Econ. 410
Spring 2008
Tauchen/Biglaiser
Practice Problems
The Consumer Theory Model, Consumer Surplus, and the Edgeworth Box Model
1. In class, we determined the income and substitution effects of the increase in the price of good X and will now work through the income and substitution effects for a price decrease. We will need the handout on income and substitution effects which was distributed in class. (If you no longer have the sheet, a copy is available at
http://www.unc.edu/courses/2008spring/econ/410/007/HO-Reading.html ) The fourth page of the handout deals with the income and substitution effects of a price decrease.
a. Show the substitution effect on the top graph. Is the substitution effect necessarily in the direction of more of good X? Suppose that both X and Y are normal goods. Show more of the individual’s indifference map consistent with both goods being normal goods. What is the direction of the income effect on the consumption of good X for this case?
b. Show the substitution effect on the bottom graph. Now assume that X is an inferior good and that Y is a normal good. Show more of the individual’s indifference map consistent with X being an inferior good and Y being a normal good. What is the direction of the income effect on the consumption of good X for this case?
c. Complete the chart which is on the last page of the handout and which is repeated below.
Effect of a Decrease in the Price of Good X on the Optimal Consumption of Good XNormal Good / Inferior Good
Substitution Effect
Income Effect
Total Effect
2. Determine the intercepts of the budget lines for the following cases. Also determine the "no borrow, no lend" bundle.
Case A: I1 = $10,000, I2 = $12,000, p1 = $1, p2 = $1, r = .10
Case B: I1 = $10,000, I2 = $13,800, p1 = $1, p2 = $1.15, r = .265
Case C: I1 = $10,000, I2 = $13,800, p1 = $1, p2 = $1.15, r = .10
3. Explain how an increase in the interest rate affects the budget line (ceteris paribus). Explain how an increase in period 1 income affects the budget line (ceteris paribus).
4. We define consumption in period 1 as a normal good if an individual chooses to consume more in period 1 when there is a parallel shift out in the budget line. Consumption in period 1 is an inferior good if an individual chooses to consume less in period 1 when there is a parallel shift out in the budget line. (The corresponding definitions for consumption in period 2 are exactly analogous.)
a. What changes in incomes, prices, or interest rates would cause a parallel shift out in the budget line?
b. Construct a graph in which you show consumption in period 1 as a normal good and an example in which you show consumption in period 1 as an inferior good.
4. Construct a graph to show that with an increase in the interest rate, ceteris paribus, an individual may switch from being a borrower in period 1 to being a saver in period 1.
5. Given her initial two-period, intertemporal budget line, Bernadette chooses to save some of her income in period 1.
a. Construct a graph on which you show Bernadette’s budget line and an indifference curve consistent with her initial choice. Determine the income and substitution effects of an increase in the interest rate. [Hint: Determine the income and substitution effects in the same way as for two goods consumed in the same period. Construct a hypothetical budget line with the same slope as the new budget line and tangent to the initial indifference curve.] Is the substitution effect for consumption in period 1 positive or negative? Is the substitution effect for saving in period 1 positive or negative? Is the substitution effect for consumption in period 2 positive or negative?
b. Rather than finding the income and substitution effects of an interest rate increase for a specific example, we want to think about whether or not the direction of the income and substitution effects is unambiguous for all preferences satisfying the usual assumptions about preferences. As in part a., we assume that the individual was a saver in the initial period. We will also assume that consumption in period 1 is measured on the horizontal and consumption in period 2 on the vertical axis.
b1. Is the new budget line steeper or flatter than the initial budget line? Is the hypothetical budget line steeper or flatter than the initial budget line? Given the curvature of the indifference curve does the tangency of the hypothetical budget line with the initial indifference curve occur down and to the right of the initial optimum or up and to the left of the initial optimum? What is the direction of the substitution effect on consumption in period 1? on savings (which is income minus consumption in period 1)? on consumption in period 2?
b2 From the individual’s point of view, is the new budget set better or worse than the initial budget set? Does the shift from the hypothetical budget line represent a shift out in the budget line or a shift back in the budget line? If consumption in period 1 is a normal good, then how does such a shift affect consumption in period 1? How does such a shift affect consumption in period 2?
b3. Let’s assume that consumption in period 1 and in period 2 are normal goods. Does consumption in period 1 necessarily increase when the interest rate increases (assuming that the individual was a saver in period 1)? Use the income and substitution effects from parts b1 and b2 to explain your answer.
6. Thus far we have assumed that individuals may borrow and lend at the same interest rate. Assume instead that the interest rate at which individuals borrow is higher than the interest rate that individuals receive on their savings. Specifically, I1 = $20,000, I2 = $10,000, p1 = $1, p2 = $1. Jan can borrow at r = .8 and receives r = .1 on savings.
a. Construct the budget line. [Hint: Construct the budget lines for r=.8 and the budget line for r=.1 . For each budget line, identify the segment for which the individual is a borrower and the segment for which the individual is a saver. As a borrower, the relevant BL is the one for which r=.8. As a saver, the relevant BL is the one for which r=.1 .
b. Show an example of Jan's indifference map (i) for which she chooses to neither borrow nor lend since she receives only r = .1 on her savings but (ii) for which she would have saved if she received the higher rate r = .8.
