Cost Minimization


We can conceptually divide the profit maximization problem into two sub-problems:

Cost minimization is a necessary (but not sufficient) condition for profit maximization. Even when a producer is not a price taker in the output market, or when the solution to the profit maximization problem is not well defined (say, due to increasing returns), the producer must still minimize costs.

Assumptions:

The producer chooses x1 and x2 (the choice variables) to minimize C(x1,x2) (the objective function)

minimize C(x1,x2) = w1 x1 + w2 x2
subject to the constraint
f(x1,x2) = y
The parameters of this problem are w1, w2, and y.

Form the Lagrangian function:

L = w1 x1 + w2 x2 + λ ( y - f(x1,x2) )
The variable λ is called the Lagrange multiplier for the constraint.

The FOC for this minimization problem is:

∂L/∂x1 = w1 - λ f1 (x1,x2) = 0
∂L/∂x2 = w2 - λ f2 (x1,x2) = 0
∂L/∂λ = y - f(x1,x2) = 0
Sufficient SOC for minimization is that all the border-preserving principle minor determinants of the matrix
	    [ -λ f11   -λ f12  -f1 ]
	H = [ -λ f21   -λ f22  -f2 ]
	    [ -f1      -f2     0  ]
are negative.

Interpretation: The FOC imply w1/w2 = f1/f2. That is, the ratio of factor prices is equal to the marginal rate of technical substitution.

The assumption of quasi-concavity guarantees that the SOC is satisfied. Alternatively, if we abandon the assumption of quasi-concavity, the SOC implies that the production function is locally quasi-concave at the optimal input combination. Here, the quasi-concavity condition is equivalent to the condition of diminishing MRTS.

The SOC does not say anything about returns to scale. Even if we have an increasing returns to scale production function (e.g., y = x1 x2), the cost-minimization problem still has a well defined solution.

The FOC can be considered as three equations in three unknowns (x1, x2, λ). Given any parameter values for w1, w2, and y, we can solve these equations for the unknowns. Of course, the solution values depend on the values of the parameters. If x1~, x2~, and λ~ are the solutions to the cost minimization problem, we emphasize the dependence of these optimal choice functions on the parameters by writing x1~ = x1~ (w1,w2,y) and x2~ = x2~ (w1,w2,y). These functions are different from the factor demand functions derived from the profit maximization problem. We call them cost-minimizing factor demand functions or conditional factor demand functions.

By definition, if you substitute the optimal choice functions into the FOC equations, the equations are always satisfied. We therefore write

w1 - λ~ (w1,w2,y) f1( x1~ (w1,w2,y), x2~ (w1,w2,y) ) ≡ 0
w2 - λ~ (w1,w2,y) f2( x1~ (w1,w2,y), x2~ (w1,w2,y) ) ≡ 0
y - f( x1~ (w1,w2,y), x2~ (w1,w2,y) ) ≡ 0

Consider how w1 affects factor demand. Use the identities above, differentiate with respect to w1, and employ the chain rule of differentiation, we get

1 - λ f11 (∂x1~/ ∂w1) - λ f12 (∂x2~ /∂w1) - f1 (∂λ~ /∂w1) = 0
0 - λ f21 (∂x1~/ ∂w1) - λ f22 (∂x2~ /∂w1) - f2 (∂λ~ /∂w1) = 0
0 - f1 (∂x1~ /∂w1) - f2 (∂x2~ /∂w1) - 0 (∂λ~ /∂w1) = 0
In matrix notation:
	[ -λ f11   -λ f12   -f1 ] [ ∂x1~/∂w1 ] = [ -1 ]
	[ -λ f21   -λ f22   -f2 ] [ ∂x2~/∂w1 ] = [ 0  ]
	[ -f1      -f2      0  ] [ ∂λ~/∂w1  ] = [ 0  ]

Use Cramer's rule to get

 ∂x1~/∂w1 = | -1   -λ f12   -f1 |   / 
           | 0    -λ f22   -f2 |  /  | H |
           | 0    -f2      0  | /
            
         = -1 | -λ f22   -f2 |  / | H |
              | -f2      0  | /
The determinant in the numerator is a border-preserving principal minor determinant, so by the SOC it is negative (verify!). Similarly, the SOC also requires the | H | < 0. Therefore ∂x1~ /∂w1 < 0 (downward sloping conditional factor demand curve).

We can also solve for ∂x2~/ ∂w1:

 ∂x2~/∂w1 = | -λ f11   -1   -f1 |   / 
           | -λ f21   0    -f2 |  /  | H |
           | -f1      0    0  | /
            
        = 1 | -λ f21   -f2 |  / | H |
            | -f1      0  | /
The determinant in the numerator is not a border-preserving principal minor determinant, so in general its sign is undetermined. But for the two-input case, this determinant is negative, so ∂x2~/ ∂w1 > 0. (Why? Theory implies ∂x1~ /∂w1 < 0. So when w1 increases, we use less x1, but we still want to produce the same amount of output y. This is achieved by increasing the use of x2.)

