Micro 8 - Cost Minimization Microeconmics

Lecture in UGM

1. Cost Minimization
Modified from Nicholson, Varian, & Perloff’s Microeconomics
Textbook
Prepared by Dea Yustisia
Universitas Gadjah Mada

2. Contents Coverage
▶So far we looked at the profit maximizing problem of the firm
▶Now, we focus our attention on costs.
▶Profit maximizing⇒Cost minimization
▶That is, if there was a cheaper way of production, profits were
not maximized
▶We break down profit maximization into two stages:
1. y(output)
2. ythat maximizes profits

3. Motivation
▶Contents Coverage:
▶We know thattotal costscan be written as the sum offixed
costsandvariable costs, that is:
TC(y) =FC+VC(y)
▶We will now derive these cost functions as functions of the
level of productiony(as well as of the prices of inputs and
outputs)
▶They embody an optimal use of factors of production.
▶→Cost minimization

4. The Isocost Curve
▶The isocost curve contains all of the input bundles that cost
an equal amount.
▶Generally, givenw1andw2, the equation of an isocost curve
consisting of all input bundles that have the costcis linear:
c=w1x1+w2x2
⇒x2=
c
w2
−
w1
w2
x1
▶The slope is−
w1
w2
▶The vertical intercept is
c
w2

5. Geometrical solution
▶Geometric solution: slope of isoquant equals slope of isocost
curve
▶The solution is theconditional factor demand(conditional
on outputy).

6. Conditional Input Demand Curves
▶Fixedw1andw2.

7. Conditional Input Demand Curves
▶Fixedw1andw2.

8. Conditional Input Demand Curves
▶Fixedw1andw2.

9. Conditional Input Demand Curves
▶Fixedw1andw2.

10. Conditional Input Demand Curves
▶Fixedw1andw2.

11. Conditional Input Demand Curves
▶Fixedw1andw2.

12. How LR Cost Varies with Output
▶As a firm increases output, theexpansion pathtraces out
the cost-minimizing combinations of inputs employed.

13. LR Cost Varies with Output
▶The expansion path enables construction of a LR cost curve
that relates output to the least cost way of producing each
level of output.

14. Return to scale and cost minimization
▶Average Cost of Production:
▶For positive output levelsy, a firm’s average total cost of
producingyunits is:
AC(w1,w2,y) =
c(w1,w2,y)
y

15. Returns to Scale and Cost Minimization
▶The returns-to-scale properties of a firm’s technology
determine how average production costs change with output
level
▶A firm is presently producingy
′
output units
▶How does the firm’s average production cost change if it
instead produces 2·y
′
units of output?

16. Returns to Scale and Cost Minimization
▶If a firm’s technology exhibitsconstantreturns-to-scale then:
▶Doubling its output level fromy
′
to 2·y
′
requiresdoublingof
all input levels
▶Total production cost doubles:
c(2·y
′
) = 2·c(y
′
)
▶Average production cost does not change
AC(2·y
′
) =AC(y
′
)

17. Returns to Scale and Cost Minimization
▶If a firm’s technology exhibitsincreasingreturns-to-scale
then:
▶Doubling its output level fromy
′
to 2·y
′
requiresless than
doublingof all input levels
▶Total production cost less than doubles:
c(2·y
′
)<2·c(y
′
)
▶Average production cost decreases
AC(2·y
′
)<AC(y
′
)

18. Returns to Scale and Cost Minimization
▶If a firm’s technology exhibitsdecreasingreturns-to-scale
then:
▶Doubling its output level fromy
′
to 2·y
′
requiresmore than
doublingof all input levels
▶Total production cost more than doubles:
c(2·y
′
)>2·c(y
′
)
▶Average production cost increases
AC(2·y
′
)>AC(y
′
)

19. Returns to Scale and Cost Minimization
▶To sum it up...
▶Constantreturns impliesconstantAC
▶Increasingreturns to scale impliesdecreasingAC
▶Decreasingreturns impliesincreasingAC

20. Returns to Scale and Cost Minimization
▶Decreasing returns to scale
(r.t.s.): Average cost
increases as output
increases.
▶Constant returns to scale
(r.t.s.): Average cost
remains constant as output
increases.
▶Increasing returns to scale
(r.t.s.): Average cost
decreases as output
increases.

