Cost Minimization PDF

10 Cost Minimization In the last notes, we introduced the idea of a production function and some key properties (Returns to scale, MRTS, isoquants) of these functions. In these notes, we will connect the concepts from the last notes to a cost minimization problem for a...

10 Cost Minimization In the last notes, we introduced the idea of a production function and some key properties (Returns to scale, MRTS, isoquants) of these functions. In these notes, we will connect the concepts from the last notes to a cost minimization problem for a firm. This problem will be closely related to the consumer’s expenditure minimization problem. Instead of a consumer trying to find the cheapest way to reach a level of utility, we will look for the firm’s least expensive way to reach a certain level of production. Concepts Covered The Firm’s Cost Minimization Problem Graphical Analysis Contingent Demand Cost Functions 10.1 The Firm’s Cost Minimization Problem Setup Each firm will choose the quantity of labor and capital they want to hire to reach a certain level of production Q̄. In order to hire a unit of labor, the firm has to pay a wage rate w (this could be measured in hours or per worker wages) and to rent a unit of capital, they have to pay a rental rate v. We can then write the firm’s cost of inputs as wL + vK The consumer will try to minimize this cost given that they produce at least quantity Q̄. Formally, we can set up the consumer’s problem as min wL + vK K,L subject to Q̄ = F (K, L) Setting up the Lagrangian we have L = wL + vK + λ(Q̄ − F (K, L)) 1 Solving the Problem Notice that this problem is set up almost the same way as the expenditure minimization problem from consumer theory. The only differences are the variable names and interpretations. As a result, we can solve the problem in the exact same way. Taking first order conditions ∂L ∂F ∂F =w−λ = 0 =⇒ w = λ ∂L ∂L ∂L ∂L ∂F ∂F =v−λ = 0 =⇒ v = λ ∂K ∂K ∂K ∂L = Q̄ − F (K, L) = 0 =⇒ Q̄ = F (K, L) ∂λ Hopefully you can already guess what happens next. Let’s divide the first two equations to get ∂F w ∂L = ∂F v ∂K Thinking of w as the price of labor and v as the price of capital, all this equation says is the price ratio should be equal to the MRTS. 10.2 Graphical Analysis Although we can interpret the firm’s cost minimization problem as a close relative of the con- sumer’s expenditure minimization problem, it might be helpful to review some of the results using our production related language. First, we can plot an isoquant that represents the amount of output the firm wants to produce. We will stick with the Cobb-Douglas production function, F = K 1/2 L1/2 and assume the firm wants to produce 4 units of output. Then we plot cost lines (assume w = 1 and v = 4). The solution once again occurs at the tangent point. 10 8 6 K 4 2 0 0 2 4 6 8 10 12 14 16 L Note that these graphs are exactly the same as those in the expenditure minimization notes except for the variable names (x has been replaced by L and y by K). 2 10.3 Contingent Demand The next step is to find the firm’s demand for capital and labor as a function of wages, rental rate of capital, and the production level they want to achieve. We will call this function the firm’s contingent demand for an input L∗ = L(w, v, Q̄) K ∗ = K(w, v, Q̄) We will solve for these in the same way as we did our Hicksian demand. Returning to the results of the Lagrangian, we had that w M RT S = v And Q̄ = F (K, L) Which gives us two equations to solve for our contingent demand functions 10.4 Cost Functions Definitions In our consumer problem, we calculated the expenditure function by plugging in Hicksian demands into expenditure. Here, we calculate a total cost function by plugging in contingent demands into total cost. This process gives us total cost as a function of wage, rental rate, and output. C = C(w, v, Q) = vK ∗ (w, v, Q̄) + wL∗ (w, v, Q̄) However, unlike the consumer problem, the quantity we choose now has some physical meaning. Therefore, we can do a bit more with this function. We will define the average cost curve as the total cost divided by the quantity produced C(w, v, Q) AC = Q And the marginal cost curve as the derivative of the total cost curve ∂C(w, v, Q) MC = ∂Q Marginal cost can be thought of as the cost of producing the next unit of a good. Example Let’s assume that total cost is given by the equation C = Q2 + 2Q + 4 Find the marginal cost and average cost functions. Find the value of Q that minimizes the average cost curve (assuming Q is strictly positive). What is the marginal cost at this value? What is the average cost? (Solution on next page) 3 Example Solution First let’s find the marginal cost curve ∂C MC = = 2Q + 2 ∂Q And the average cost curve C AC = = Q + 2 + 4/Q Q To minimize average cost, take the derivative and set it equal to 0 4 1− = 0 =⇒ Q2 = 4 =⇒ Q = 2 Q2 Plug this into marginal cost M C = 2(2) + 2 = 6 And average cost AC = 2 + 2 + 4/2 = 6 In this example, average cost is at its minimum when it is equal to marginal cost. This relationship is not a coincidence. To see why, think about what marginal cost means. If marginal cost is below average cost, it means the next unit of production costs less than the previous units, so it pulls the average down and when marginal is above average, the next unit costs more, pulling the average up. The minimum must then occur when they are equal 4