1. In a simple but delicious world, Joey eats only sandwiches, s, and jam, j. He has a Cobb-Douglas utility functionU(j, s) = Nj1-asª, where 0 < a <1 and N > 0. The price of jam is pj, the price of sandwiches is Ps, and Joeyhas a monthly budget Y to spend on lunch.a. Explain why you can safely use a simpler Cobb-Douglas utility function, V(j, s), to represent Joey'spreferences, which is the same as U(j, s) except for replacing N with 1.b. Transform V(j,s) by taking natural logs and bringing down exponents. Explain why it is useful to do thisfor a Cobb-Douglas utility function, but not for a quasi-linear utility function.Use In(V(j, s)) 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. Simplify your answers so that you arrive at theС.(1-a)YaYformulas j* =and s*Psd. What fraction of his income does Joey spend on jam, and what fraction on sandwiches?

Answered: 1. In a simple but delicious world,…

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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1. In a simple but delicious world, Joey eats only sandwiches, s, and jam, j. He has a Cobb-Douglas utility function
U(j, s) = Nj1-asª, where 0 < a <1 and N > 0. The price of jam is pj, the price of sandwiches is Ps, and Joey
has a monthly budget Y to spend on lunch.
a. Explain why you can safely use a simpler Cobb-Douglas utility function, V(j, s), to represent Joey's
preferences, which is the same as U(j, s) except for replacing N with 1.
b. Transform V(j,s) 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.
Use In(V(j, s)) 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. Simplify your answers so that you arrive at the
С.
(1-a)Y
aY
formulas j* =
and s*
Ps
d. What fraction of his income does Joey spend on jam, and what fraction on sandwiches?

Expert Solution

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Since we answer a maximum of 3 subparts, a-c will be answered here. Please reupload the question mentioning the parts you would like answered.

a. Since N>0, for any N in that, the order of the bundles will be preserved. That mean if bundle A is preferred to bundle B for N=m, it will also be preferred to bundle B if N=n. Since utility is an ordinal concept we can replace the utility function with N=1.

b. By taking log transformation, our utility function is

$\ln (V (j, s)) = (1 - α) \ln (j) + α \ln (s)$ ..........(1)

Any monotone increasing transformation of utility function preserves the order of the bundles. So here, the log transformation separates two good in the utility function and helps with the calculation.

Quasilinear utility functions are already separate in two goods so taking log will make it more complex. Sometimes taking log in quasilinear utility transforms a risk neutral utility function into a risk averse utility function.

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