21. RETURNS-TO-SCALE AND TOTAL COSTS
▶What does this imply for the shapes of total cost functions?

22. RETURNS-TO-SCALE AND TOTAL COSTS
Av. cost increases with y if the firm’s technology exhibits
decreasing r.t.s.

23. RETURNS-TO-SCALE AND TOTAL COSTS
Av. cost decreases with y if the firm’s technology exhibits
increasing r.t.s.

24. RETURNS-TO-SCALE AND TOTAL COSTS
Slope = c(2y’) / 2y’
= 2c(y’) / 2y’
= c(y’) / y’
so
AC(y’) = AC(2y’).
Av. cost is constant when the firm’s technology exhibits
constant r.t.s.

25. SR and LR Expansion Paths
▶Firms have more flexibility in the LR.
▶Expanding output is cheaper in LR than in SR because of the
ability to move away from fixed capital choice.

26. Cost Minimization
This firm is seeking the least cost way of producing 100 units
of output.

27. Isocost Line
▶Isocost linesummarizes all combinations of inputs that require
the same total expenditure
▶If the firm hiresx1hours of labor at a wage ofw1per hour,
total labor cost isw1x1.
▶If the firm rentsx2hours of machine services at a rental rate
ofw2per hour, total capital cost isw2x2
▶Cost is fixed at a particular level along a given isocost line:
C=w1x1+w2x2
▶Rewrite the isocost equation for easier graphing:
x2=
C
w2
−
w1
w2
x1

28. Properties of Isocost Lines
▶Three properties of isocost lines:
1. C, and input prices determine where the
isocost line hits the axes.
2.
closer to the origin.
3.
relative prices of the inputs.
dx2
dx1
=−
w1
w2

29. Cost Minimization
▶Three equivalent approaches to minimizing cost:
1.Lowest-isocost rule: Pick the bundle of inputs where the
lowest isocost line touches the isoquant associated with desired
level of output.
2.Tangency rule: Pick the bundle of inputs where the desired
isoquant is tangent to the budget line.
MRTS=−
w1
w2
3.Last-dollar rule: Pick the bundle of inputs where the last
dollar spent on one input yields as much additional output as
the last dollar spent on any other input.
MPx1
MPx2
=
w1
w2
Or rewrite as
MPx1
w1
=
MPx2
w2

30. Cost Minimization with Calculus
▶Minimizing cost subject to a production constraint yields the
Lagrangian and its first-order conditions:
min
x1,x2,λ
L=w1x1+w2x2+λ(q−f(x1,x2))
∂L
∂x1
=w1−λ
∂f
∂x1
= 0
∂L
∂x2
=w2−λ
∂f
∂x2
= 0
∂L
∂λ
=q−f(x1,x2) = 0
▶Rearranging terms reveals the last-dollar rule:
w1
w2
=
∂f
∂x1
∂f
∂x2
=
MPx1
MPx2

31. Cost Minimization with Calculus
▶Minimizing cost subject to a production constraint yields the
Lagrangian and its first-order conditions:
min
x1,x2,λ
L=w1x1+w2x2+λ(q−f(x1,x2))
▶First-order condition with respect tox1:
∂L
∂x1
=w1−λ
∂f
∂x1
= 0
▶First-order condition with respect tox2:
∂L
∂x2
=w2−λ
∂f
∂x2
= 0
▶First-order condition with respect toλ:
∂L
∂λ
=q−f(x1,x2) = 0

32. Cost Minimization with Calculus (continued)
▶Rearranging the first-order conditions with respect tox1and
x2:
λ=
w1
∂f
∂x1
andλ=
w2
∂f
∂x2
▶Setting these two expressions forλequal to each other:
w1
w2
=
∂f
∂x1
∂f
∂x2
▶This can be rewritten as the last-dollar rule:
w1
w2
=
∂f
∂x1
∂f
∂x2
=
MPx1
MPx2

