浙江大学 课件 厂商理论（2）_高级微观经济学第七章

Lecture 7

1 Lecture 7

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 Wi 8Z Z,aw, 2

2 • 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 j j j j W Z Z W  

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 pO-W oZ, Differentiation gives us:of oz1 ofaz OZ,oW P Z,aW

3 • 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: j j f P W Z  =  j j 1 i j j j j j f f Z Z Z W P Z W      = −     

Or using the first order condition and apig意-！之影g 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). 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

4 • Or using the first order condition and manipulating: • 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). • 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. j j 1 i j j j j j f f Z Z Z W P Z W      = −     

Just plugging into the formula gives us: W aL 1 RK W OK L aw L WL K aW It is more useful to also use the two first order conditions and note that +% aL and fu aL K=0. P听aW+Pism Manipulating these equations yields: aL <0 ow Pfufkx-Pf足 and 6K f aw Pfufkx-Pf是 5

5 • Just plugging into the formula gives us: • It is more useful to also use the two first order conditions and note that and • Manipulating these equations yields: and W L RK W K 1 L W L WL K W   = −   1 0. LL KL KL KK L K f Pf W W L K Pf Pf W W   + =     + =   2 2 0 KK LL KK KL KL LL KK KL L f W Pf f Pf K f W Pf f Pf  =   −  =  −

This still isn't all that helpful-let's try a separable production function:f(K,L)=aka+bL W oL w 1 PbBL- L owL PbB(B-1)1-3-LPbB(B-1)15B-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:f(K,L)=K“L In the CRS case,you can't solve for scale, only for factor proportions

6 • This still isn’t all that helpful– let’s try a separable production function: • Well– that isn’t all that interesting. It certainly tells us that the unimportance result can be general (Hicks’ point). • How about Cobb-Douglas: • In the CRS case, you can’t solve for scale, only for factor proportions. 1 2 2 1 1 ( 1) ( 1) 1 W L W Pb L L W L Pb L LPb L          − − −  = = =  − − − f K L aK bL ( , )   = + f K L K L ( , )   =

Use cost minimization to solve: min RK+WL+(KD-@-) K,L,2 This gives us:aRK =(1-a)WL,using the Q constraint, K-e0- Ra ·And L- W aL or Law 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

7 • Use cost minimization to solve: • This gives us: , using the Q constraint, • And or • 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. 1 , , min ( ) K L RK WL K L Q     − + + −   RK WL = − (1 ) 1 (1 ) ( ) W K Q R    − − = ( ) (1 ) R L Q W    = − W L L W   = − 

How about the more interesting CES production function f(K,L)=(aKo+bL)° Again,as long as this is CRS,we can only do cost minimization not profit maximization. .This gives us R=Aak-(ak+bL) W=AbL-(ak+borK bR from which W、 L=(al-obi-G(")-0+b)2 R 1-0 k=(a+bl-oal-o()1o) 8

8 • How about the more interesting CES production function • Again, as long as this is CRS, we can only do cost minimization not profit maximization. • This gives us or from which 1 1 1 W bL aK bL ( )      − − = + 1 1 ( ) aW K L bR − = 1 1 1 1 1 ( ( ) ) W L a b b Q R       − − − − − = + 1 1 1 1 1 ( ( ) ) R K a b a Q W       − − − − − = + 1 f K L aK bL ( , ) ( )    = + 1 1 1 R aK aK bL ( )      − − = +

.Holding quantity constant-the elasticity of labor with respect to wages equals: L=a六6号5)品+b)0 R 票6wog50 _1+g R and w aL 1 、1 Zaw1-oab+Rw品+l This is increasing with a,decreasing with b, increasing with W and decreasing with R. If the terms in the denominator don't move too much with o,then as o rises,the elasticity rises-this is Marshall's substitutability point

9 • Holding quantity constant– the elasticity of labor with respect to wages equals: and • This is increasing with a, decreasing with b, increasing with W and decreasing with R. • If the terms in the denominator don’t move too much with , then as rises, the elasticity rises– this is Marshall’s substitutability point. 1 1 1 1 1 ( ( ) ) W L a b b Q R       − − − − − = + 1 2 1 1 1 1 1 1 1 1 1 1 1 ( ( ) ) 1 L W a b R W a b b Q W R                − − − − + − − − − − − − −  = +  − 1 1 1 1 1 1 1 1 1 1 W L L W a b R W        − − − − − −  =  − +  

To get law #3:we need to allow R to be a function of K,and then we get 品片股能Q aL 1-0 where a5ob▣形品号 。0。 1-σR R 1+ This a little artificial,since we are holding overall output constant. But nonetheless,from this it should be clear that as gets bigger,the value ofgets smaller. That is Marshall's third law. 10

10 • To get law # 3: we need to allow R to be a function of K, and then we get where • This a little artificial, since we are holding overall output constant. • But nonetheless, from this it should be clear that as gets bigger, the value of gets smaller. • That is Marshall’s third law. 1 1 1 1 1 1 1 1 1 ( ) 1 ( ) ( ( ) ) a b W W R R W a b b R               − − − − − + − − − − = + R K   L W   1 1 ( )[ ] L W R K Q W R W R K W     = −   