Question

5. A perfectly competitive firm is trying to adjust to a recent decrease in consumer demand by adjusting their levels of production. The firm has more than enough existing capital to meet any reasonable production levels, so must only consider how much labour to hire. Assume the cost of using this capital is zero. The price of a unit of labour (the wage rate) is z, the quantity of labour used is O, and the price of a unit of output is s. Labour use is transformed into output at a rate of { = ln(O), where { is the quantity of output.

(a) What are the firm’s total cost, total revenue, marginal cost, and marginal revenue functions?

(b) What is the profit maximising level of labour use? How do we know for sure this is the profit maximising level?

(c) How does the profit maximising level of labour use change with wages and prices?

(d) Write down the indirect profit function of the firm, and investigate the eect of increases in s and z upon the value of optimal profits. Are the signs of those eects what you would expect?

(e) Is the indirect profit function concave, convex or linear in price? Is it concave, convex or linear in the wage rate? From your answers to these questions, can you conclude whether the indirect profit function is concave or convex in the vector (s> z)? (f) Assuming that the indirect profit function is convex in the vector (s> z), sketch a contour of the indirect profit function in the space of the output price and the wage rate, and indicate the corresponding better set.

Answer 1

a)  The Total Cost is the actual cost incurred in the production of a given level of output. In other words, the total expenses (cost) incurred, both explicit and implicit, on the resources to obtain a certain level of output is called the total cost.

Total revenue is the total receipts a seller can obtain from selling goods or services to buyers. It can be written as P × Q, which is the price of the goods multiplied by the quantity of the sold goods. Cost received by sellers from buyer by selling number of products is known as TOTAL REVENUE.

The marginal cost is the cost of producing one more unit of a good. Marginal cost includes all of the costs that vary with the level of production. For example, if a company needs to build a new factory in order to produce more goods, the cost of building the factory is a marginal cost.

Marginal revenue is the incremental revenue generated from each additional unit. It is the rate at which total revenue changes. It equals the slope of the revenue curve and first derivative of the revenue function.

b) The marginal revenue productivity theory states that a profit maximizing firm will hire workers up to the point where the marginal revenue product is equal to the wage rate. The change in output from hiring one more employee is not limited to that directly attributable to the additional worker.

This equilibrium price is determined by finding the profit maximizing level of output where marginal revenue equals marginal cost (point c)and then looking at the demand curve to find the price at which the profit maximizing level of output will be demanded. Monopoly profits and losses.

Marginal product of labor is a measurement of a change in output when additional labor is added. However, all other factors remain constant. To calculate marginal product of labor you simply divide the change in total product by the change in labor.

c)

Marginal revenue product of labour (MRPL) is the extra revenue generated when an additional worker is employed

Formula for MRPL is:

MRPL = marginal product of labour x marginal revenue

Marginal Revenue Product Calculation

In this example, we are assuming here that the firm employing labour is operating in a perfectly competitive market so that each unit of output sold generates a revenue of $20.

Firms are assumed to be profit maximisers and they will choose a level of employment that maximises profit. The MRPL curve is the demand curve for labour. MRPL falls when diminishing returns set in.

d) Producer engaging in indirect exchange ... We should interpret this as a one-output, one-input production function, thus x is the only input and ... The profit function maps particular factor prices to the maximum profit levels achievable at those .

The indirect revenue function is characterized and its duality relationship with the indirect output distance function is presented. Similarly, the indirect cost function and its duality relationship with the indirect input distance function is given. In Section 5.2, various envelope results are reported including a cost indirect version of Shephard’s lemma and some additional shadowing pricing formulae. In Section 5.3, a nonparametric cost indirect revenue function is applied to cost benefit analysis.

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