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1 Cost Minimization: An Example

1.1 Setup

Suppose that Johns TV Shop has production function Q = F(K, L) = KL. The wage paidto labor is w = 10 and the rental price of capital is r = 20. Given Q = F(K, L) = KL, wecalculate the marginal product of labor as M PL = K and similarly the marginal productof capital M PK = L (You wont need to calculate these for this class; we will provide themfor you). Therefore, the marginal rate of technical substitution of K for L is

MRTSLK = M PL

M PK= K

L

Note that the MRTS is diminishing because the MRTS falls as L rises. This meansthat the isoquants are convex and the solution will lie in the interior of the positive orthant.

1.2 The LONG-RUN Problem

Johns problem is to minimize his costs subject to a quantity constraint:

M in(K,L)wL + rK s.t. Q = KL

The result of Johns problem is a labor and capital demand function, L(Q) and K(Q),which tells him how much labor and capital to hire depending upon how much Q he decidesto produce. We can then use L(Q) and K(Q) to find his total cost function, T C(Q) =wL(Q) + rK(Q).

Here is a graphical representation of the problem. Notice that the isoquant is fixed atlevel Q, and the isocost line shifts downward until it is just tangent to the isoquant.

(Notice that this is a long-run problem because John can choose both K and L. In theshort-run, at least one of these factors is fixed, which greatly simplifies the problem)

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1.3 The LONG-RUN Solution

The interior solution to the cost minimization problem is characterized by two equations:

(1) MRTS= w

r

K

L = 1

2

Recall that this equation means that the gradient of the isocost line and the isoquant curveare the same. This means that John is hiring labor and capital in the optimal ratio.

(2) Q = KL

Recall that this equation means that Johns TV Shop is producing a set level of quantityusing its production function.

Together, these two equations say that John is producing a fixed quantity Q using the

most cost effective allocation of K and L.Now we have a system of two equations and two unknowns. We can solve this system

by solving for one variable using one equation, and then plugging into the other equation.Solving equation (1) for K yields L = 2K.

Plugging that into equation (2) yields Q = 2K2. Solving for K we find that K(Q) =Q2 . To find L(Q), simply plug K(Q) =

Q2 into either equation. Using equation (1)

yields L(Q) = 2

Q2 =

2Q.

We can now write the total cost function for Johns TV Shop. It is

T C(Q) = wL(Q) + rK(Q) = 102Q + 20Q

2

= 102Q + 102Q

T C(Q) = 20

2Q

Note that there are no fixed costs here because all inputs are variable in the long-run.

1.4 The SHORT-RUN Problem

In the short run we generally assume that one factor is fixed. Usually it is capital. Thuslets assume that K is fixed at 5. Johns cost minimization problem is then

M inL 10L + 20(5) s.t. Q = 5L

The problem is greatly simplified because John can only choose how many laborers tohire. There is no substitutability between K and L because K is fixed.

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1.5 The SHORT-RUN Solution

Using the constraint, we solve for L and find that L(Q) = Q5 . Plugging this into the cost

function we find that T C(Q) = 10Q5 + 20(5) = 2Q +100. Note that the $100 in fixed costs

comes from the fixed capital level. All the variable costs come from labor being hired.

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