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School
University of California, Los Angeles *
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Course
11
Subject
Economics
Date
May 24, 2024
Type
Pages
13
Uploaded by ChancellorKoalaMaster567
Economics 11
Week 5
Sungwoo Cho
*
April 2024
I
Duality
¯
U
=
V
(
p
x
, p
y
, I
)
and
E
(
p
x
, p
y
,
¯
U
) =
I
Marshallian to Hicksian
g
x
(
p
x
, p
y
,
E
(
p
x
,p
y
,
¯
U
)
) =
h
x
(
p
x
,p
y
,
¯
U
)
and
g
y
(
p
x
, p
y
,
E
(
p
x
,p
y
,
¯
U
)
) =
h
y
(
p
x
,p
y
,
¯
U
)
Hicksian to Marshallian
h
x
(
p
x
,p
y
,
V
(
p
x
, p
y
, I
)
) =
g
x
(
p
x
, p
y
,I
)
and
h
y
(
p
x
,p
y
,
V
(
p
x
, p
y
, I
)
) =
g
y
(
p
x
, p
y
,I
)
*
Contact:
chosw13@ucla.edu
. If you find any typo or error please send me an email.
1
II
Income and Substitution Effects
Remark
:
A change in price causes two effects known as the substitution and income effect. As the
price of x decreases the consumer becomes richer, this is known as the income effect. Also as the price of
x decreases x becomes relatively cheaper than y, this is known as the substitution effect.
Large price changes
Total effect
The difference between the new consumption bundle and the old consumption bundle is
called the total effect. The next graph shows that the budget constraint rotates to the right. The
new consumption bundle is marked by point B.
Total effect
x
=
g
x
(
p
0
x
,p
y
,I
)
-
g
x
(
p
x
,p
y
,I
)
p
0
x
-
p
x
Total effect
y
=
g
y
(
p
0
x
,p
y
,I
)
-
g
y
(
p
x
,p
y
,I
)
p
0
x
-
p
x
Substitution effect
The substitution effect measures how much the consumer would substitute x for y given
the price change holding utility constant. How do we compute demand while holding utility
constant? The substitution effect is the movement from point A to point C.
2
Note that point A is the initial optimal consumption bundle. To get point C, we shift the
new budget constraint (which slope is the new price ratio) towards the indifference curve with
the initial level of utility. That is, towards the indifference curve where point A is located. Note
that A and C belong to the same indifference curve, thus we are holding constant the utility
level. The Hicksian demand functions do precisely this!
Substitution effect
x
=
h
x
(
p
0
x
,p
y
,
¯
U
)
-
h
x
(
p
x
,p
y
,
¯
U
)
p
0
x
-
p
x
Substitution effect
y
=
h
y
(
p
0
x
,p
y
,
¯
U
)
-
h
y
(
p
x
,p
y
,
¯
U
)
p
0
x
-
p
x
Income effect
The income effect measures the change in demand given an increase in income holding the
price ratio constant. How do we compute demand holding the price ratio constant?
The income effect is the movement from point C to point B. Note that point B is determined by
the tangency of the new budget constraint (slope
p
0
x
p
y
) to the indifference curve
U
2
. The point C is
determined by the tangency of the auxiliary budget constraint that has slope
p
0
x
p
y
and is tangent
to the indifference curve
U
1
. That is, we are holding constant the price ratio, by comparing two
points were the slope is equal to
p
0
x
p
y
.
