Assignment 1
1. Given the production function as:
y = 6.8 + 1.35x1 + 7.7x2 - 8.5x3 + 0.4x4 - 0.06x12 - 1.25x22 + 0.2x1x2 and assuming x3 = 0.32 and x4 = 4.25
i. Form production function in iso-quant set.
ii. Given the iso-quant equation and prices of inputs W1 = 0.4 and W2 = 4.4 , find the least cost combination of inputs.
iii. Find elasticity of substitution and explain its implications.
2. Consider a firm which purchases inputs x = (x1 x2 x3 ) in competitive markets at prices w = (w1 w2 w3). Let w1; w2; w3> 0. This firm's production function is y = x13/4 x21/2x31/2 be the input prices for good x1 and x2 respectively.
Then, answer the following questions.
a. Set up the cost minimization problem.
b. Solve this cost minimization problem you describe in (a), and derive the cost function.
c. What are properties of cost function?
d. Let p be the output price. Set up the profit maximization problem using your answer in (b).
e. Solve the profit maximization problem in (c), and derive the profit function, π (p; W1; W2).
f. What are the properties of profit function?
g. Derive the conditional factor demand equation for input x2 .
h. Is this firm facing: constant, increasing, decreasing or indeterminate returns to scale? Explain
3. Explain the relationship between average physical productivity and marginal physical productivity given the following equation. y = 25 + 2.5x + 0.5x2 - 0.12x3
4. Here is a possible cost function for a price-taking firm in standard notation:
c * (w1 ,w2 , y) = 8y2(w1w2 )1/2. Assume the input prices are w1 = 2 and w2 = 8 . Derive and graph the marginal and average cost curves as functions of output y.