Exam 5: Different Types of Tastes

All homogeneous functions (of any degree)are homothetic but not all homothetic functions are homogeneous (of some degree).

Tastes for perfect substitutes are both homothetic and quasilinear.

Suppose you are very picky about your outdoor BBQ experiences --- and you need exactly 1 cup of lighter fluid for each bag of charcoal you use.If you have either leftover charcoal or leftover lighter fluid,you simply discard it. a.With cups of lighter fluid on the horizontal and bags of charcoal on the vertical axis,illustrate some of your indifference curves. b.Suppose that your favorite charcoal has just gotten better because the producer has infused the charcoal with half a cup of lighter fluid per bag.How does your answer to (a)change? c.How could you change the units in which lighter fluid is measured on the horizontal axis to get your graph from (b)to look the same as you original graph in (a)?

Suppose our tastes can be represented by the function $u \left( x _ { 1 } , x _ { 2 } \right) = \left( \alpha x _ { 1 } ^ { - \rho } + ( 1 - \alpha ) x _ { 2 } ^ { - \rho } \right) ^ { - 1 / \rho }$ .It is often observed that people become more rigid --- more set in their ways --- as they get older.What parameter is changing as we get older --- and how is it changing? (Explain.)

Suppose the only characteristic of beer that a consumer cares about is alcohol content.Currently,Bud Light and Miller Lite have the same alcohol content. a.Using $x _ { 1 }$ to denote ounces of Miller Lite and $x _ { 2 }$ to denote ounces of Bud Light,what's the simplest possible utility function that can describe this consumer's tastes over the two products. b.Suppose Bud Light lowers its alcohol content by 50%.How would you change the utility function to account for this? c.Derive the MRS for the functions in (a)and (b)--- and interpret your answer.

Consider the utility function $u \left( x _ { 1 } , x _ { 2 } \right) = \left( \alpha x _ { 1 } ^ { - \rho } + ( 1 - \alpha ) x _ { 2 } ^ { - \rho } \right) ^ { - 1 / \rho }$ .If $\rho = 0$ ,the elasticity of substitution is equal to 1. The elasticity of substitution for CES utility functions is $\frac { 1 } { 1 + \rho }$ .

Suppose you are very picky about your outdoor BBQ experiences --- and you need exactly 1 cup of lighter fluid for each bag of charcoal you use.If you have either leftover charcoal or leftover lighter fluid,you simply discard it. a.Letting cups of lighter fluid be denoted as $x _ { 1 }$ and bags of charcoal as $x _ { 2 }$ ,give the simplest possible utility function that captures your tastes. b.Suppose that your favorite charcoal has just gotten better because the producer has infused the charcoal with half a cup of lighter fluid per bag.How does your answer to (a)change?

Suppose the only characteristic of beer that a consumer cares about is alcohol content.Currently,Bud Light and Miller Lite both have the same alcohol content. a.Illustrate the consumer's indifference curves in a graph with ounces of Miller Lite on the horizontal and ounces of Bud Light on the vertical axis. b.Suppose that the producers of Bud Light lower the price of Bud Light.How will your answer to (a)change? c.Suppose that the producers of Bud Light lower the alcohol content of their beer by 50%.How will your answer to (a)change? d.Since we identify tastes with indifference maps,would you say that the consumer's tastes have changed in (b)or (c)? e.How could we change the units we use to measure Miller Lite in order to get the indifference map in (c)to again look like the one in (a)?

Suppose consumer cannot taste the difference between Miller Lite and Bud Light,but Miller Lite is sold in 12 ounce cans while Bud Light is sold in 8 ounce cans.In a graph with cans of Miller Lite on the horizontal and cans of Bud Light on the vertical axis,which of the following is the correct slope for this consumer's indifference curves: a. $- \frac { 2 } { 3 }$ b. -1 c. $- \frac { 3 } { 2 }$ d. There is not enough information to tell.

There are no quasilinear tastes that have constant elasticity of substitution.

Suppose our tastes are homothetic.It is often observed that people become more rigid --- more set in their ways --- as they get older.Can you translate this observation into "economics-speak" by discussing which feature of our tastes is likely the be changing as we get older?

Consider the utility function $u \left( x _ { 1 } , x _ { 2 } \right) = \left( \alpha x _ { 1 } ^ { - \rho } + ( 1 - \alpha ) x _ { 2 } ^ { - \rho } \right) ^ { - 1 / \rho }$ .If $\rho = 2$ ,the elasticity of substitution is equal to $\frac { 1 } { 2 }$ .

There is no elasticity of substitution that is inconsistent with tastes being homothetic.

Tastes for perfect complements are both homothetic and quasilinear.

Homothetic tastes are always tastes over essential goods.

Consider the utility function $u \left( x _ { 1 } , x _ { 2 } \right) = x _ { 1 } ^ { \alpha } x _ { 2 } ^ { \beta }$ .Which of the following are true statements about the indifference maps represented by this function. a. MRS=-1 along the 45 degree line if and only if $\alpha = \beta$ . b. MRS=-1 along a ray steeper than the 45 degree line if and only if $\alpha > \beta$ . c. MRS=-1 along a ray shallower than the 45 degree line if and only if $\alpha < \beta$ . d. All of the above. e. None of the above.

There are no quasilinear tastes that have elasticity of substitution equal to 1 everywhere.

If tastes are Cobb-Douglas,they can be represented by a utility function that is homogeneous of degree k where k can take on any positive value.