If you are curious, you can do a similar comparative statics analysis for w2. You will then verify that

∂x1~/ ∂w2 = ∂x2~/ ∂w1
Let's also try the comparative statics for y. Differentiate the FOC with respect to y and write in matrix notation, we have
	[ -λ f11   -λ f12   -f1 ] [ ∂x1~/∂y ] = [ 0  ]
	[ -λ f21   -λ f22   -f2 ] [ ∂x2~/∂y ] = [ 0  ]
	[ -f1      -f2      0  ] [ ∂λ~/∂y  ] = [ -1 ]

Therefore,

 ∂x1~/∂y = | 0    -λ f12   -f1 |   / 
          | 0    -λ f22   -f2 |  /  | H |
          | -1   -f2      0  | /
            
       = -1 | -λ f12   -f1 |  / | H |
            | -λ f22   -f2 | /
The determinant in the numerator is not a border-preserving principal minor determinant, so this derivative cannot be signed. If ∂x1~/∂y > 0, then x1 is a "normal factor." If ∂x1~/∂y < 0, then we call it an "inferior factor."

We have not derived any comparative statics for the λ~ (w1,w2,y) function. But if we do it, we will see that all the partial derivatives have ambiguous signs.

Finally, if we look at the FOC equations again, and consider the effect of changing all input prices from (w1, w2) to (tw1, tw2) while keeping the parameter y unchanged. The FOC become

tw1 - λ f1 (x1,x2) = 0
tw2 - λ f2 (x1,x2) = 0
y - f(x1,x2) = 0
If ( x1~, x2~, λ~ ) solve the original FOC equations, then ( x1~, x2~, tλ~ ) must solve the new set of equations. We therefore conclude that
x1~ (tw1, tw2, y) ≡ x1~ (w1, w2, y)
x2~ (tw1, tw2, y) ≡ x2~ (w1, w2, y)
That is, the conditional input demand functions are homogeneous of degree 0 in w1 w2 (but not in y). (Since λ~ (tw1, tw2, y) ≡ t λ~(w1, w2,y), we also see that the λ~() function is homogeneous of degree 1.)

SUMMARY: Properties of conditional input demand functions

Relationship between cost minimization and profit maximization

If we compare the FOCs for the profit-maximization problem with the FOCs for the cost minimization problem, we can see that they will give the same solution values for x1 and x2 if the value of the Lagrange multiplier is λ = p. From introductory microeconomics, we know that a condition for profit maximization is Marginal Cost = p. This is a hint that the Lagrange multiplier can be interpreted as Marginal Cost.

The FOC implies λ = w1/f1 = w2/f2. What is the marginal cost of producing one unit of output? Well, we can produce more outputs by using more x1. If the marginal product of x1 is f1, we need 1/f1 units of x1 to produce one more unit of output. Each unit of x1 costs w1, so the cost of 1/f1 units of x1 is w1/f1. This is another hint that λ can be interpreted as Marginal Cost. More on this later.

If you want to produce y = y*(p,w1,w2) units of output, the cost-minimizing input bundle must be the same as the profit-maximizing input bundle. So we must have

x1~ (w1, w2, y*(p,w1,w2) ) ≡ x1* (p,w1,w2)
Differentiate the identity with respect to, say w1, we get
∂x1~/ ∂w1 + (∂x1~ /∂y) (∂y*/∂w1) = ∂x1*/∂w1
To find out what ∂x1~/∂y is, we differentiate the identity with respect to p to get
(∂x1~ /∂y) (∂y*/∂p) = ∂x1*/∂p
So ∂x1~ /∂y = (∂x1*/∂p) / (∂y*/∂p). Substitute this expression back in:
∂x1~/ ∂w1 - ∂x1*/ ∂w1 = - (∂x1*/∂p) (∂y*/∂w1) / (∂y*/∂p)
Since the symmetry condition ensures that the ∂x1*/∂p = - ∂y*/∂w1, the numerator is a square must be positive. The denominator is also shown to be positive, so the whole term is positive. I.e., ∂x1~/ ∂w1 > ∂x1*/∂w1. But because demand curves are negatively sloped, this means that the derivative of the profit-maximizing demand function is larger in absolute value than is the derivative of the cost-minimizing demand function. Consider the effect of an increase in w1 on the profit-maximizing choice of x1:
	∂x1*/∂w1 = ∂x1~/∂w1 + (∂x1~/∂y) (∂y*/∂w1) 
                    ^                ^
               substitution        scale
                  effect           effect
The substitution effect is always negative. The scale effect is also always negative ( for "normal factors," ∂x1~/∂y > 0 and ∂y*/∂w1 < 0; for "inferior factors," ∂x1~/∂y < 0 and ∂y*/∂w1 > 0). The two effects reinforce each other.


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