y’>0, Y’’<0 The firm may have market power. Market power is measured by the elasticity, where P(Y) is an isoelastic inverse demand function:. p Y. P (Y ) Y P (Y ). Under perfect competition the firm is price-taker . p Y. Under imperfect competition the price is affected by Y . 0 p Y. 0. 14.

(15) Prices. Effect of the wage on the demand for labor. 15.

(16) Appendix 2: The scale effect under homogeneous production function (proof). 16.

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