In a simple but delicious world, Joey eats only sandwiches, s, and jam, j. He has a Cobb-Douglas utility function ?(?,?) = ??!"#?#, where 0 < ? < 1 and ? > 0. The price of jam is ?$, the price of sandwiches is ?%, and Joey has a monthly budget ? to spend on lunch. a. Explain why you can safely use a simpler Cobb-Douglas utility function, ?(?,?), to represent Joey’s preferences, which is the same as ?(?,?) except for replacing N with 1. b. Transform ?(?,?) by taking natural logs and bringing down exponents. Explain why it is useful to do this for a Cobb-Douglas utility function, but not for a quasi-linear utility function.
In a simple but delicious world, Joey eats only sandwiches, s, and jam, j. He has a Cobb-Douglas utility function
?(?,?) = ??!"#?#, where 0 < ? < 1 and ? > 0. The price of jam is ?$, the price of sandwiches is ?%, and Joey
has a monthly budget ? to spend on lunch.
a. Explain why you can safely use a simpler Cobb-Douglas utility function, ?(?,?), to represent Joey’s
preferences, which is the same as ?(?,?) except for replacing N with 1.
b. Transform ?(?,?) by taking natural logs and bringing down exponents. Explain why it is useful to do this
for a Cobb-Douglas utility function, but not for a quasi-linear utility function.
c. Use ln(?(?,?)) and the substitution method to derive the formulas for Joey’s optimal amount of jam, j*,
and sandwiches, s*, to buy and consume per month.
d. What fraction of his income does Joey spend on jam, and what fraction on sandwiches?
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