33. Output Maximization with Calculus
▶The “dual” problem to cost minimization isoutput
maximization.
▶Maximizing output subject to a cost constraint yields the
Lagrangian and its first-order conditions:
max
x1,x2,λ
L=f(x1,x2) +λ
Γ
C−w1x1−w2x2
∆
∂L
∂x1
=
∂f
∂x1
−λw1= 0
∂L
∂x2
=
∂f
∂x2
−λw2= 0
∂L
∂λ
=C−w1x1−w2x2= 0
▶Rearranging terms reveals the tangency rule:
w1
w2
=
∂f
∂x1
∂f
∂x2
=
MPx1
MPx2

34. Output Maximization with Calculus
▶Maximizing output subject to a cost constraint yields the
Lagrangian and its first-order conditions:
max
x1,x2,λ
L=f(x1,x2) +λ
Γ
C−w1x1−w2x2
∆
▶First-order condition with respect tox1:
∂L
∂x1
=
∂f
∂x1
−λw1= 0
▶First-order condition with respect tox2:
∂L
∂x2
=
∂f
∂x2
−λw2= 0
▶First-order condition with respect toλ:
∂L
∂λ
=C−w1x1−w2x2= 0

35. Output Maximization with Calculus (continued)
▶Rearranging the first-order conditions with respect tox1and
x2:
λ=
∂f
∂x1
w1
andλ=
∂f
∂x2
w2
▶Setting these two expressions forλequal to each other:
w1
w2
=
∂f
∂x1
∂f
∂x2
▶This can be rewritten as the tangency rule:
w1
w2
=
∂f
∂x1
∂f
∂x2
=
MPx1
MPx2

36. Cost Minimization Condition
The cost minimization condition occurs when a firm chooses inputs
x1(e.g., labor) andx2(e.g., capital) in such a way that it
minimizes its total cost for a given level of output, given input
pricesw1andw2.
The condition for cost minimization requires that the ratio of the
marginal products of each input equals the ratio of their prices.
This relationship is based on the idea that each dollar spent on
either input should produce the same marginal output.
Mathematically, this condition can be written as:
w1
w2
=
Marginal Product ofx1
Marginal Product ofx2

37. The Cost Minimization Problem
▶Consider a firm using two inputs to make one output:
▶The production function isy=f(x1,x2).
▶Assume a positive output levely>0.
▶Given the input pricesw1andw2, the cost of an input bundle
(x1,x2) is
c=w1x1+w2x2
▶The firm’scost minimization problemis:
min
x1,x2
w1x1+w2x2
subject tof(x1,x2) =y

38. The Cost Minimization Problem
▶The cost is minimized when:
|TRS|or|MRTS|=
MP1(x
∗
1
,x
∗
2
)
MP2(x
∗
1
,x
∗
2
)
=
w1
w2
▶This condition holds when the firm chooses its inputs optimally.
▶The firm’sconditional factor demandfor inputs 1 and 2:
x
∗
1(w1,w2,y) andx
∗
2(w1,w2,y)
▶These are the optimal choices of factorsgiven output level
▶In the previous lecture, optimal choices were conditional on
output price
▶The minimum cost for a given outputyis:
c(w1,w2,y) =w1x
∗
1(w1,w2,y) +w2x
∗
2(w1,w2,y)

39. A Cobb-Douglas Example of Cost Minimization
▶A firm’s Cobb-Douglas production function is
y=f(x1,x2) =x
1/3
1
x
2/3
2
▶Input prices arew1= 5 andw2= 10.
▶What are theconditional input demandsfor the two
factors?
▶What is the firm’s totalcost function?
▶Need to solve forx
∗
1
(y),x
∗
2
(y), andc(y).

40. A Cobb-Douglas Example of Cost Minimization
▶A firm’s Cobb-Douglas production function is
y=f(x1,x2) =x
1/3
1
x
2/3
2
▶Input prices are noww1andw2.
▶What are theconditional input demand functionsfor the
two factors?
▶What is the firm’s totalcost function?
▶Need to solve forx
∗
1
(w1,w2,y),x
∗
2
(w1,w2,y) and
c(w1,w2,y).