3
Income effect
x
=
Total effect
x
-
Substitution effect
x
Income effect
x
=
g
x
(
p
0
x
,p
y
,I
)
-
g
x
(
p
x
,p
y
,I
)
p
0
x
-
p
x
-
h
x
(
p
0
x
,p
y
,
¯
U
)
-
h
x
(
p
x
,p
y
,
¯
U
)
p
0
x
-
p
x
Income effect
y
=
Total effect
y
-
Substitution effect
y
Income effect
y
=
g
y
(
p
0
x
,p
y
,I
)
-
g
y
(
p
x
,p
y
,I
)
p
0
x
-
p
x
-
h
y
(
p
0
x
,p
y
,
¯
U
)
-
h
y
(
p
x
,p
y
,
¯
U
)
p
0
x
-
p
x
Small price changes
Slutsky Equation
Start from the duality result:
g
x
(
p
x
, p
y
, E
(
p
x
,p
y
,
¯
U
)) =
h
x
(
p
x
,p
y
,
¯
U
)
Let’s take the partial derivative with respect to
p
x
and use the fact :
I
=
E
(
p
x
,p
y
,
¯
U
)
∂g
x
(
p
x
, p
y
, I
)
∂p
x
+
∂g
x
(
p
x
, p
y
, I
)
∂I
∂E
(
p
x
,p
y
,
¯
U
)
∂p
x
=
∂h
x
(
p
x
,p
y
,
¯
U
)
∂p
x
By Shephard’s Lemma:
∂g
x
(
p
x
, p
y
, I
)
∂p
x
+
∂g
x
(
p
x
, p
y
, I
)
∂I
h
x
(
p
x
,p
y
,
¯
U
)
=
∂h
x
(
p
x
,p
y
,
¯
U
)
∂p
x
By duality:
∂g
x
(
p
x
, p
y
, I
)
∂p
x
+
∂g
x
(
p
x
, p
y
, I
)
∂I
g
x
(
p
x
,p
y
,I
)
=
∂h
x
(
p
x
,p
y
,
¯
U
)
∂p
x
Re-arrange terms:
∂g
x
(
p
x
, p
y
, I
)
∂p
x
|
{z
}
Total Effect
=
∂h
x
(
p
x
,p
y
,
¯
U
)
∂p
x
|
¯
U
=
V
|
{z
}
Substitution Effect
-
∂g
x
(
p
x
, p
y
, I
)
∂I
g
x
(
p
x
,p
y
,I
)
|
{z
}
Income Effect
4
Note:
We evaluate the derivative of the Hicksian demand at the indirect utility function to make the
equation consistent and to express everything in terms of prices and income.
Derivation: cross-price Slutsky Equation
Start from the duality result:
g
x
(
p
x
, p
y
, E
(
p
x
,p
y
,
¯
U
)) =
h
x
(
p
x
,p
y
,
¯
U
)
Let’s take the partial derivative with respect to
p
y
and use the fact :
I
=
E
(
p
x
,p
y
,
¯
U
)
∂g
x
(
p
x
, p
y
, I
)
∂p
y
+
∂g
x
(
p
x
, p
y
, I
)
∂I
∂E
(
p
x
,p
y
,
¯
U
)
∂p
y
=
∂h
x
(
p
x
,p
y
,
¯
U
)
∂p
y
By Shephard’s Lemma:
∂g
x
(
p
x
, p
y
, I
)
∂p
y
+
∂g
x
(
p
x
, p
y
, I
)
∂I
h
y
(
p
x
,p
y
,
¯
U
)
=
∂h
x
(
p
x
,p
y
,
¯
U
)
∂p
y
By duality:
∂g
x
(
p
x
, p
y
, I
)
∂p
y
+
∂g
x
(
p
x
, p
y
, I
)
∂I
g
y
(
p
x
,p
y
,I
)
=
∂h
x
(
p
x
,p
y
,
¯
U
)
∂p
x
y
Re-arrange terms:
∂g
x
(
p
x
, p
y
, I
)
∂p
y
|
{z
}
Total Effect
=
∂h
x
(
p
x
,p
y
,
¯
U
)
∂p
y
|
¯
U
=
V
|
{z
}
Substitution Effect
-
∂g
x
(
p
x
, p
y
, I
)
∂I
g
y
(
p
x
,p
y
,I
)
|
{z
}
Income Effect
Price changes for normal goods:
If a good is normal, substitution and income effects reinforce one another: when the price falls,
both effects lead to a rise in quantity demanded.
Price changes for inferior goods:
If a good is inferior, substitution and income effects move in opposite directions: the combined
effect is indeterminate. When the price falls, the substitution effect leads to a rise in quantity
demanded, but the income effect moves in the opposite direction
Giffen’s paradox
For an inferior good, if the income effect dominates the substitution effect an increase in price
generates an increase in demand:
•
An increase in price leads to a drop in real income.
•
Since the good is inferior, a drop in income causes the quantity demanded to rise.
•
If the income effect is greater than the substitution effect consumption increases.
