# Consider the utility function u(x_1,x_2) = x_1 x_2. Suppose that the prices are given 1 for each...

## Question:

Consider the utility function {eq}u(x_1,x_2) = x_1 x_2 {/eq}. Suppose that the prices are given 1 for each good and the income is 10.

(1) Find out the optimal choice(s). Figure out the utility level at this optimal choice.

(2) Suppose now that the price of good 2 increases from 1 to 2. Figure out the value of the bundle that you obtained in (1) under the new prices. Draw the budget set associated with this value and the new prices.

(3) Find out the optimal choice(s) given the budget set in (2).

(4) Find the optimal choice(s) under the new prices and the initial income of 10.

(5) What is the (Slutsky) substitution effect on good 2? What is the income effect on good 2?

(6) Consider the utility level you calculated in (1). What is the bundle that gives you the same utility while minimizing the expenditure under the new prices?

(7) What is the (Hicksian) substitution effect on good 2?

(8) Suppose that the price of good 2 is now given {eq}p_2 > 0 {/eq} while the price of good 1 is fixed at 1. Figure out the Marshallian demand curve of good 2.

(9) Continuing from (8), figure out the Hicksian demand curve of good 2 associated with utility level 25.

(10) Is the Hicksian demand curve from (9) always steeper than the Marshallian demand curve from (9) at {eq}x_2 = 5 {/eq} in this example?

## Utility Maximization:

Utility is a fundamental concept in microeconomic theory. It measure the level of satisfaction of consumers while choosing and consuming a set of good, i.e. a product or a service. It measures preferences over a set of consumer choices. Utility function measures the satisfaction of a consumer as a function of consumption. It is widely used in microeconomics in order to analyse consumers' behaviours and consumers' preferences.