Problem 4 (Demand function, elasticities, substitution and in- come effects)

Zeno, a BEMACS student, is convinced that the mind can be healthy and works well only conditional on the body being healthy (as the ancient Latin dictum said Mens sana in corpore sano). For this reason, every week he combines2 hours of martial arts, x1, with 3 hours of intensive private lessons of microeconomics with professor Botticelli, x2. Let the price of martial arts be p1, the price of microeconomics lessons be p2, and Zenos income be m.

1. What type of goods are x1 and x2 for Zeno? Explain your answer.

2. Derive analytically Zenos utility function for the two goods.

3. On a properly labeled graph, show three indifference curves.

4. Calculate Zenos MRS.

5. Derive analytically the demand curves for martial arts lessons and microeconomics lessons, Q1 and Q2, as functions of the prices and Zenos income.

6. Calculate own price and cross-price elasticities of the demand function for martial arts.

7. Is the sign of the cross-price elasticity in line with your expectations given Zenos utility function? Explain why.

8. Suppose that this month, Zeno has a budget of 240 e available to spend on these two activities. Also, suppose p1 = 15 e p2 = 10 e. Write Zenos budget constraint equation.

9. On the graph above with the indifference curves, draw the budget constraint clearly indicating the intercepts and the slope.

10. Derive analytically Zenos optimal consumption bundle x = (x1, x2).

11. On the graph show the optimal bundle x.

12. The new Bocconi Sport Center has just opened and they are offering special prices to Bocconi students. Zeno can now practice martial arts at a price p1 = 10 e for each lesson. Derive analytically Zenos new optimal consumption bundle x = (x, x).

13. To which effectsubstitution or incomedo you attribute the change in the optimal consumption bundle due to the price decrease in martial arts lessons?

14. Derive analytically the compensated (Hicksian) demand curves for the two goods, H1 and H2.

15. What happens to the compensated demand functions H1 and H2 when p1 decreases? Explain your answer.

16. Given the discount on martial arts lessons, Zeno realizes that he could keep his previous level of happiness and save money. Which bundle x = (x, x) would guarantee to him the same level 12 of utility he had before the above mentioned decrease in price? Hint: use the Hicksian demand functions to get the answer.

17. Which is the amount of money Zeno can save keeping his utility constant at the initial level?