Question

Suppose the process of producing lightweight parkas by Polly’s Parkas is described by the function q = 10K^0.7(L - 40)^0.3 where q is the number of parkas produced, K the number of computerized stitching-machine hours, and L the number of person-hours of labor. In addition to capital and labor, $15 worth of raw materials is used in the production of each parka.

1. Write total cost function of producing parkas in terms of the quantity produced, and labor and capital used in production.

2. By minimizing cost subject to the production function, derive the cost-minimizing demands for K and L as a function of output (q), wage rates (w), and rental rates on machines (r). Use these results to derive the total cost function: that is, costs as a function of q, r, w, and the constant $10 per unit materials cost.

(Hint: to find the cost function: step 1: set up the minimization problem, with total cost as the objective function, and the constraint production to achieve the quantity q. step 2: Solve the minimization problem using the Lagrange to find the optimal L and K that would achieve the minimum cost. step 3: Substitute the optimal L and K back into the total cost function.)

Answer 1

1. The total cost function-

We are given that the quantity produced is q. We also know that it costs $15 in terms of raw materials for each Parka. Which means 15q is the total raw material cost.

Wage rate is given as w, and rental cost r (which is basically costing capital). So, the total cost function is

15q+wL+rK.

2. Cost Minimization demands for K and L-

To solve part 2, we first have to set-up a Lagrangian. What is a Lagrangian, in economic terms? Its simply a function that combines the objective function that you have to optimize for (lets say cost) and the constraints (lets say constraints on labor or capital etc.). When we solve the lagrangian, we are optimizing the objective function while keeping the constraints in mind.

So first we need to identify the objective function. What is our objective function? Its the total cost (as given in hint). This, as already esablished in part 1, is

10q+wL+rK (here its 10q rather than 15q because the materials cost has been changed to 10q in the second part of the question)

The second part of the lagrangian are the constraints. What are our constraints? Its the maximum we can produce. We are given that

q = 10K^0.7(L - 40)^0.3

So our constraint becomes 10K^0.7(L - 40)^0.3 -q After all, our production cant be negative, can it?

The next part I will solve on paper and post the image as it will involve lots of functions. Before that, you will notice that our lagrangian contains a . What the hell is that? Thats the lagrangian multiplier. What it tells you is that how much the objective being optimized will change when one unit of constraints is changed.

So our Lagrangian becomes-