Question

. This problem examines the effects of an increase in θon the optimal choice of the fraction of risky assets (λ) in the portfolio choice model presented in Lecture 10. Recall that the problem is what fraction (λ) of wealth (W) a consumer should hold in risky assets. The risk-free asset has a gross return equal to r, and the risky asset has a gross return equal to θ with probability p (and 0 with probability 1-p). The consumer’s utility of wealth is given by U(W) = lnW. Recall that we showed that the optimal level of λ is given by: λ*=(pθ– r) /(θ– r)

Pick what you consider to be realistic values for the parameters p and r, and then see

if λ* increases as you increase θ using Excel. Submit an Excel spreadsheet containing a scatterplot of (x, y) = (θ,λ*) to demonstrate whether increasing the gross return causes people to invest more heavily in the risky asset. Decision Theory Problem with a Risk Neutral Firm

3. Consider a risk-neutral firm that operates for two periods with a production function that depends only on the amount of labor hired: f(L) = 100L^ 1/2 Assume that the

interest rate (r) is equal to 5%. The wage in the first period is equal to $10 per hour, but the second period’s wage is either $10 (with probability 0.4) or $20 (with probability 0.6). The current price for the firm’s output is Po =$20. In the second period, the price is either P1=$20 (with probability ½) or P1=$30 (with probability ½).

a) Assuming the firm cannot store output over time, what is the expected value of the presented discounted value of lifetime profits for this firm?

(Hint: Recall that if you receive a cash flow = CF in T years in the future, the value of this cash flow today (i.e. the present discounted value of CF) is equal to CF/(1+r)^T.)

b) Find the expected profit maximizing choice of labor in both periods.

c) Given the levels of labor you found in part b, what are the corresponding levels of output?

d) Now suppose the firm hires union workers that demand a two period contract. In other words, the same amount of labor must be hired in both periods (but the wages may be different in the two periods). How will the firm’s optimal choice of labor be affected by this labor market rigidity?

e) Now suppose the firm has access to inventory storage that costs h(s) = s2 in the second period, where s is the amount stored in inventories. Assuming labor market rigidity as in part d, how does the introduction of inventory storage change the firm’s behavior?

f) Explain the intuition behind the firm’s behavior in part e

Answer 1

Following data points were used to calculate the value of the λ

Values used:

p = 0.66

r = 2.74

θ	λ
0	1
1	1.195402
2	1.918919
3	-2.92308
4	-0.07937
5	0.247788
6	0.374233
7	0.441315
8	0.48289
9	0.511182
10	0.53168
11	0.547215
12	0.559395
13	0.569201
14	0.577265
15	0.584013
16	0.589744
17	0.59467
18	0.598952
19	0.602706
20	0.606025
21	0.608981
22	0.61163
23	0.614018
24	0.616181
25	0.618149
26	0.619948
27	0.621599
28	0.62312
29	0.624524
30	0.625825

Steps followed for scatter plot:

1. Enter the value of theta from 0-30

2. Get the value of λ from the formula = =(0.66*D19 - 2.74)/(D19 -2.74) [ D19 is the address of the theta cell]

3. Go to Insert> Charts> Scatterplot(last icon)

4. Following Scatter plot will form

We can analyze that the lambda will increase to max and then decline to negative and then will increase continuously.