Nir Dagan, Esther Hauk, and Albrecht Ritschl

**Due Monday, 25 May, 1998**

**1.** A firm has the production function:
`f(x,y)=x ^{1/4}y^{3/4}`. The prices of
the inputs are

- Find the conditional demand functions for the inputs.
- 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`w`._{x}=2, w_{y}=1 - 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 `x _{1}` or Kuwaiti oil

- Write the production function. Does it have decreasing, constant or increasing returns to scale?
- Find the conditional demand function when the inputs' prices are
`(w`._{1},w_{2})

**3.** A firm has the production function:
`f(x,y)=x ^{1/5}y^{4/5}`. The prices of
the inputs are

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