1. 11. Cost minimization
Econ 494
Spring 2013

2. Agenda Cost minimization Readings Primal approach
Envelope theorem for constrained optimization
Link to profit maximization
Readings
Silberberg Chapter 8
Also review constrained optimization

3. Quick optimization review
For the constrained optimization problem:
Set up the Lagrangian function:
To get correct sign of l,
For maximization problems, write the constraints to be non-negative
For minimization problems, write the constraints to be non-positive

4. FONC Note that l is a choice variable, hence there are 3 FONC.
If the SOSC are satisfied, you can use the IFT to solve the FONC simultaneously for x1*, x2* and l*.

5. See Silb table 6-1, p. 138; Chiang table 12.1, p. 362
SOSC
With one constraint, the SOSC require that all border-preserving principal minors be:
Sign (–1)r, r = 2,…n for a maximum
starts with positive second order principal minor
All minors of order r will have same sign (no need to check all)
Sign is negative for a minimum
A border-preserving principal minor of order r is the determinant of the matrix obtained by deleting n-r rows and corresponding columns – but you cannot delete the row/column with the border

6. Border-preserving principal minors (one constraint)
n =2, r = 1  n – r =1  delete one row and column.
Sign unknown
n =2, r = 2  n – r =0  delete no rows or columns.
Max: >0 Min: <0

7. Question
For a profit maximization problem, what must we assume about the shape of the production function?
How do you know this?
Profit max requires that the production function be strictly concave. This is implied by the SOSC: |H|=f11·f22-(f12)^2>0

8. Cost minimization
We want to determine the properties of a function that specifies the total cost of producing any given level of output.
Note that this cost function differs from the “cost function” we used for P-max. How?

9. Cost minimization
To assert the existence of a well-defined cost function, we need a theory of the firm.
The cost function depends upon what the firm intends to do, and the constraints they face.
The production function is a constraint
We cannot directly derive the cost function from the postulate of P-max. (see Silb p. 176)
Need y to enter as a parameter, not a variable

10. Cost minimization
Cost functions must be derived from a model with output y as a parameter
Clearly, in order to maximize profits, the firm must minimize the cost producing the optimal level of output.
This cost min assertion is consistent with P-max

11. Cost minimization
We assert that the firm will minimize the costs of producing a given level of output
Costs are simply the sum of all input levels times their respective (fixed) prices
Constraint: have to produce some level of output, y, which is related to inputs by the production function

12. Graphical illustration
Choose some combination of x1 and x2 to produce y0 output  Isoquant

13. Graphical illustration
Choose (x1, x2) such that the firm produces y0 at least cost.
Remember w1 and w2 are fixed, as is y0
Isocost line – “same cost”
There are an infinite number of isocost lines given factor prices w1 and w2, but only one that will produce y0

14. Graphical illustration
Cost is minimized where isocost line is tangent to isoquant

15. Graphical illustration
What if isocost and isoquant were not tangent?
Consider:
C2 > C0 > C1

16. Mathematics of cost minimization
Set up Lagrangian:
Adding zero

17. Optimality conditions

18. Evaluate SOSC

19. Evaluate SOSC (cont.) SHAPE OF PRODUCTION FUNCTION:
Note that this result is the same as the slope of the isoquant dx2/dx1. We derived this earlier in the semester when we talked about production functions.
SHAPE OF PRODUCTION FUNCTION:
Cost min requires quasi-concavity
Profit max requires strict concavity (stronger requirement)
Isoquants are convex

20. Conditional factor demands
There are 3 parameters in the FONC (y,w1,w2)
By the IFT, assuming the SOSC hold, we can solve the FONC simultaneously for the explicit choice functions:
x1c(y,w1,w2), x2c (y,w1,w2), lc (y,w1,w2)
xic are known as conditional factor demands
The factor demands from a cost min problem are conditioned on the level of output y.
To distinguish between conditional factor demands and profit-max factor demands, look at the arguments:
Conditional factor demands: xic (y,w1,w2)
Profit max factor demands: xi* (p,w1,w2)

21. Interpreting lc
Later, using the Envelope Thm, we will see that lc(y,w1,w2) is the marginal cost function
From the first two FONC:

22. lc(y,w1,w2) is marginal cost function
At cost min input mix, if the firm were to increase xi “a little bit” Dxi
Total cost would rise by wi×Dxi
Output would also rise by MPi×Dxi = fi×Dxi
Hence l represents the incremental cost of increasing output through the use of xi.

23. Cost min comparative statics
The approach is the same as always.
Substitute the explicit choice functions x1c(y,w1,w2), x2c (y,w1,w2), lc (y,w1,w2) back into the FONC to get the identities:

24. Differentiate wrt w1
If we differentiate the identities with respect to w1, and then express this system of equations using matrix notation, we get:

25. Use Cramer’s rule to solve

26. Interpretations ¶ x1c / ¶w1 < 0
Law of demand – but different from P-max
Cost min
move along isoquant (y fixed)
pure substitution effect
P-max
also move to new isoquant (y varies)
substitution and output effects

27. Interpretations ¶ x2c / ¶w1 > 0
The firm will use more x2 when w1 increases.
We already know that x1 will decrease when w1 increases
If output is held constant, and x1 decreases, then x2 must increase
This result will not necessarily hold if there are more than two inputs.