7. The rate of inflation is defined as (p2 - p1 )/p1 . Macroeconomists commonly use the symbol π for the rate of inflation. The real interest rate is defined as (r-π)/(1+π). For the intertemporal problems with only one good in each period, we often think of the price as being a measure of the general price level. The real income in each period is the income for the period divided by the price level.
a. Compute the rate of inflation for each case in question 2. Also compute the real income for each period and the real interest rate. Determine the budget line for each case.
b. Is the individual better off for Case A in which there is no inflation or Case B in which there is inflation? Explain.
c. In comparing Cases A and B, note that the real incomes and the real interest rate are the same for the two cases although the dollar incomes and the interest rates differ. The budget lines are also the same. We want to show that all cases for which the real incomes and the real interest rates are the same yield the same budget lines.
Consider the following two cases and show that the budget lines are the same.
Case with No Inflation: The incomes in the two periods are and . The prices in both period are and the interest rate at which the individual borrows and lends is .
Case with Inflation at Rate : The income in period 1 is and the income in period 2 is
(1+ ). The price in period 1 is and the price in period 2 is (1+ ). The interest rate for this case is + + .
8. The market for good X is competitive and the price is determined by supply and demand. Construct a graph and show the effect of a per unit tax on the equilibrium price and quantity. Then show the effect on consumer surplus. Does the effect of the tax on consumer surplus depend upon whether producers pay the tax out of their revenues or consumers pay the tax (in addition to paying the seller for the good)? Use the graph to explain your answer.
9. The next page shows the graph that we used in the class discussion of compensating and equivalent variation. The individual’s income is $32/time period and the price of good Y is $4. The price of good X is initially $1 and then increases to $4. The individual has quasilinear preferences and the indifference curves are parallel relative to the horizontal axis.
The budget line for the initial prices is denoted BL:LP and for the higher price of good X is denoted BL:HP. The hypothetical budget lines used to determine the compensating and equivalent variation are denoted BL:CV and BL:EV respectively.
a. The individual initially chose the bundle (10, 5.5). What is the slope of the indifference curve at that point? What is the slope of the indifference curve that goes through every other bundle with 10 units of good X? [Hint: Remember that the individual has quasilinear preferences.]
b. At the higher price for good X, the individual selected the bundle (1.5,6.7). What is the slope of the indifference curve that goes through this point? What is the slope of the indifference curve that goes through any other bundle which contains 1.5 units of good X?
c. The budget line BL:CV is constructed for the new prices and is tangent to the indifference curve that the individual attained with the initial lower price for good X. Let’s refer to this indifference curve as IC1. What is the slope of IC1 at the tangency to the budget line BL:CV? Given that the preferences are quasilinear and the indifference curves are parallel, what is the amount of X at the bundle where the budget line BL:CV is tangent to IC1?
d. The budget line BL:EV is constructed for the initial prices and is tangent to the indifference curve that the individual attained with the new higher price for good X. Let’s refer to this indifference curve as IC2. What is the slope of IC2 at the tangency to the budget line BL:EV? Given that the preferences are quasilinear and the indifference curves are parallel, what is the amount of X at the bundle where the budget line BL:EV is tangent to IC2?
e. Given the parallel indifference curves what can you conclude regarding the distance between BL:HP & BL:CV versus the distance between BL:LP & BL:EV? What does your answer to this question imply regarding the relative values of the compensating and the equivalent variation?
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10. Condoleezza consumes two goods – X and Y. Her income is $27 and the price of good Y is $3. The budget line for px=$1 is denoted BL1 and the budget line for px=$3 is denoted BL2.
a. Use the graph below to determine the compensating variation for the price increase. To do so, first construct a hypothetical budget line for the new higher price of good X and for which Condoleezza obtains the same level of well-being as for the initial price of good X, but no higher level of well-being. How much additional income would be required for Condoleezza to have this budget line? The compensating variation is the change in income required for her to have the hypothetical budget line that you constructed.
b. Suppose that you provided Condoleezza with enough additional income to purchase C1, which is the optimum for the initial lower price for good X, rather than the income required for her to obtain the same level of well-being, but no greater level of well-being, as at the initial lower price. Would she be better off, equally well off, or better off than with the compensating variation income increase? Explain.
c. Use the graph below to determine the equivalent variation for the price increase. To do so, first construct the budget line that is for the initial prices and by which Condoleezza achieves the same level as well-being as with BL2 (the budget line with the higher price for good X) but no higher level of well-being. The equivalent variation is the reduction in income that would give Condoleezza the budget line that you have just constructed.
d. Compare the compensating and equivalent variation.
11. Use the graphs above to identify the compensating and equivalent variation for a decrease in the price of good X from $3 to $1.
12. An individual’s endowment is 6 units of good X and 6 units of good Y. The individual may buy and sell the goods at the market prices. Construct the budget lines for the following combinations.
a. px =$1 and py=$1
b. px =$10 and py=$10
c. px =$2 and py=$1
d. px =$10 and py=$40
e. px =$10 million and py=$40 million
13. Mitt’s endownment is 6 units of good X and 6 units of good Y.