A Producer Has Two Alternative Techniques to Produce a Unit

question 43

Essay

A producer has two alternative techniques to produce a unit of output out of labour. In the first, the production function is given by y₁(l₁)=l₁^0.5, and in the second it is given by y₂(l₂)=2l₂^0.5. The cost function c(y,w)of the producer gives the minimal expenditure required to produce a given total amount of y of the output (i.e.: y₁ +y₂
=y)at input price w.
a)Set up the cost minimisation problem and solve for the optimal choice of l₁ and l₂ as a function of the required output. Finally, derive the cost function for the producer.
b)Determine the firm supply under price- taking behaviour of the firm as a function of p and w.
c)Finally, suppose that p=10 and w=5. Determine the profit- maximising quantity produced and the associated level of profits.

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Definitions:

Slope

Slope is a measure of the steepness, incline, or grade of a line, often represented mathematically as the ratio of rise over run in the context of a graph.

Isocost Line

Represents all combinations of inputs that cost the same amount for a firm, illustrating budget constraints in production theory.

Price of Capital

The cost of using capital assets in production, which can include interest rates on loans or opportunity costs of using owned capital.

Price of Labor

The wage or salary paid for the services of an employee or worker; essentially the cost of hiring labor.