Microeconomic Theory

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Quiz 4 :
Income and Substitution Effects

Quiz 4 :
Income and Substitution Effects

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Thirsty Ed drinks only pure spring water, but he can purchase it in two different-sized containers: 0.75 liter and 2 liter. Because the water itself is identical, he regards these two ''goods'' as perfect substitutes. a. Assuming Ed's utility depends only on the quantity of water consumed and that the containers themselves yield no utility, express this utility function in terms of quantities of 0.75-liter containers (x) and 2-liter containers (y). b. State Ed's demand function for x in terms of px, py, and I. c. Graph the demand curve for x, holding I and py constant. d. How do changes in I and py shift the demand curve for x? e. What would the compensated demand curve for x look like in this situation?
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a )
Let x denote 0.75-liter containers of water and y denote 2-liter containers of water. The two goods are perfect substitutes such that one unit of
img gives as much utility as
img If
img and
img are perfect substitutes such that one unit of
img gives as much utility as
img then the utility function can be written as:
img Therefore, the utility function for the given scenario can be written as:
img b )
From the utility function, calculate the marginal utilities of the two goods.
img This is the case of perfect substitutes. Hence, the consumer will spend entire income on
img if the price ratio
img is less than or equal to the ratio of marginal utilities; otherwise, zero. That is, the consumer will buy
img units of
img if
img Therefore, the demand function for
img can be written as:
img c)
The demand curve for good
img can be represented as the red curve in Figure 1.
img d)
There are two portions of the demand curve - the portion that intersects the horizontal axis and the potion that coincides the vertical axis.
An increase in income will shift the former to the right.
An increase in the price of good
img will increase the length of the former and reduce the length of the latter.
e)
The compensated demand curve shows only the substitution effect and shows no income effect. In the case of perfect substitutes, there is no income effect. Therefore, the compensated demand curve is the same as the demand curve shown in Figure 1.

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David N. gets $3 per week as an allowance to spend any way he pleases. Because he likes only peanut butter and jelly sandwiches, he spends the entire amount on peanut butter (at $0.05 per ounce) and jelly (at $0.10 per ounce). Bread is provided free of charge by a concerned neighbor. David is a particular eater and makes his sandwiches with exactly 1 ounce of jelly and 2 ounces of peanut butter. He is set in his ways and will never change these proportions. a. How much peanut butter and jelly will David buy with his $3 allowance in a week? b. Suppose the price of jelly were to increase to $0.15 an ounce. How much of each commodity would be bought? c. By how much should David's allowance be increased to compensate for the increase in the price of jelly in part (b)? d. Graph your results in parts (a) to (c). e. In what sense does this problem involve only a single commodity, peanut butter and jelly sandwiches? Graph the demand curve for this single commodity. f. Discuss the results of this problem in terms of the income and substitution effects involved in the demand for jelly.
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Complementary goods are the goods that are consumed together by a consumer. Consumption of a single commodity do not add to the utility and thus, when the price of good increases, the demand for its complement good falls. The income of the consumer is the total budget that the individual has to spend on all the available commodities.
It is given that the person D likes to consume peanut butter and jelly sandwiches using 1 ounce of jelly and 2 ounces of peanut butter. Thus, the utility and demand function of the consumer is given as perfect complements.
Now it is assumed that x stands for peanut butter and y stand for jelly.
The utility function for the complementary goods is as follows:
img The money income ( M ) of $3 and the price of peanut butter to be $0.05 and jelly to be $0.10.
The budget constraint is as follows:
img Since the two goods are used in fixed proportions, therefore
img . Now, it is needed to calculate the demand function.
Substitute the value of
img in the equation (1) as follows:
img Calculate the value of x as follows:
img Hence, person D would buy peanut butter and
img of jelly.
b.
When the price of jelly increases to $0.15 and it is needed to find the quantity the consumer would buy.
As the price of jelly has increased to $0.15 and thus it is costlier than before for the consumer.
The new budget constraint is as follows:
img Substituting the value of
img to get the new consumption bundle. The new consumption bundle for the consumer will be seen as done below:
img Calculate the value of as follows:
img Due to increase in price, person D consumes
img of peanut butter and
img of jelly.
c.
It is required to find the increasing amount of D's allowance in order to compensate for the rise in the price.
The hike in the price of jelly has caused the consumption to fall. Thus, D must be compensated for the sacrificed units.
Calculate the person D's allowance to compensation given a new income level and the change in the income level as follows:
img Hence, person D 's allowance to compensate would be
img .
d.
Represent the graph as follows:
img All the above results are graphed in the above graph.
e.
It is required to graph the demand curve for the single commodity. The peanut butter and jelly are complements that are consumed together in Bread. Thus, bread consumption is the consumption of a single commodity.
The units of bread consumed will stay fixed given the ounces of butter and jelly used. The demand curve shows the relationship between price and units of consumption pattern of an individual with a particular income level.
Represent the graph as follows:
img The units of bread consumed will stay fixed given the ounces of butter and jelly used. The demand curve shows the relationship between price and units of consumption pattern of an individual with a particular income level.
f.
Substitution effect refers to the consumption of units in such a way such that the original bundle is also affordable as income is substituted to the consumer.
When the price rises to $0.15, the new income is given to be $3.75.
Calculate the new bundle purchased as follows:
img Calculate the value of x as follows:
img Hence, the Substitution effect is
img . There will be no change in the quantity of goods demanded at the new income level as they are used in fixed proportions and thus, no price hike can lead to a change unless income level is not substituted in such a way that the original bundle is affordable.
Income Effect refers to the change in the purchasing power of the consumer due to the change in the price level of a commodity. The consumer has the same income level, thus a hike in price level leads to a fall in purchasing power.
Calculate the new consumption bundle as follows:
img Calculate the value of x as follows:
img Calculate the income effect ( I.E.) as follows:
img Hence, the income effect is
img .

