Microeconomics cost function question.

[itsmylifenow]

q = 3K^0.5 L^0.5

where q is the number of engines per week, K is the number of machines, and L is the number of labor. Each assembly machine rents
for r = $9000 per week, and each team costs w = $4000 per week. Engine
costs are given by the cost of labor teams and machines.

Suppose the plant is cost minimizing.

What is the cost function?

How much would it cost to produce q engines?

What are average and marginal costs for producing q engines?

What is the cost minimizing input combination of producing q =
1800?

Does the production function for this plant exhibit increasing, con-
stant or decreasing returns to scale?

 

[talk2glenn]

This may be a little late, but in case you still need help...

This is a pretty simply problem. You are trying to satisfy the tangency condition, eg, find the point at which the slope of the cost curve is equal to the slope of the production function. The cost function will be linear in paramters, and is just a straight line, with a slope given by the ratio of the prices of the factors of production (capital and labor). The production function will be concaved with a variable slope, equal to the ratio of the marginal products of the factors of production.

That is to say, a firm with a fixed budget optimizes profit by manufacturing on the highest production curve possible, without exceeding its budget constraint. A firm with a target output, as here, minimizes costs by operating on the lowest budget line possible, given its production curve. These are called iso-cost and isoquant curves in your textbook, for reference. You can graph these visually to help understand the problem, but only need to set up the equalities, algebraically, to solve the problem.

Marginal in economics always refers to the first derivative of some function, with respect to some variable. It is the equivalent of velocity in mathematics. Eg, marginal cost of engines is the change in total costs from a one unit increase in q, and the marginal product of labor is the change in total production Q from a one unit increase in L.