ENGR.ECONOMIC ANALYSIS
14th Edition
ISBN: 9780190931919
Author: NEWNAN
Publisher: Oxford University Press
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Transcribed Image Text:Suppose you go to Trader Joe's to buy fruit for the week. You only like apples (A) and bananas
(B) and your weekly fruit budget is $11. When you arrive at Trader Joe's you notice that the
price of an apple is $1.00 and the price of a banana is $0.25.
QUESTION #1: How many apples and bananas should you buy?
QUESTION #2: When you have found the answer, draw a diagram that shows the outcome.
Step #1. Determine your preferences. Let's suppose that your preferences can be represented
by the following utility function: U(A, B) = AªBß = A0.40 B0.60
FYI: This utility function is known as a Cobb-Douglas utility function. It is the most commonly used function used in
economics! The reason we like it so much is that it has:
1. Constant returns (double your consumption of A and B and your utility doubles); a + B = 1
2. Diminishing marginal utility (the extra utility gained from consuming A (or B) decreases as you consume more of
the A good (or B good); a < 1, p<1.
The way I described the utility function, you like bananas more than apples; that is, all things equal, eating a banana
gives you more utility than eating an apple (0.60 > 0.40); B > a.
Step #2: Determine your budget constraint (pê + PÂB = Y). Given your income and the prices
of the two goods, your budget constraint is: 1.A +0.25 B = 11
FYI: The budget constraint shows that the amount spent on apples (1. A) and bananas (0.25 B) cannot add up to
more than $11. Clearly, more money spent on apples means less spent on bananas, and vice versa.
Step #3: Set up the constrained utility maximization problem. That is, set up a problem that
shows that you need to maximize utility, given your budget constraint.
max U(A, B) = A0.40 B0.60
A,B
s. t. 1. A+0.25 · B = 11
FYI: the two letters s. t. _ stands for "subject to"
Step #4: Solve the constrained utility maximization problem. For this pre-problem, please use
the substitution method. *
Substitution method: max U (A, B) = A⁰.4⁰ (44 – 4A)⁰.60
A
FYI: Substitution method - use the budget constraint to substitute away the B so that you maximize one equation in
one unknown. That is, after removing the B, take the derivative with respect to A, set it equal to zero, and solve for
A.
*
In general, there are two common ways to solve the constrained optimization problem: (1) the substitution
method, and (2) the Lagrangian method.
Expert Solution
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Step 1: Describe the problem
Preference is over apples and bananas.
Income = $11
Price of Apples = $1
Price of Bananas = 0.25
We assume preferences to be Cobb Douglas with utility function ,
U = A0.4 B0.6
Step by stepSolved in 3 steps with 1 images
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