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As defined in Chapter 3, a utility function is homothetic if any straight line through the origin cuts all indifference curves at points of equal slope: The MRS depends on the ratio y/x. a. Prove that, in this case, img is constant. b. Prove that if an individual's tastes can be represented by a homothetic indifference map then price and quantity must move in opposite directions; that is, prove that Giffen's paradox cannot occur.
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a)
It is required to be shown that if a straight line through the origin cuts all indifference curves at the points of equal slope, then
img is constant; that is, for any given set of prices, a given change in income always causes the same change in the quantity of good
img bought/consumed.
A straight line through the origin has a constant ratio of two goods at every point on the line. This means that if a straight line through the origin cuts all indifference cuts at the points of equal slope, then, MRS depends on the ratio
img of the two goods. In this case, the following expression can be written:
img where
img is some constant
For optimality, MRS should equal the price ratios. Hence,
img Substitute the above expression in the budget equation.
img img Find the partial derivative of the above equation with respect to
img .
img Hence, given a set of prices, the expression
img is constant. Therefore,
img is constant.
(b)
Find the partial derivative of the Equation (1) with respect to
img .
img Since
img is negative. An increase in price of
img causes a decrease in the quantity demanded of good
img . Hence, Giffen's paradox cannot occur if a straight line through the origin cuts all indifference curves at the points of equal slope. That is, Giffen's paradox cannot occur in the case of a homothetic utility function.

