Isoquants and Isocost Lines; Cost Function etc.

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1. Suppose that Natash'a utility function is given by

U(I)= Sq. Root of I

where U represents utility and I represents annual income in thousands of dollars.

A) Is natasha risk loving, risk neutral, risk averse?
B) Suppose that Natasha is currently earning an income of 10,000, and can earn that income next year with certainty. She is offered a chance to take a new job that offers 0.5 probability of earning 16,000 and a 0.5 probability of earning 5000. Should she take the new job.. explain.
C) in part B, would natasha be willing to buy insurance to protect against the variable income associated with the new job? If so, how much is she willing to pay for that insurance?

2. Suppose the production function for widgets has the form

Q=q(K,L)= 50K^0.3 L^0.7

Where q is the quantity of widgets per day, K is the quantity of capital input, and L is the quantity of labor input per day.

A) Does this production function have increasing, constant, or decreasing returns scale?
B) What is the marginal product of capital when the firm is suing 5 units of capital and 1 unit of labor.

C) What is the marginal product labor when the firm is using 5 units of labor and 1 unit of capital?

D) What is the total output when the firm is using 5 units of capital and 5 units of labor? Draw a isoquant representing this level of output?

E) what additional information would you need to determine which of the combinations on your isoquant is " best"? That is what would you want to know to choose the best combination of labor and capital to use in your production process?

3. Suppose that a firms production function is q= 10L^1/2K^1/2. The cost of a unit of labor (W) is $20 and the cost of a unit of capital (r) is $80.

A) The firm is currently producting 100 units of output, and has determined that the cost minimizing quantities of labr and capital are 20 and 5. Respectively. Graphically illustrate this situation on a graph using isoquants and isocost lines.

B) The firm now wants to increase output to 140 units. If capital is fixed in the short run, how much labor will the firm require? Illustrate this point on you graph and find the new cost

C) Graphically identiy the cost minimizing level of capital and labor in the long run if the firm wants to produce 140 units.

D) If the marginal rate of tenical substitution is K/L, find the optimal level of capital and labor required to produce the 140 units ofoutput

4.You manage a plant that mass produces engines by teams of workers using assembly machines. The technology is summarized by the production function.

q= 5 K L

where q is the number of engines per week, K is the number of assembly machines, and L is the number of labor teams. Each assembly machine rents for r=10,000 per week and each team costs w=5,000 per week. Engine costs are given by the cost of labor teams and machines, plus 2,000 per engine for raw materials. Your plant has a fixed installation of 5 assembly machines as part of its design.

A) What is the cost function for your plant- namely , how much would it cost to produce q engines? What are average and marginal costs for producing q engines? How do average costs vary with output?

B) How many teams are required to produce 250 engines? What is the average cost per engine?

C) You are asked to make recommendations for the design of a new production facility. What capital/labor ratio (K/L) should the new plant accommodate if it wants to minimize the total cost of producing any level of output q?