28. Graphical illustration (w1 increases)

29. Interpretations ¶ lc / ¶w1 indeterminate
Marginal costs may increase or decrease when a factor price increases
May seem counter-intuitive
Later, we will show that this depends upon whether x1 is a normal or inferior input
See Silb fig 8-11

30. Comparative statics for y
Differentiate identities wrt y and express in matrix notation, apply Cramer’s rule

31. Interpretation ¶ x1c / ¶y indeterminate
If x1c is normal (¶ x1c / ¶y > 0), then MC increase as the price of factor 1 rises (¶ lc / ¶w1 > 0)
If x1c is inferior (¶ x1c / ¶y < 0), then MC decreases as the price of factor 1 rises (¶ lc / ¶w1 < 0)
Does this mean that an increase in w1 is beneficial for the firm?
At least one factor must be normal
motivation: see Silb p. 200 “digging ditches” and Figures 8-10 and 8-11
Fig 8-10  when the price of an inferior factor increases, MC decreases, but AC increases. The firm does NOT view a factor price increase as beneficial

32. Interpretation Look familiar? ¶ lc / ¶y is slope of MCMR $ y p
Cost min allows for either,
P-max requires ¶lc / ¶y > 0
¶ lc / ¶y is slope of MC
Cost min allows for possibility of U-shaped MC
f11f22 – f122 > 0 for P-max
MC can only slope up for P-max
Cost min is “weaker” than P-max
Only require quasi-concave production fctn
P-max has stronger restrictions and is not implied by cost min
Requires strictly concave production function
Some students have asked whether this graph violates the quasi-concavity condition in the SOSC. It is true that this graph is NOT quasi-concave. Does this violate the SOSC? No. The SOSC state that the production function must be quasi-concave, the graph above is the marginal cost function, not the production function.

33. Envelope Theorem for general constrained problems
Consider the two-variable, one constraint problem:
By the IFT, assuming the SOSC hold, we can solve the FONC simultaneously for the explicit choice functions
x1*(a), x2*(a), l*(a)
Form the Lagrangian:
FONC

34. Envelope Theorem for one constraint
Define the indirect objective function
Substitute the explicit choice functions back into the objective function to get the indirect objective function
Note that the indirect objective function, g(x*(a)), by definition incorporates the constraint. x*(a) is the optimal x that solves the constrained optimization problem. So when writing the indirect objective function do NOT make the mistake of saying the indirect obj fctn is “g(x*(a)) subject to h(x*(a)).”

35. Proof
Note that we are differentiating the indirect obj function with respect to a; we are NOT differentiating L evaluated at the optimal solution.

36. Proof (cont)

37. Apply ET to cost minimization
Define the indirect cost function
We already solved the FONC for the explicit choice functions.
Substitute these back into the objective function to get the indirect cost function

38. Envelope Theorem Results (y)
By definition, marginal cost is the change in cost when output changes: ¶C* / ¶y
 lc = ¶C* / ¶y must be the marginal cost function

39. Envelope Theorem Results (w1)
Derivative of indirect cost function wrt an input price yields conditional factor demand
Shephard’s Lemma

40. Symmetry and reciprocity

41. Conditional factor demands are HOD(0) in factor prices
Multiply factor prices in FONC by t:
Clearly, the t will cancel out. Since these FONC, with prices (tw1, tw2) are identical to the FONC with prices (w1, w2), the solutions must also be the same. Thus: xic(tw1, tw2) = xic(w1, w2)

42. Indirect cost function is HOD(1) in factor prices
For MC is HOD(1), use FONC: lambda* = w1 / f(x1*,x2*) and xi* are HOD(0). Stranlund 712 notes, p or Silb p. 213.
On your own…show that Avg Cost and Marginal Cost are HOD(1). Silb p. 213

43. Relationship between Cost min and profit max
Recall the profit max problem
Instead consider the following:
Clearly, a solution to P-max, requires a solution to cost min
Lagrangian for cost min problem

44. Relationship between factor demands from cost min and profit max
The solutions to the profit max and cost min FONC are the factor demand functions
The profit max factor demand functions:
xi*(p, w1, w2)
The conditional factor demands from cost min:
xic(y, w1, w2)
These functions are different. As w1 increases:
P-max xi* holds w2 and p constant
 movement to new isoquant
Cost-min xic holds w2 and y constant
 movement along isoquant
How can we tie these two demand functions together?

45. Identity linking cost-min and profit max factors demands
Same idea as Le Châtlier
Says that the value of the profit maximizing and cost minimizing factor demand functions are the same (when the conditional factor demand is evaluated at the P-max output level).
Does not say that the functions are the same

46. Interpretation Economic interpretation: Alternatively:
The value of the profit maximizing demand function is identically equal to the value of the cost minimizing demand function, if the parametric level of output in the cost min problem is taken as the profit maximizing level of supply, given identical input and output prices for each model
Alternatively:
If, in the cost min model, output is allowed to adjust to price changes so as to maintain a profit max (i.e., adjust via profit maximizing supply response), then the value of profit max and cost min factor demands will be identical

47. Differentiate wrt w1
Using Young’s theorem and the Envelope theorem for the P-max model, we showed:
Now differentiate identity wrt p:

48. Combine
<0
>0

49. Interpret x1* is more negatively sloped in own price than x1c
Multiply last inequality by w1/x1  elasticities
P-max factor demands are more price elastic

50. Graph The cost min and profit max factor demand functions intersect w1
Be careful…
Note that this graph has w1 on the vertical axis, so x1* more price elastic means x1* is flatter than x1c. w1 x1
x1*(p, w1, w2)
x1c(y, w1 , w2)

51. Decomposing the effect
OUTPUT EFFECT
Output responds to changes in w1.
 movement to new isoquant
SUBSTITUTION EFFECT
Holds output constant.
 movement along isoquant