Page 310 - Microeconomics, Fourth Edition

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284 CHAPTER 7 COSTS AND COST MINIMIZATION
Conditions (A7.6) and (A7.7) are two equations in two unknowns, L and K. They are
identical to the conditions that we derived for an interior solution to the cost-
minimization problem using graphical arguments. The solution to these conditions is
found in the long-run input demand functions, L*(Q, w, r) and K *(Q, w, r).
For more on the use of Lagrange multipliers to solve problems of constrained op-
timization, see the Mathematical Appendix in this book.

DUALITY: “BACKING OUT” THE PRODUCTION
FUNCTION FROM THE INPUT DEMAND FUNCTIONS

This chapter has shown how we can start with a production function and derive the
input demand functions. But we can also reverse directions: If we start with input de-
mand functions, we can characterize the properties of a production function and
sometimes even write down the equation of the production function. This is because
duality The correspon- of duality, which refers to the correspondence between the production function and
dence between the produc- the input demand functions.
tion function and the input We will illustrate duality by backing out the production function from the input
demand functions. demand curves that we derived in Learning-By-Doing Exercise 7.4. We use that ex-
ample because we already know what the underlying production function is, and we
can thus confirm whether the production function we derive is correct. We will pro-
ceed in three steps.
• Step 1. Start with the labor demand function and solve for w in terms of Q, r,
and L:
Q r
L
50 A w
Q 2
w a b r
50L
• Step 2. Substitute the solution for w into the capital demand function
K (Q
50)1(w
r):
Q 2 1
Q ( 50L ) r 2
K a b
50 r
Q 2
which simplifies to K .
2500 L
1 1
• Step 3. Solve this expression for Q in terms of L and K: Q 50K L 2 .
2
If you go back to Learning-By-Doing Exercise 7.4, you will see that this is indeed the
production function from which we derived the input demand functions.
You might wonder why duality is important. Why would we care about deriving pro-
duction functions from input demand functions? We will discuss the significance of du-
ality in Chapter 8, after we have introduced the concept of a long-run total cost function.