Reference no: EM132884186

1 The Gauntlet

Suppose Justin exclusively derives utility from the consumption of two goods, namely, x1 and x2 with prices p1 and p2 respectively. His utility function is given as U (x₁, x₂) = (x^0.5 + x^0.5)². The domain of the function is that x₁ > 0, x₂ > 0, p₁ > 0, and p₂ > 0.

1. Calculate the marginal utility derived from consuming x1 and x2. Do these marginal utilities support the assumption of non-satiation?

2. Does Justin's consumption preference also indicate that his preferences obey the Law of Diminishing Marginal Utility? Show your solution.

3. Obtain the marginal rate of substitution for good 1 and good 2 (MRS12)

4. Solve for the values of the Marshallian Demand Functions for good 1 and good 2

5. Demonstrate that the Marshallian Demand Functions are homogenous of degree zero in prices and income.

6. Demonstrate that the Marshallian Demand Functions satisfy Walras' Law.

7. Obtain Justin's Indirect Utility Function V (p1, p2, m).

8. Demonstrate that the Indirect Utility Function is homogenous of degree zero in prices and income.

9. Demonstrate that the Indirect Utility Function is strictly increasing in m.

10. Demonstrate that the Indirect Utility Function is strictly decreasing in p1.

11. Demonstrate that the Indirect Utility Function satisfies Roy's Identity for good 1.

12. Using the Indirect Utility Function obtained in 7, derive the expendi- ture function e(p1, p2, U ).

13. Using the expenditure function e(p1, p2, U ), derive the Hicksian De- mand Functions for good 1 and good 2 using Shephard's Lemma.

14. Verify that the value of the expenditure function e(p1, p2, U ) is equal to zero when U takes on the lowest level of utility.

15. Demonstrate that the expenditure function is homogenous of degree one in prices.

16. Demonstrate that the expenditure function e(p1, p2, U ) is strictly in- creasing in U .

17. Demonstrate that the expenditure function e(p1, p2, U ) is strictly in- creasing in p1.

Suppose Justin has an income of Php 200 and the current prices of good 1 and good 2 are Php 10 and Php 5, respectively.

18. How much of good 1 will Justin buy?

19. Calculate the uncompensated own-price elasticity of demand for good 1. Is the demand for good 1 price elastic, inelastic, or unit elastic?

20. Calculate the income elasticity of demand for good 1. Is good 1 normal or inferior? Why? If good 1 is normal, is it a luxury good or a necessity good? Why?

21. Calculate her compensated own-price elasticity of demand for good 1. Interpret this result. Compare this to your answer in part 19 and explain the difference.

22. Demonstrate that the Marshallian Demand Functions satisfy Cournot Aggregation

23. Demonstrate that the Marshallian Demand Functions satisfy Engel Ag- gregation Suppose the price of good 1 increases by Php 5

24. Given this price increase, calculate the Total Effect, Substitution Effect, and Income effect using the (Discrete) Slutsky equation

25. Calculate her compensating variation for the increase in p1

26. Calculate her equivalent variation for the increase in p1

27. Find the equation of her Marshallian demand curve for good 1 given that the price of good 2 is Php 10 and her income is 200. Based on this equation, calculate the change in her Marshallian consumer surplus due to the increase in p1.

2 A Bit of Irrationality

Suppose we have a consumer who has unique preferences in which his/her entire preference is just contingent on the quantity of good 1 consumed. Sup- pose he/she faces the choice of two bundles, namely, A( 1 , 100) and B(0, +∞) where n → ∞.

1. Prove that the consumer's preferences are not continous. Explain this intuitively with accompanying solutions

2. If a third (average) bundle, say C( 1 , 1000) were to be added to the comparison, would the preferences now be continuous?

3 Bonus

Which is steeper, the Marshallian Demand Curve or the Hicksian Demand Curve? Use the Slutsky Equation, words, and sound economic intuition to argue your point.

Attachment:- The Gauntlet.rar