# Problem set 3 of Economic Theory I

Nir Dagan, Esther Hauk, and Albrecht Ritschl

Due Monday, 4 May, 1998

1. Jaume has the utility function:

```
__
u(x1,x2) = \/x1 + x2 ((Square root of x1) plus x2)
```
1. Are his preferences convex?
2. Calculate the MRS at the bundles (4,1) and (1,5)
3. Draw a map of indifference curves of Juame's preference.
4. Find the optimal choice when p1 = p2 and m = 40

2. A consumer has the utility function: u(x1,x2) = log(x1) + x2.

Find the demand functions and the consumer's optimal choices in the following cases:

1. p1 = 1, p2 = 1, and m = 20
2. p1 = 1, p2 = 1, and m = 10
3. p1 = 2, p2 = 1, and m = 10

3. Nuria has 10000 pesetas. She may spend them on books (x1) or on beer (x2). Her utility function is u(x1,x2) = 2x1*x2.

1. Find her optimal choice when the market prices are p1 = p2 = 100
2. A certain bookstore has a different pricing system. There is an entry fee of 5000 pesetas, but every book costs only 25 pesetas. Would she prefer to enter this store or to buy in the market?

4. A consumer has the utility function: u(x1,x2) = log(x1) + log(x2). And p1 = p2 = 1, and m = 20.

1. Find the demand functions, and the optimal choice for the prices and income mentioned above.
2. The government subsidizes every unit of good 1 beyond the first 6 ones. The subsidy is \$0.5 per unit. Find the new constraints that the individual faces, and his optimal choice.

5. Jordi has the utility function:

```
1    1
u(x1,x2) = - -- - --
x1   x2
```

While Juan's utility function is u(x1,x2) = x1*x2 + 2x1.

1. Find analytically and draw the indifference curves of Jordi and Juan.
2. Find all the demand functions.
3. For both Jordi and Juan, and for each good find our whether the good is:
1. Normal or inferior
2. Ordinary or Giffen
3. A substitute or a complement of the other
4. Find analytically and draw Engel curves of both goods for both Jordi and Juan.