## Review

Key Concepts comparative statics envelope theorem expenditure function fundamental equation Hotelling's lemma implicit function implicit function theorem Roy's identity shadow wage rate Shephard's lemma Slutsky equation value function

1. (a) What does comparative-statics analysis do? (b) How does it do it?

2. What are the sufficient conditions under which we can earn out the comparative-statics analysis of a general economic model?

3. In a model based on an optimization problem, how may the second-order conditions help us in the comparative-statics analysis ?

4. How do we proceed if the sign of the comparative-statics effect cannot be determined unambiguously?

5. What is the value function in a constrained optimization problem? Name some economic examples of value functions.

6. State, prove, and explain the envelope theorem.

7. What does the envelope theorem tell us about the interpretation of a Lagrange multiplier in a constrained optimization problem? Give some examples of such Lagrange multipliers and their interpretations.

### Review Exercises

1. For each of the following constrained optimization problems, find the comparative-statics effects of a change in the «-variables, and derive and sketch the value functions:

(a) |
— JT, x2 |
subject to 2x\ + 4r2 = a | |

(b) |
max y |
= 2x) + 3X2 |
subject to aixf -1- = a2 |

(c) |
max y |
_ ,.0.25 ,.0.75 — 1 -l2 |
subject to 2xf + 5x| = a |

(d) |
min y |
- 2x: + 4x2 |
subject to x^x™5 = cc |

(e) |
max y |
= (jti + 2)(x2 4- 1 ) subject to + x2 = u | |

(f) |
min y |
= atxi +x2 |
subject to (xi 4- 2)(*2 + 0 = «2 |

Compare and explain the difference in the comparative-statics results for the following IS-LM models:

(a) Consumption function: C = 0.8 V + 30r Investment function: / = 7° + 0.1Y - Wr Demand for money: 30 + 0.2)' - 10.5r

(b) Consumption function: C = 0.8y - 30r Investment function: 7 = 7° + 0.1 Y - lOr Demand for money: 30 + 0.06Y - 60r

(c) Consumption function: C = 0.8y - 30r Investment function: 7 = 7° + 0.1Y - lOr Demand for money: 50 + 0.5 Y - 0.11 r

3. A consumer has the utility function u = + Pu(X21,X22), o < p < 1

where x,t is the amount of good «"=1,2 consumed in period / = 1,2. The prices of the goods are p\ and p2, and are the same in each period. The consumer's income in period t is m, and not necessarily equal in both periods.

(a) Assume first that the consumer has separate budget constraints in each period. Derive the indirect utility function and comment on its form. Interpret the Lagrange multipliers in this problem. Under what conditions •are they equal?

(b) Now assume it is possible to borrow or lend income between the two periods at a fixed interest rate r. Show that the consumer cannot be worse off as a result of this. Give conditions under which she is strictly better off. Obtain the indirect utility function in this case.

4. The demand functions for two goods are given by

where p\ and p2 are prices and y is consumers' income. The supply functions are

Carry out a general comparative-statics analysis of the effect of a change in consumers' income on the equilibrium prices of the goods.

5. A consumer has the strictly quasiconcave indirect utility function

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