41. A Cobb-Douglas Example of Cost Minimization
At the input bundle (x
∗
1
,x
∗
2
) which minimizes the cost of producing
youtput units:
(a)y= (x
∗
1
)
1/3
(x
∗
2
)
2/3
(b)−
w1
w2
=−
∂y
∂x
1
∂y
∂x
2
=−
(
1
3)(x
∗
1
)
−2/3
(x
∗
2
)
2/3
(
2
3)(x
∗
1
)
1/3
(x
∗
2
)
−1/3
=−
x
∗
2
2x
∗
1

42. Cost Minimization Condition
For a cost-minimizing firm, the ratio of input prices should equal
the ratio of marginal products of the inputs:
w1
w2
=
∂y
∂x1
∂y
∂x2
or equivalently:
−
w1
w2
=−
∂y
∂x1
∂y
∂x2
For a Cobb-Douglas production function:
y=x
1/3
1
x
2/3
2
calculate the partial derivatives (marginal products):
∂y
∂x1
=
1
3
x
−2/3
1
x
2/3
2
∂y
∂x2
=
2
3
x
1/3
1
x
−1/3
2

43. Setting MRTS Equal to Input Price Ratio and
Simplification
Substitute into MRTS and set it equal to−
w1
w2
:
−
w1
w2
=−
1
3
x
−2/3
1
x
2/3
2
2
3
x
1/3
1
x
−1/3
2
Simplify by canceling terms:
−
w1
w2
=−
−
1
3
·
x
−2/3
1
x
2/3
2
−
2
3
·
x
1/3
1
x
−1/3
2
=−
x2
2x1
Thus, we arrive at:
−
w1
w2
=−
x
∗
2
2x
∗
1
This expression shows the relationship between the input quantities
and prices.

44. Cobb-Douglas Production Function
We start with the production function and the cost-minimization
conditions given by:
▶Production function:
y= (x
∗
1)
1/3
(x
∗
2)
2/3
▶Cost-minimization condition:
w1
w2
=
x
∗
2
2x
∗
1
Our goal is to find the conditional demand for inputx1as a
function ofy,w1, andw2.

45. Calculate Conditional Demand for Input 1 (Step 1
1. x
∗
2
in terms of
x
∗
1
:
x
∗
2= 2
w1
w2
x
∗
1
2. x
∗
2
into the production function:
y= (x
∗
1)
1/3
`
2
w1
w2
x
∗
1
´
2/3

46. Calculate Conditional Demand for Input 1 (Step 2
1.
y= 2
2/3
⊆
w1
w2
⊇
2/3
(x
∗
1)
1/3+2/3
y= 2
2/3
⊆
w1
w2
⊇
2/3
x
∗
1
2. x
∗
1
:
x
∗
1=
y
2
2/3
≍
w1
w2
≡
2/3
Or equivalently:
x
∗
1=y·
⊆
w2
2w1
⊇
2/3
This is the conditional input demand function forx1as a function
ofy,w1, andw2.

47. Calculate Conditional Demand for Input 2 (Step 1)
Given:
x
∗
2=
2w1
w2
x
∗
1andx
∗
1=
`
w2
2w1
´
2/3
y
Substitutex
∗
1
into the expression forx
∗
2
:
x
∗
2=
2w1
w2
`
w2
2w1
´
2/3
y
Now, let’s simplify the expression step by step in the next slide.

48. Calculate Conditional Demand for Input 2 (Step 2)
Starting with:
x
∗
2=
2w1
w2
`
w2
2w1
´
2/3
y
Further simplification:
x
∗
2=
`
2w1
w2
´
1/3
y
Thus, we have the final expression:
x
∗
2=
`
2w1
w2
´
1/3
y
This is the firm’s conditional demand for input 2.

49. A Cobb-Douglas Example of Cost Minimization
The cheapest input bundle yieldingyoutput units is:
(x
∗
1(w1,w2,y),x
∗
2(w1,w2,y))
=

θ
w2
2w1
ι
2/3
y,
θ
2w1
w2
ι
1/3
y
!
In this Cobb-Douglas production function example, the firm
minimizes its cost by choosing input quantitiesx1andx2that yield
the desired outputyat the lowest cost. The expressions forx
∗
1
and
x
∗
2
as functions of input pricesw1andw2and outputyrepresent
the cost-minimizing bundle of inputs.

