哈佛大学：《中级微观经济学》（英文版）Lecture10 More on production

Economics 2010a Fa2003 Edward L. Glaeser Lecture 10

Economics 2010a Fall 2003 Edward L. Glaeser Lecture 10

10. More on production a. Derived demand-Marshall's Laws b. Long Run/Short Run, LeChatelier, Dynamics C. Aggregating Supply d. Theory of the Firm, the Holdup problem Agency Issues f. Application The Coase Theorem

10. More on Production a. Derived Demand—Marshall’s Laws b. Long Run/Short Run, LeChatelier, Dynamics c. Aggregating Supply d. Theory of the Firm, the Holdup Problem e. Agency Issues f. Application: The Coase Theorem

Marshall-Hicks laws of derived demand (1) The demand for a good is more elastic the more readily substitutes can be obtained (2) The more important the good, the more elastic the derived demand(Hicks addition- if substitutes are readily available) 3)The demand for an input is higher the more elastic is the supply of other inputs (4)The more elastic the demand for the final good-the more elastic is the demand for the input

Marshall-Hicks laws of Derived Demand (1) The demand for a good is more elastic the more readily substitutes can be obtained. (2) The more important the good, the more elastic the derived demand (Hicks’ addition– if substitutes are readily available). (3) The demand for an input is higher, the more elastic is the supply of other inputs. (4) The more elastic the demand for the final good– the more elastic is the demand for the input

All of these statements are supposed to be about Some of these comparative statics we know what to do with: (1)and(2). To get (3 and(4), we need some new ingredients (1)The demand for a good is more elastic the more readily substitutes can be obtained In the limit, this is obvious-if there exists a perfect substitute, then the derived demand elasticity is infinite Start with the Foc for an input j Differentiation gives us:花角=?>两m

All of these statements are supposed to be about Wj Zj /Zj /Wj Some of these comparative statics we know what to do with: (1) and (2). To get (3) and (4), we need some new ingredients. (1) The demand for a good is more elastic the more readily substitutes can be obtained. In the limit, this is obvious– if there exists a perfect substitute, then the derived demand elasticity is infinite. Start with the FOC for an input j P /f /Zj = Wj. Differentiation gives us: /f /Zj /Zj /Wj = 1 P ? >i®j /f /Zi /Zi /Wj

Or using the first order condition and manipulating W/Z 2/W Unrigorously-just looking at the equation gives you the unimportance result (i.e, when z is small you expect this expression to be bigger) and the substitutes result (when the other goods are able to adjust a lot- you expect a bigger demand elasticity)

Or using the first order condition and manipulating: Wj Zj /Zj /Wj = 1 Zj ? > i®j WiZi WjZj Wj Zi /Zi /Wj Unrigorously– just looking at the equation gives you the unimportance result (i.e., when z is small you expect this expression to be bigger) and the substitutes result (when the other goods are able to adjust a lot– you expect a bigger demand elasticity)

et's make this rigorous with two good k and l, where we look at labor demand, the price of labor is w, the price of capital is R Just plugging into the formula gives us 斧=?爷斧 is more useful to also use the two first order conditions and note that 作+B and PkK /K=0. /W Manipulating these equations yields PfulfKK? Ph o

Let’s make this rigorous with two good K and L, where we look at labor demand, the price of labor is W, the price of capital is R. Just plugging into the formula gives us: W L /L /W = 1 L ? RK WL W K /K /W It is more useful to also use the two first order conditions and note that fLL /L /W + PfKL /K /W = 1 and PfKL /L /W + PfKK /K /W = 0. Manipulating these equations yields: /L /W = fKK PfLLfKK ? PfKL 2 < 0

and /K PfiLfkx pfx KL

and /K /W = fKL PfLLfKK ? PfKL 2

This still isn,'t all that helpful- let's try a separable production function YK, Lp=ak+ bLk 里=" PALK? L PbKYK? 1pLK?2 LPbKYK? 1PL K?2 K? Well-that isnt all that interesting. It certainly tells us that the unimportance result can be general(Hicks point)

This still isn’t all that helpful– let’s try a separable production function: fÝK,LÞ = aKJ + bLK W L /L /W = W L 1 PbKÝK ? 1ÞL K?2 = PbKL K?1 LPbKÝK ? 1ÞL K?2 = 1 K ? 1 Well– that isn’t all that interesting. It certainly tells us that the unimportance result can be general (Hicks’ point)

How about Cobb-Douglas: N\K, LP=KLK In the Crs case, you cant solve for scale, only for factor proportions Use cost minimization to solve min RK+WL+VKL??0 This gives uS: JRK= Y1? JpWL or, using the g constraint, K=O W1?p11?J RJ And L=Q WY1?Jp )0r"h=?J Thats a little bit better - in this case the elasticity of demand for labor is equal to one minus labors share in the production function

How about Cobb-Douglas: fÝK,LÞ = KJL K In the CRS case, you can’t solve for scale, only for factor proportions. Use cost minimization to solve: min K,L,V RK + WL + V KJL 1?J ? Q This gives us: JRK = Ý1 ? JÞWL or, using the Q constraint, K = Q WÝ1?JÞ RJ 1?J And L = Q RJ WÝ1?JÞ J or W L /L /W = ?J That’s a little bit better– in this case, the elasticity of demand for labor is equal to one minus labor’s share in the production function

Of course, we can't say anything about the degree of substitutability

Of course, we can’t say anything about the degree of substitutability