Nir Dagan, Esther Hauk, and Albrecht Ritschl

**Due Monday, 4 May, 1998**

**1.** Jaume has the utility function:

__ u(x_{1},x_{2}) = \/x_{1}+ x_{2}((Square root of x_{1}) plus x_{2})

- Are his preferences convex?
- Calculate the MRS at the bundles
`(4,1)`and`(1,5)` - Draw a map of indifference curves of Juame's preference.
- Find the optimal choice when
`p`and_{1}= p_{2}`m = 40`

**2.** A consumer has the utility function:
`u(x _{1},x_{2}) = log(x_{1}) +
x_{2}`.

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

`p`,_{1}= 1`p`, and_{2}= 1`m = 20``p`,_{1}= 1`p`, and_{2}= 1`m = 10``p`,_{1}= 2`p`, and_{2}= 1`m = 10`

**3.** Nuria has `10000` pesetas. She may spend them on
books (`x _{1}`) or on beer (

- Find her optimal choice when the market prices are
`p`_{1}= p_{2}= 100 - 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(x _{1},x_{2}) = log(x_{1}) +
log(x_{2})`. And

- Find the demand functions, and the optimal choice for the prices and income mentioned above.
- 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(x_{1},x_{2}) = - -- - -- x_{1}x_{2}

While Juan's utility function is `u(x _{1},x_{2}) =
x_{1}*x_{2} + 2x_{1}`.

- Find analytically and draw the indifference curves of Jordi and Juan.
- Find all the demand functions.
- For both Jordi and Juan, and for each good find our whether the good is:

- Normal or inferior
- Ordinary or Giffen
- A substitute or a complement of the other

- Find analytically and draw Engel curves of both goods for both Jordi and Juan.