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As in Example 5.1, assume that utility is given by img a. Use the uncompensated demand functions given in Example 5.1 to compute the indirect utility function and the expenditure function for this case. b. Use the expenditure function calculated in part (a) together with Shephard's lemma to compute the compensated demand function for good x. c. Use the results from part (b) together with the uncompensated demand function for good x to show that the Slutsky equation holds for this case.
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Suppose the utility function for goods x and y is given by img a. Calculate the uncompensated (Marshallian) demand functions for x and y, and describe how the demand curves for x and y are shifted by changes in I or the price of the other good. b. Calculate the expenditure function for x and y. c. Use the expenditure function calculated in part (b) to compute the compensated demand functions for goods x and y. Describe how the compensated demand curves for x and y are shifted by changes in income or by changes in the price of the other good.
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Over a three-year period, an individual exhibits the following consumption behavior: img Is this behavior consistent with the axioms of revealed preference?
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Suppose that a person regards ham and cheese as pure complements-he or she will always use one slice of ham in combination with one slice of cheese to make a ham and cheese sandwich. Suppose also that ham and cheese are the only goods that this person buys and that bread is free. a. If the price of ham is equal to the price of cheese, show that the own-price elasticity of demand for ham is _0.5 and that the cross-price elasticity of demand for ham with respect to the price of cheese is also _0.5. b. Explain why the results from part (a) reflect only income effects, not substitution effects. What are the compensated price elasticities in this problem? c. Use the results from part (b) to show how your answers to part (a) would change if a slice of ham cost twice the price of a slice of cheese. d. Explain how this problem could be solved intuitively by assuming this person consumes only one good-a ham and cheese sandwich.
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Show that the share of income spent on a good x is img where E is total expenditure.
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Share elasticities In the Extensions to Chapter 4 we showed that most empirical work in demand theory focuses on income shares. For any good, x, the income share is defined as img . In this problem we show that most demand elasticities can be derived from corresponding share elasticities. a. Show that the elasticity of a good's budget share with respect to income img . Interpret this conclusion with a few numerical examples. b. Show that the elasticity of a good's budget share with respect to its own price img is equal to img . Again, interpret this finding with a few numerical examples. c. Use your results from part (b) to show that the ''expenditure elasticity'' of good x with respect to its own price img d. Show that the elasticity of a good's budget share with respect to a change in the price of some other good img e. In the Extensions to Chapter 4 we showed that with a CES utility function, the share of income devoted to good x is given img Hint: This problem can be simplified by assuming p x = p y , in which case s x = 0.5
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More on elasticities Part (e) of Problem 5.9 has a number of useful applications because it shows how price responses depend ultimately on the underlying parameters of the utility function. Specifically, use that result together with the Slutsky equation in elasticity terms to show: a. In the Cobb-Douglas case (s ¼ 1), the following relationship holds between the own-price elasticities of x and img b. If img Provide an intuitive explanation for this result. c. How would you generalize this result to cases of more than two goods? Discuss whether such a generalization would be especially meaningful. Reference: Problem 5.9 Share elasticities In the Extensions to Chapter 4 we showed that most empirical work in demand theory focuses on income shares. For any good, x, the income share is defined as img . In this problem we show that most demand elasticities can be derived from corresponding share elasticities. a. Show that the elasticity of a good's budget share with respect to income img . Interpret this conclusion with a few numerical examples. b. Show that the elasticity of a good's budget share with respect to its own price img is equal to img . Again, interpret this finding with a few numerical examples. c. Use your results from part (b) to show that the ''expenditure elasticity'' of good x with respect to its own price img d. Show that the elasticity of a good's budget share with respect to a change in the price of some other good img e. In the Extensions to Chapter 4 we showed that with a CES utility function, the share of income devoted to good x is given img Hint: This problem can be simplified by assuming p x = p y , in which case s x = 0.5
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Aggregation of elasticities for many goods The three aggregation relationships presented in this chapter can be generalized to any number of goods. This problem asks you to do so. We assume that there are n goods and that the share of income devoted to good i is denoted by s i. We also define the following elasticities: img
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Quasi-linear utility (revisited) Consider a simple quasi-linear utility function of the form U(x, y) =x + ln y. a. Calculate the income effect for each good. Also calculate the income elasticity of demand for each good. b. Calculate the substitution effect for each good. Also calculate the compensated own-price elasticity of demand for each good. c. Show that the Slutsky equation applies to this function. d. Show that the elasticity form of the Slutsky equation also applies to this function. Describe any special features you observe.
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The almost ideal demand system The general form for the expenditure function of the almost ideal demand system (AIDS) is given by img img a. Derive the AIDS functional form for a two-goods case. b. Given the previous restrictions, show that this expenditure function is homogeneous of degree 1 in all prices. This, along with the fact that this function resembles closely the actual data, makes it an ''ideal'' function. c. Using the fact that img (see Problem 5.8), calculate the income share of each of the two goods. Reference: Problem 5.8 img
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Price indifference curves Price indifference curves are iso-utility curves with the prices of two goods on the X- and Y-axes, respectively. Thus, they have the following general form: img a. Derive the formula for the price indifference curves for the Cobb-Douglas case with img Sketch one of them. b. What does the slope of the curve show? c. What is the direction of increasing utility in your graph?
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