4. Marshall's Rules of Derived Demand (Chapter 3) We will now prove the first three of Marshall's rules of derived demand and, in doing so, also derive a Slutsky-type equation that decomposes the industry-level elasticity of demand into scale and substitution effects. The proof of Marshall's fourth rule is much messier, and little is learned from the added complexity. Labor economists often assume a specific functional form for the production function. A common assumption in modern labor economics is that the industry can be charac- terized in terms of a constant elasticity of substitution (CES) production function. This industry-level production function is given by: Q = [ak + (1 -0)1/6 (A-23) As an exercise, it is worth showing that the CES production function has constant returns to scale (that is, a doubling of all inputs doubles output). The CES functional form is useful because it allows for a wide array of possibilities that describe the extent of substitution between labor and capital. The parameter o is less than or equal to one (and can be negative). If 8 = 1, it is easy to see that the CES production function is linear, and that is the case where labor and capital are perfectly substitutable (so that the isoquants are straight lines). It can be shown that if o goes to minus infinity, the isoquants associated with the CES production function become right-angled isoquants, so that there is no substitution possible between labor and capital. The elasticity of substitut tion between labor and capital is defined by o = 1/(1 -6). Note that if 8 = 1, the elasticity of substitution goes to infinity (perfect substitution), and if 8 = -co, the elasticity of sub- stitution goes to zero (perfect complements). If the industry is competitive, the price of labor and capital must equal the respective values of marginal product. It is easy to verify that these conditions can be written as: r= pa Q'- K -1 w = p(1 - a)(1-5 ES-1 (A-24) As an exercise, it is instructive to derive: SK rK _ aK pQ SE = WE _ (1 - Q) ES PQ Qo (A-25) goes to labor. where sx gives the share of industry income that goes to capital and s gives the share that By totally differentiating the production function in equation (A-23) and rearranging terms, it follows that: d log E = d log Q - Sk(d log K - d log E) (A-26) Changes in the scale of the industry (d log Q) depend on the demand for the industry's output. Define the absolute value of the elasticity of demand for the output as: n = |d log Q d log p (A-27)Note that although the demand curve for the output is downward sloping, the elasticity n arro vanidabbs is defined to be a positive number. Equation (A-26) can then be rewritten as: dlog E= -n d log p - Sk(d log K - d log E) (A-28) We now need to find out by how much the price of the output changes when the wage changes (note that we are holding r constant throughout the exercise). In a competitive industry, the output price must equal the marginal cost, which must equal the average cost (there are zero profits). We can write the zero-profit condition as: park + WE (A-29) Note that equation (A-23) implies that d log Q = Sk d log K + SEd log E. By totally dif- ferentiating equation (A-29) and rearranging terms, we can derive that: d log p = SEd log w (A-30) Finally, the ratio of first-order conditions in equation (A-24) implies that: W_ (1 - 0) E5-1 (A-31) Totally differentiating equation (A-31) implies that the (percent) change in the capital/ labor ratio is: di nwond d log K - dlog E = (1 - 8) d log w = od log w unt nordx (A-32) Substituting equations (A-30) and (A-32) into equation (A-28) yields: d log E d log w = -[sEn + (1 -s)] (A-33) The elasticity of demand for labor can be written as a weighted average of the elasticity of product demand and the elasticity of substitution between capital and labor. The first term of equation (A-33) gives the scale effect that depends on the elasticity of demand for the industry's output, while the second term gives the substitution effect that depends on how easily substitutable labor and capital are along a single isoquant. The first three of Marshall's rules of derived demand state that: 1. The labor demand curve is more elastic the greater the elasticity of substitution. 2. The labor demand curve is more elastic the greater the elasticity of demand for the output. 3. The labor demand curve is more elastic the greater labor's share in total costs (but this holds only when the absolute value of the elasticity of product demand exceeds the elasticity of substitution). As an exercise, it is worth verifying these rules directly from equation (A-33)