a) Do the following production functions exhibit constant returns to scale, increasing returns to scale, or decreasing returns to scale? For full credit, show why.

1) Q= 10L^ 0.5K^0.3

2) Q= 10L^0.5K^0.5

3) Q= 10L^0.5K^0.7

4) Q= min{K, L}

b) Which objects pin down a_LC and a_KC? Explain carefully.

c) Why does labor being mobile across sectors automatically imply revenue maximization for firms? Explain carefully.

Suppose that we have a single input variable X and all possible values of X is {0, 1, 2}. Also, suppose that we have two groups for outcomes (i.e., Y = 1, 2). It is known that the X|Y = 1 has Binomial(2, 0.7) and X|Y = 2 has the discrete uniform. In addition, P(Y = 1) = 0.3.

(1) Predict Y for X = 0, 1, 2, respectively, using the Bayes classifier.

(2) Compute the overall Bayes error rate.

In a recent 5-year period, mutual fund manager Goldie Touch produced the following percentage rates of return for the Mesozoic Fund. Rates of return on the market index are given for comparison. Calculate:

1. the average return on both the fund and the index
2. the standard deviation of the returns on each.
3. Did Ms. Goldie Touch do better or worse than the market index on these measures?   

   	1	2	3	4	5
   Fund	-1.2	+24.8	+40.7	+11.1	+0.3
   Market Index	-0.9	+16.0	+31.7	+10.9	-0.7

Consider a portfolio dependent on the price of a single asset that is delta neutral, with a gamma of -6000 and a vega of -9600. Suppose that a traded option (called Option 1) with a delta of 0.3, a gamma of 0.5 and a vega of 1.0 is available.

1. How could the portfolio be made delta and gamma neutral?
2. How could the portfolio be made delta and vega neutral?
3. If another traded option (called Option 2) with a delta of 0.7, a gamma of 1.2 and a vega of 1.6 is available. How could the portfolio be made delta, gamma and vega neutral?

Sally measures the amount of CO₂ in a pint soda bottle by weight loss; she shakes the bottle vigorously and then slowly opens the cap, letting the gas escape. Three measurements on Brand X indicate a loss of 2.2 ± 0.2 g of gas, while three measurements on Brand Y shows a loss 2.4 ± 0.3 g. (The ± indicates standard error.) Do Brands X and Y have different amount of CO₂?

The demand, D, for parts at a repair bench per day can be described by the following discrete PMF:

Generate the demand for the first 4 days using the following numbers (starting with the first row)

A point charge q = 40 nC has a velocity 3* 106 m/s in the direction av = 0.2 ax + 0.75 ay – 0.3 az. Calculate;
a) The magnitude of the force on the charge due to the field B = 3ax + - 4ay + 6az mT.
b) The magnitude of the force on the charge due to the field E = -2ax + 4ay + 5az kV/m
c) The Lorentz force due to the two fields in (i) and (ii) above.

Fill out table and show all calculations

Final             Volume of              Volume of               Volume of       Final volume (uL)

[PNPP] mM   0.5mM PNPP(ul)   0.2M Tris-HCl(ul)   enzyme (uL)

0.01                                                                            100                        1500

0.02                                                                             100                       1500

0.04                                                                             100                         1500

0.06                                                                            100                            1500

0.08                                                                            100                              1500

0.1                                                                           100                             1500

0.2                                                                              100                               1500

0.3                                                                           100                               1500

0.4                                                                             100                                1500

Consider three scenarios with the probabilities given below. Let the returns on two different stocks in these scenarios be as follows: Scenario Probability return K1 return K2 ω1 0.2 −10% −30% ω2 0.5 0% 20% ω3 0.3 10% 50% If a portfolio has 60% of funds invested in stock 1 and 40% of funds invested in stock 2, find the risk σV for this portfolio. (Need explanation not just spreadsheet)

you are evaluating a new project and need an estimate for your project's beta. You have identified the following information about three firms with comparable projects.

The unlevered beta for Nod is closest to :

A.1.00

B.0.90

C.0.95

D.1.10