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 )
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*
decreases and z* incre2ses. That is, the firm partially
substitutes the input that became more costly with the other
● 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,
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
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
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.
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
However, if MRTS(z ,z )= 1 /2 , th2n 1uch a function is NOT
Econ 2G03/2X03 Oct 17, 24 Chapter 7 7/16 Homothetic Production Functions
● Claim: for homothetic production functions, the output expansion