# Problem 69

**Problem 69**

In the electricity pricing model, the demand functions have positive and negative coefficients of prices. The negative coefficients indicate that as the price of a product increases, demand for that product decreases. thatthatThe positive coefficients indicate that as the price of a product increases, demand for the other product other other increases.

- Increase the magnitudes of the negative coefficients from –0.013 and –0.015 to –0.018 and –0.023, and rerun Solver. Are the changes in the optimal solution intuitive? Explain.
- Increase the magnitudes of the positive coefficients from 0.005 and 0.003 to 0.007 and 0.005, and rerun Solver. Are the changes in the optimal solution intuitive? Explain.
- Make the changes in parts a and b simultaneously and rerun Solver. What happens now?

**Problem 70**

In the electricity pricing model, we assumed that the capacity level is a decision variable. Assume now that capacity has already been set at 0.65 million of mWh . (Note that the cost of capacity is now a sunk cost, so it is irrelevant to the decision problem.) Change the model appropriately and run Solver. Then use SolverTable to see how sensitive the optimal solution is to the capacity level, letting it vary over some relevant range. Does it appear that the optimal prices will be set so that demand is always equal to capacity for at least one of the two periods of the day?

**Problem 72**

Add a new stock, stock 4, to the portfolio optimiza-tion model. Assume that the estimated mean and standard deviation of return for stock 4 are 0.125 and 0.175, respectively. Also, assume the correlations between stock 4 and the original three stocks are 0.3, 0.5, and 0.8. Run Solver on the modified model, where the required expected portfolio return is again 0.12. Is stock 4 in the optimal portfolio? Then run SolverTable as in the example. Is stock 4 in any of the optimal portfolios on the efficient frontier?

**Problem 94**

You have $50,000 to invest in three stocks. Let Ri be the random variable representing the annual return on $1 invested in stock i. For example, if Ri=0.12, then $1 invested in stock i at the beginning of a year is worth $1.12 at the end of the year. The means are E(R1)=0.14, E(R2)=0.11, and E(R3)=0.10. The variances are VaVVrR1=0.20, VaVVrR2=0.08, and VaVVrR3=0.18. The correlations are r12=0.8, r13=0.7, and r23=0.9. Determine the minimum-variance portfolio that attains an expected annual return of at least 0.12.

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