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 x_{1} and x_{2} (the choice variables) to minimize C(x_{1},x_{2}) (the objective function)
Form the Lagrangian function:
The FOC for this minimization problem is:
[ -λ f_{11} -λ f_{12} -f_{1} ] H = [ -λ f_{21} -λ f_{22} -f_{2} ] [ -f_{1} -f_{2} 0 ]are negative.
Interpretation: The FOC imply w_{1}/w_{2} = f_{1}/f_{2}. 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 = x_{1} x_{2}), the cost-minimization problem still has a well defined solution.
The FOC can be considered as three equations in three unknowns (x_{1}, x_{2}, λ). Given any parameter values for w_{1}, w_{2}, and y, we can solve these equations for the unknowns. Of course, the solution values depend on the values of the parameters. If x_{1}^{~}, x_{2}^{~}, and λ^{~} are the solutions to the cost minimization problem, we emphasize the dependence of these optimal choice functions on the parameters by writing x_{1}^{~} = x_{1}^{~} (w_{1},w_{2},y) and x_{2}^{~} = x_{2}^{~} (w_{1},w_{2},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
Consider how w_{1} affects factor demand. Use the identities above, differentiate with respect to w_{1}, and employ the chain rule of differentiation, we get
[ -λ f_{11} -λ f_{12} -f_{1} ] [ ∂x_{1}^{~}/∂w_{1} ] = [ -1 ] [ -λ f_{21} -λ f_{22} -f_{2} ] [ ∂x_{2}^{~}/∂w_{1} ] = [ 0 ] [ -f_{1} -f_{2} 0 ] [ ∂λ^{~}/∂w_{1} ] = [ 0 ]
Use Cramer's rule to get
∂x_{1}^{~}/∂w_{1} = | -1 -λ f_{12} -f_{1} | / | 0 -λ f_{22} -f_{2} | / | H | | 0 -f_{2} 0 | / = -1 | -λ f_{22} -f_{2} | / | H | | -f_{2} 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 ∂x_{1}^{~} /∂w_{1} < 0 (downward sloping conditional factor demand curve).
We can also solve for ∂x_{2}^{~}/ ∂w_{1}:
∂x_{2}^{~}/∂w_{1} = | -λ f_{11} -1 -f_{1} | / | -λ f_{21} 0 -f_{2} | / | H | | -f_{1} 0 0 | / = 1 | -λ f_{21} -f_{2} | / | H | | -f_{1} 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 ∂x_{2}^{~}/ ∂w_{1} > 0. (Why? Theory implies ∂x_{1}^{~} /∂w_{1} < 0. So when w_{1} increases, we use less x_{1}, but we still want to produce the same amount of output y. This is achieved by increasing the use of x_{2}.)
If you are curious, you can do a similar comparative statics analysis for w_{2}. You will then verify that
[ -λ f_{11} -λ f_{12} -f_{1} ] [ ∂x_{1}^{~}/∂y ] = [ 0 ] [ -λ f_{21} -λ f_{22} -f_{2} ] [ ∂x_{2}^{~}/∂y ] = [ 0 ] [ -f_{1} -f_{2} 0 ] [ ∂λ^{~}/∂y ] = [ -1 ]
Therefore,
∂x_{1}^{~}/∂y = | 0 -λ f_{12} -f_{1} | / | 0 -λ f_{22} -f_{2} | / | H | | -1 -f_{2} 0 | / = -1 | -λ f_{12} -f_{1} | / | H | | -λ f_{22} -f_{2} | /The determinant in the numerator is not a border-preserving principal minor determinant, so this derivative cannot be signed. If ∂x_{1}^{~}/∂y > 0, then x_{1} is a "normal factor." If ∂x_{1}^{~}/∂y < 0, then we call it an "inferior factor."
We have not derived any comparative statics for the λ^{~} (w_{1},w_{2},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 (w_{1}, w_{2}) to (tw_{1}, tw_{2}) while keeping the parameter y unchanged. The FOC become
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 x_{1} and x_{2} 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 = w_{1}/f_{1} = w_{2}/f_{2}. What is the marginal cost of producing one unit of output? Well, we can produce more outputs by using more x_{1}. If the marginal product of x_{1} is f_{1}, we need 1/f_{1} units of x_{1} to produce one more unit of output. Each unit of x_{1} costs w_{1}, so the cost of 1/f_{1} units of x_{1} is w_{1}/f_{1}. This is another hint that λ can be interpreted as Marginal Cost. More on this later.
If you want to produce y = y*(p,w_{1},w_{2}) units of output, the cost-minimizing input bundle must be the same as the profit-maximizing input bundle. So we must have
∂x_{1}*/∂w_{1} = ∂x_{1}^{~}/∂w_{1} + (∂x_{1}^{~}/∂y) (∂y*/∂w_{1}) ^ ^ substitution scale effect effectThe substitution effect is always negative. The scale effect is also always negative ( for "normal factors," ∂x_{1}^{~}/∂y > 0 and ∂y*/∂w_{1} < 0; for "inferior factors," ∂x_{1}^{~}/∂y < 0 and ∂y*/∂w_{1} > 0). The two effects reinforce each other.