# Problem set 6 of Economic Theory I

Nir Dagan, Esther Hauk, and Albrecht Ritschl

Due Monday, 25 May, 1998

1. A firm has the production function: f(x,y)=x1/4y3/4. The prices of the inputs are (wx,wy).

1. Find the conditional demand functions for the inputs.
2. Find the quantity demanded of the inputs, and the quantity of the output that maximize profits when the price of the output is p=8 and of the inputs wx=2, wy=1.
3. Does the production function exhibit increasing, constant, or decreasing returns to scale?

2. A refinery produces gasoline from crude oil. It may use Norwegian oil x1 or Kuwaiti oil x2. From one litre of Norwegian oil one can extract one tenth of a litre of gasoline, and from one litre of Kuwaiti oil one can extract half a litre of gasoline.

1. Write the production function. Does it have decreasing, constant or increasing returns to scale?
2. Find the conditional demand function when the inputs' prices are (w1,w2).

3. A firm has the production function: f(x,y)=x1/5y4/5. The prices of the inputs are (wx,wy).

1. What is the technical rate of substitution in (x,y)=(50,30)?
2. What are the inputs' and output quantities that maximize profits when p=7 and wx=1, wy=3.
3. The government taxes every unit of x with t dollars. How this policy will affect the firms optimal decision?
4. In the short run the firm cannot change the quantity of y. Assume y=4. Derive and draw the firms short run production function.