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# maximize utility subject to the budget constraint

1. Consider the utility functions of the form . Show that the implied
demand curves are

(see attached file for equations)

2. Suppose a consumer will have income this year and next year. He or she consumes this year and next year, being able to borrow and lend at interest rate . Assume the consumer maximizes the utility of consumption over these two years.

a. Derive the comparative statics for this problem. Will an increase in this year's income necessarily lead to an increase in consumption this year?

b. Prove that the consumer will be better off or worse off if the interest rate rises if he/she was net saver or dissever this year.

#### Solution Preview

Please see the attached file.

1. Consider the utility functions of the form . Show that the implied
demand curves are

Find and and verify that

Mathematically, we can show this result using Lagrangian technique - maximize utility subject to the budget constraint

We can set up Lagrangian for utility max as:

Setting partial derivatives equal to zero yields system of n + 1 equations that define interior solution

Optimal xi is given by solution to usual first-order condition if FOC holds with equality

Here we are giving an example, which will be helpful to solve the above problem
Here the problem is to maximize,

Forming the Langrangian equation we get,

Where x and y are the two commodities, a and c are the price of the two commodities.
b is the money income &  is the Lanrangian multiplier.
First order conditions are,

Dividing (i) by (ii) yields

Using (iv) to substitute out y from (iii) yields x

and ...

#### Solution Summary

Using Lagrangian technique, the expert maximizes the utility subject to the budget constraint. The interest rates for consumer maximization are determined.

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