# Slutsky and Pareto Equations

1. Consider the problem of maximizing the utility function u=f(c)+g(s) subject to the budget constraint pc+s=y, where c is consumption and s is savings . Derive and interpret the Slutsky equations and explain what happens if g”(s)=0 .

We can make sense of the substitution and the income effects by this intuitive story. Suppose a consumer is consuming the optimal amount of two goods x and y, given his income and suddenly the price of x drops. The consumer will respond to this price change in two ways. First, as x becomes relatively cheaper the consumer will shift some of his consumption of y to x (assume x and y are not perfect complement). Second, as the price of x drops, even if the consumer does not make any consumption shift from y to x, he has
more purchasing power because of the savings that results from the price drop in x. This savings allows the consumer to buy more goods (x or y). The shift in consumption from y to x is the substitution effect, and the increase in purchasing power due to the savings is the income effect.

2. (a) Explain why the Pareto criterion does not provide a complete ordering of the ordinal utility space.

(b) “The competitive equilibrium is the only allocation where the gains from trade are exhausted”True or False? Explain your answer.

(a) Pareto Principle: Resource allocation Y is socially preferred to allocation Z if all individuals are at least as well off in Y as in Z, and at least one individual strictly prefers Y to Z.
Define a utility function to represent an individual’s preferences over resource allocations.
This is an ORDINAL utility function
a. Ordinally measurable
i. utility numbers for an individual convey information about that individual’s welfare BUT only convey information on the order, not intensity

b. Noncomparable (across individuals)
i. utility numbers generated by different individuals’ utility functions cannot be compared

An ordinal utility function is unique up to a positive, monotonic transformation.

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