5
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- Assume a Heckscher-Ohlin model. Home and Foreign produce food (F) and machinery (M) using labor (L) and capital (K) as the unique inputs in both sectors. Both countries face the same technologies. In both countries the input requirements are fixed at akf = 2, aLF = 1, aKM = 2, and aLM = 3 (i.e., 3 units of labor are required to produce one unit of machinery etc.). Home is endowed with K = 800 units of capital-hours and L= 600 units of labor-hours. Foreign is endowed with K* = 600 units of capital-hours and L* = 400 units of labor-hours. The output of food is denoted by QF. The output of machinery is denoted by QM- a) Determine Home's capital constraint. Home's capital constraint is b) Which of the two goods is capital-intensive in the production? is capital-intensive. c) Which output mix will be realized in Home if it uses all its resources? The optimal output mix is units of food and units of machinery. d) Which good will be exported by Home? Home will exportarrow_forwardQuestion 3 Krugman Model: Suppose that the elasticity of substitution between vari- eties of a differentiated good is o = 3. The marginal cost of production is c = 2, the fixed cost of operating a business is f = 10. There are two countries, H and F, that initally do not trade. The countries differ only in the size of their economies with EH 100 > EF = 50. All the Krugman model assumptions hold regarding preferences, monopolistic competition, free entry, etc. = Starting from the closed economy, answer the following. a. Profits: Using A to denote the level of demand preceived by a typical firm in H, write the typical firm's profit function. Plug in the relevant numbers for the elasticity of substitution, the marginal cost, and the fixed cost, leaving AH as a constant. b. Profit Maximization: Take the derivative of the profit function and solve the first order condition for the optimal price. What is the optimal price charged? c. Free Entry: Now solve for the total number of entrants…arrow_forward3. Consider a Factory town model. Draw a market area graph for a single factory (hint: similar to figure 2-1) with a shirt cost at the factory of 0.5 bread and a shirt cost at home of 1 bread. Now, suppose that gluten-addicted aliens have invaded our factory town, and with a small probability, will abduct people walking to the factory and steal half of their bread. This makes traveling to the factory more costly on average. What will this do to the overall size of the factory town? Copy your graph from before, but this time add the change to the graph.arrow_forward
- Q6 Homework Answered Due Sep 7th, 11:59 PM . After much consideration, you have chosen Ireland over Spain for your Study Abroad program next year. However, the deadline for your final decision is still months away and you may reverse this decision. Which of the following events could prompt you to reverse this decision? Select an answer and submit. For keyboard navigation, use the up/down arrow keys to select an answer. The marginal benefit of going to Spain increases. b The marginal cost of going to Spain increases. The marginal benefit of going to Ireland increases. The marginal cost of going to Ireland decreases. X Your answerarrow_forwardConsider the specific-factors model with 2 goods: manufacturing and agriculture. Manufacturing requires labour and sector-specific capital, while agriculture requires labour and sector-specific land. Labour is freely mobile across sectors. The government is considering to impose an import tariff on agricultural products. Which of the following is true? (a)The relative price of agricultural products will go down. (b)Capitalists will support this initiative. (c)Agricultural production will increase. (d)Marginal product of labour will decrease in manufacturing. Explain your answer clearly. Limit your explanation to 200 words.arrow_forwardConsider the specific-factors model with 2 goods: manufacturing and agriculture. Manufacturing requires labour and sector-specific capital, while agriculture requires labour and sector-specific land. Labour is freely mobile across sectors. The government is considering to impose an import tariff on agricultural products. Which of the following is true? (a) The relative price of agricultural products will go down. (b) Capitalists will support this initiative. (c) Agricultural production will increase. (d) Marginal product of labour will decrease in manufacturing. Explain your answer clearly.arrow_forward
- Suppose the outputs of beauty shops and pet-grooming salons are complementary, providing one-stop shopping for personal and pet maintenance. Betty Beehive is thinking about moving her beauty shop from an isolated location to a vacant building next to Peter’s pet grooming shop.In making her decision, she makes the following assumptions: If Betty moves, she will keep all her current customers (20 people per week) and attract 25% of Peter’s current customers. Peter currently has 60 customers per week. 3. Excluding rent, Betty’s profit per beauty treatment is $10. 4.The weekly rent at the new location is $200 higher than Betty’s current rent. Question = If Betty moves her beauty shop, will her profits increase or decrease? By how much? Question =Suppose that if Betty makes the move, 50 percent of her original customerswill switch from George’s grooming salon to Peter’s. If Peter’s profit pertreatment is $8, how much would he be willing to pay Betty to make themove? Will the…arrow_forward11.2 Suppose P(x) represents the profit on the sale of x Blu-ray discs. If P(1,000) = 4,000 and P'(1,000) = −6, what do these values tell you about the profit? P(1,000) represents the profit on the sale of Blu-ray discs. P(1,000) = 4,000, so the profit on the sale of Blu-ray discs is $ . P'(x) represents the as a function of x. P'(1,000) = −6, so the profit is decreasing at the rate of $ per additional Blu-ray disc sold.arrow_forwardNancy Lerner is trying to decide how to allocate her time in studying for her economics course. There are two examinations in this course. Her overall score for the course will be the minimum of her scores on the two examinations. She has decided to devote a total of 1,200 minutes to studying for these two exams, and she wants to get as high an overall score as possible. She knows that on the first examination if she doesn’t study at all, she will get a score of zero on it. For every 10 minutes that she spends studying for the first examination, she will increase her score by one point. If she doesn’t study at all for the second examination she will get a zero on it. For every 20 minutes she spends studying for the second examination, she will increase her score by one point.arrow_forward
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