Suppose your tastes are defined by the utility function 
.
a. Suppose your income is $1,000, the price of 
is 1 and the price of 
style="vertical-align: -7px;" width="15px" height="20px" align="absmiddle" /> is 
. Set up your utility maximization problem.
b. Derive the quantity of 
you will consume.
c. What happens to your consumption of 
if 
increases?
d. Now suppose that your and my tastes are captured by the utility function 
, with the parameter 
different for you than it is for me. When 
increases, you consume more 
than before and I consume less. What range of 
is consistent with your behavior, and what range is consistent with mine? Use your answer to (b) to explain.
What will be an ideal response?
b.
c. It remains the same.
d. Cobb-Douglas tastes are CES tastes with elasticity of substitution equal to 1. The elasticity of substitution in the CES function is given by
, with Cobb-Douglas tastes therefore having 
.If 
, the elasticity of substitution is less than 1, and if 
, the elasticity of substitution is greater than 1. The graphs below illustrate two scenarios --- the first with a high elasticity of substitution and the second with a low elasticity of substitution. In both cases, tastes are homothetic (as they are for CES preferences). A high elasticity of substitution implies that 
consumption increases with an increase in 
, and a low elasticity of substitution implies that 
consumption decreases with an increase in 
. In part (c), we found that Cobb-Douglas tastes fall at the border of these scenarios, which implies that 
divides the two cases. For you, 
therefore lies between 0 and 
, while for me 
lies between 0 and -1.
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