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 l₁and l₂as 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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The Answer of 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 l₁and l₂as 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.

A Producer Has Two Alternative Techniques to Produce a Unit