Microeconomics cost function question.

In summary, the problem involves finding the cost function for producing q engines in a plant that is cost minimizing. The cost function is linear and the production function is concave. The firm must satisfy the tangency condition by finding the point where the slope of the cost curve is equal to the slope of the production function. This can be graphed using iso-cost and isoquant curves. Marginal in economics refers to the first derivative of a function, and is equivalent to velocity in mathematics.
  • #1
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?
 
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  • #2
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.
 

1. What is a cost function in microeconomics?

A cost function in microeconomics is a mathematical representation of the relationship between the cost of production and the level of output. It shows the minimum cost of producing a certain level of output, taking into account the prices of inputs and the production technology.

2. How is a cost function calculated?

A cost function is typically calculated by analyzing the prices of inputs, such as labor and materials, and the production technology used to produce a certain level of output. The function is then derived by determining the most efficient way to produce that level of output at the lowest cost.

3. What is the significance of a cost function in microeconomics?

A cost function is important in microeconomics because it helps businesses and economists make decisions about production, such as determining the optimal level of output and the most cost-effective production methods. It also allows for the analysis of costs and the impact on profitability.

4. How does a cost function affect a firm's pricing decisions?

A cost function can have a significant impact on a firm's pricing decisions. By understanding the relationship between production costs and output, businesses can determine the optimal price point for their products to maximize profits. Additionally, cost functions can also influence decisions about cost-cutting measures and efficiency improvements.

5. Can a cost function change over time?

Yes, a cost function can change over time due to various factors such as changes in input prices, technological advancements, and shifts in demand. This is why it is important for businesses to regularly review and update their cost functions to stay competitive in the market.

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