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Maxim Ivanov

Lectures 6-7 Comparative Statics for Input Prices If all input prices change by the same factor a, then: 1. The cost of minimizing the input bundle for y units of output does not change. This is because the slope of the isocost line w /w will1be 2he same, thus, the equation MP /MP =w /w is unchanged. 1 2 1 2 Also, the isoquant y=F(z ,z )1is2unchanged. 2. The minimum cost of producing y units of output changes by the same factor a. This is because: TC(y,a*w ,a*1 )=a*2 *z +a*1 *1 =a*(w2*z2+w *z )1 1 2 2 =a*TC(y,a*w ,a*w ) 1 2 Econ 2G03/2X03 Oct 17, 24 Chapter 7 1/16 Comparative Statics for Input Prices ● Suppose that the cost-minimizing quantity of inputs i and j is positive, and MRTS is diminishing (in z ). positive diminishing 1 ● If w 1ncreases to aw , where a>1 and w does not,2then z* 1 1 decreases and z* incre2ses. That is, the firm partially substitutes the input that became more costly with the other input. ● This is because for new input prices, input z becomes1less productive (per dollar) compared to z : 2 MP /aw 1 and1 the quantity demanded of that input is positive, that is, z *>0, 1 then the total costs will increase: TC(y,aw ,w1) >2TC (y,w ,w ) 1 2 ● This is because the new cost-minimizing input bundle is above the initial isocost line for (z* ,z* ), that is, new input bundle is 1 2 more costly. ● See the figure below. Econ 2G03/2X03 Oct 17, 24 Chapter 7 3/16 Comparative Statics for Input Prices Econ 2G03/2X03 Oct 17, 24 Chapter 7 4/16 Comparative Statics for Level of Output ● The ooutput expansion pathshows the cost minimizing input bundles (z* 1z* 2 for all levels of output y. ● A normal input - the cost-minimizing level of that input normal input increases when output y increases. ● An iinferior input- the cost-minimizing level of that input decreases when output y increases. ● Note: the input can be normal for some level of output y , bu0 inferior for another level y . See the figure below. 1 Econ 2G03/2X03 Oct 17, 24 Chapter 7 5/16 The output expansion path Econ 2G03/2X03 Oct 17, 24 Chapter 7 6/16 Homothetic Production Functions Homothetic Production Functions ● A hhomothetic production functionis a type of function such that the MRTS is consconstantng any ray from the origin. ● A production function is homothetic if MRTS can be represented as a function of z /z : 2 1 MRTS(z ,z 1 =2f(z /z 2 1 ● Examples: if MRTS(z ,z ) = (z /z ) , then such a function is 1 2 2 1 homothetic. 3 However, if MRTS(z ,z )= 1 /2 , th2n 1uch a function is NOT homothetic. Econ 2G03/2X03 Oct 17, 24 Chapter 7 7/16 Homothetic Production Functions ● Claim: for homothetic production functions, the output expansion path
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