50. A Cobb-Douglas Example of Cost Minimization
Hence, for the production function:
y=f(x1,x2) =x
1/3
1
x
2/3
2
(const. returns to scale)
the cheapest input bundle yieldingyoutput units is:
(x
∗
1(w1,w2,y),x
∗
2(w1,w2,y)) =

θ
w2
2w1
ι
2/3
y,
θ
2w1
w2
ι
1/3
y
!

51. A Cobb-Douglas Example of Cost Minimization
and the firm’stotal cost functionis:
c(w1,w2,y) =w1x
∗
1(w1,w2,y) +w2x
∗
2(w1,w2,y)

52. A Cobb-Douglas Example of Cost Minimization
and the firm’s total cost function is:
c(w1,w2,y) =w1x
∗
1(w1,w2,y) +w2x
∗
2(w1,w2,y)
=w1
θ
w2
2w1
ι
2/3
y+w2
θ
2w1
w2
ι
1/3
y
=
θ
1
2
ι
2/3
w
1/3
1
w
2/3
2
y+ 2
1/3
w
1/3
1
w
2/3
2
y
= 3
θ
w1w
2
2
4
ι1/3
y.

53. A Cobb-Douglas Example of Cost Minimization
Given:
w1= 5,w2= 10
The firm’s total cost function is:
C(y) = 3
θ
w1w
2
2
4
ι1/3
y
Substitutew1= 5 andw2= 10 into the expression:
C(y) = 3
θ
5×10
2
4
ι1/3
y
Calculate inside the parentheses:
5×100
4
= 125
Now, take the cube root of 125:
125
1/3
= 5
Substitute back into the cost function:
C(y) = 3×5×y= 15y
Therefore,C(y) = 15y.

54. A Cobb-Douglas Example of Cost Minimization
Given:
w1= 5,w2= 10
The firm’s total cost function is:
C(y) = 3
θ
w1w
2
2
4
ι1/3
y
Substitutew1= 5 andw2= 10 into the expression:
C(y) = 3
θ
5×10
2
4
ι1/3
y
Calculate inside the parentheses:
5×100
4
= 125
Now, take the cube root of 125:
125
1/3
= 5
Substitute back into the cost function:
C(y) = 3×5×y= 15y
Therefore,C(y) = 15y.

55. Perfect Complements and Cost Minimization
The firm’s production function is
y= min{4x1,x2}.
▶Input prices arew1= 5 andw2= 10.
▶What are theconditional input demandsfor the two
factors?
▶What is the firm’stotal cost function?
Need to solve forx
∗
1
(y),x
∗
2
(y) andc(y).
▶Constant returns to scale (complements)
▶y=
p
min{4x1,x2}

56. Perfect Complements and Cost Minimization
Where is the least costly input bundle yieldingy
′
output
units?
▶Given the production functiony= min{4x1,x2}, we need to
find the values ofx
∗
1
andx
∗
2
that minimize cost.
▶From the conditions:
x
∗
1=
y
4
,x
∗
2=y

57. Perfect Complements and Cost Minimization
The firm’s production function is
y= min{4x1,x2}
and the conditional input demands are
x
∗
1(w1,w2,y) =
y
4
andx
∗
2(w1,w2,y) =y.
So the firm’s total cost function is
c(w1,w2,y) =w1x
∗
1(w1,w2,y) +w2x
∗
2(w1,w2,y).

58. Perfect Complements and Cost Minimization (continued)
c(w1,w2,y) =w1
y
4
+w2y=
ζ
w1
4
+w2
η
y.
Note:Again, with constant returns to scale, average cost is
constant in output.

