Chapter 21: Cost Minimization
cost minimization input bundle must be properties:
-f[x*1,x*2]=y* -slope isoquant=slope isocost
Model of profit max in terms of cost minimization two questions:
-how does firm choose lowest cost combo of inputs to produce any level of output? -what is profit max level of output?
slope of iso cost =
-w1/w2
Cobb Douglas slope of isoquant=slope of isocost
-w1/w2 = -x*2/2x*1
Cobb Douglas total cost fx dervied as c(w1,w2,y)=
3[w1w2^2/4]^1/3 y
SR costs Cs(y'')
= cy''
SR costs Cs(y')
> cy'
SR costs Cs(y''')
> cy'''
Decreasing Returns to Scale, doubling output level requires
MORE than double all input levels; Total prod costs MORE than doubles, avg prod cost inc
if long run choice for x2 was NOT x'2 then what is true of the relationship btwn Min LR and SR Total Costs?
SR TC>LR TC
slope of isoquant
TRS=-MP1/MP2
Cobb Douglas cheapest input bundle yielding y output units at input prices w1 and w2 is (x*1,x*2)=
[(w2/2w1)^2/3 y, (2w1,w2)^1/3 y]
Increasing returns to scale avg cost fx (doubling output) AC(2y) =
[b/2]c/y < c/y
Decreasing Returns Average cost fx (doubling output) AC(2y) =
[b/2]c/y > c/y
Cobb Douglas conditional demand for input 1 x*1=
[w2/2w1]^[2/3]y
Regarding short run total cost,
at least 1 input level fixed
Average total cost of production y units AC(w1,w2,y)=
c[w1,w2,y]/y
Constant returns to scale, if we double inputs
double outputs; if double TC and double output no change to avg costs
increasing RTS graphically
downward sloping, dec
Regarding long run total cost,
firm can vary all input levels
constant RTS graphically
horizontal line
Returns to scale of firms technology determines
how avg prod costs change w output level
Increasing returns to scale, doubling output requires
less than doubling all input levels; total prod cost LESS than doubles and avg prod cost dec
Cost minimization problem: Given production function y=f(x1,x2), choose values of (x1,x2) to
min x1,x2>=0 [w1x1+w2x2] subject to tech constraint f[x1,x2]=y*
Model of profit maximization is equivalent to
model of cost min
total cost function c(y) is
prod y u of output given input ps using lowest cost tech
SR total cost > LR total cost EXCEPT
pt on LR TC curve that is same as pt on SR TC curve (output level where SR input level restriction happens to be LR input level choice)
cost min input bundles graphically
pt where isoquant is tangent to c' and 2 pts where isoquant intersects w c''
if long run choice for x2 was x'2 then what is true of the relationship between Min LR and SR Total Costs?
same
isocost line
set of all input bundles that cost same amt
isoquant
shows all input combos that can be used to produce given level output y*
LR output expansion path
shows how opt bundle changes as output inc, when firm can vary all inputs
In the LR, when firm free to choose x1 and x2 the least costly input bundles (graphically) are
tangency pts btwn isoquants and isocosts
decreasing RTS graphically
upward sloping, inc
LR costs c(y''') =
w1x'''1 + w2x'''2
LR costs c(y'') =
w1x''1 + w2x'''2
LR costs c(y') =
w1x'1 + w2x'2
Cobb Douglas conditional demand for input x*2=
y=[2w1/w2]^[1/3]y