59. Perfect Complements and Cost Minimization (continued)
What is the cost function?
▶With input pricesw1= 5 andw2= 10:
c(y) =w1x
∗
1+w2x
∗
2
▶Substitutingx
∗
1
=
y
4
andx
∗
2
=y:
c(y) = 5·
y
4
+ 10·y=
5y
4
+ 10y= 11.25y

60. Short Run Costs
▶In the short run, the cost minimization problem is:
min
x1
w1x1+w2x2s.t.f(x1,x2) =y
▶The short-run (conditional) factor demands are:
x1=x
s
1(w1,w2,x2,y) andx2=x2
▶The short-run cost function is:
c(w1,w2,x2,y) =w1x
s
1(w1,w2,x2,y) +w2x2

61. Short Run vs Long Run Costs
▶The long-run cost-minimization problem is
min
x1,x2≥0
w1x1+w2x2
subject tof(x1,x2) =y.
▶The short-run cost-minimization problem is
min
x1≥0
w1x1+w2x
′
2
subject tof(x1,x
′
2) =y.

62. SHORT-RUN & LONG-RUN TOTAL COSTS
▶The short-run cost-minimization problem is the long-run
problem subject to the extra constraint thatx2=x
′
2
.
▶If the long-run choice forx2wasx
′
2
, then the extra constraint
x2=x
′
2
is not really a constraint at all, and so the long-run
and short-run total costs of producingyoutput units are the
same.
▶The short-run cost-minimization problem is therefore the
long-run problem subject to the extra constraint thatx2=x
′
2
.
▶But, if the long-run choice forx2̸=x
′
2
, then the extra
constraintx2=x
′
2
prevents the firm in this short-run from
achieving its long-run production cost, causing the short-run
total cost to exceed the long-run total cost of producingy
output units.

63. SHORT-RUN & LONG-RUN TOTAL COSTS
▶Consider three output
levels:y
′
,y
′′
, and
y
′′′
.
▶Each curve represents
an isoquant, showing
combinations ofx1
andx2that produce
the same level of
output.
▶Higher isoquants
represent higher
output levels.

64. SHORT-RUN & LONG-RUN TOTAL COSTS
▶In the long-run, when
the firm is free to
choose bothx1and
x2, the least-costly
input bundles are ...

65. SHORT-RUN & LONG-RUN TOTAL COSTS
Long-run costs are:
c(y
′
) =w1x
′
1+w2x
′
2
c(y
′′
) =w1x
′′
1+w2x
′′
2
c(y
′′′
) =w1x
′′′
1+w2x
′′′
2

66. SHORT-RUN & LONG-RUN TOTAL COSTS
▶Now suppose the firm becomes subject to the short-run
constraint thatx2=x
′′
2
.

67. SHORT-RUN & LONG-RUN TOTAL COSTS
▶Long-run costs are:
c(y
′
) =w1x
′
1+w2x
′
2
c(y
′′
) =w1x
′′
1+w2x
′′
2
c(y
′′′
) =w1x
′′′
1+w2x
′′′
2

68. SHORT-RUN & LONG-RUN TOTAL COSTS
Long-run costs are:
c(y
′
) =w1x
′
1+w2x
′
2
c(y
′′
) =w1x
′′
1+w2x
′′
2
c(y
′′′
) =w1x
′′′
1+w2x
′′′
2

69. SHORT-RUN & LONG-RUN TOTAL COSTS
Short-run costs are:
cs(y
′
)>c(y
′
)

70. SHORT-RUN & LONG-RUN TOTAL COSTS
Short-run costs are:
cs(y
′
)>c(y
′
)
cs(y
′′
) =c(y
′′
)

71. SHORT-RUN & LONG-RUN TOTAL COSTS
Short-run costs are:
cs(y
′
)>c(y
′
)
cs(y
′′
) =c(y
′′
)
cs(y
′′′
)>c(y
′′′
)

72. SHORT-RUN & LONG-RUN TOTAL COSTS
Short-run costs are:
cs(y
′
)>c(y
′
)
cs(y
′′
) =c(y
′′
)
cs(y
′′′
)>c(y
′′′
)

73. Short-Run and Long-Run Total Cost Curves
▶A short-run total cost curve always has one point in common
with the long-run total cost curve, and is elsewhere higher
than the long-run total cost curve.
▶cs(y) represents the short-run total cost curve.
▶c(y) represents the long-run total cost curve.
▶F=w2x
′
2
indicates the fixed cost in the short run.