Econ. 410
Spring 2008
Tauchen
Answers to Practice Problem -- Applications of Consumer Theory
Answers to Questions #5-8
5. The initial and new budget lines are labeled BL0 and BLN respectively.
a. What change in income and/or prices would cause this rotation in the budget line? Answer: A decrease in the price of good Y.
b. Determine the income and substitution effects for this change in the budget line. Is good X a normal or an inferior good (for this part of the indifference map)? Is good Y a normal or an inferior good (for this part of the indifference map)?
Answer: The hypothetical budget line is BLH. The optima for the initial, hypothetical, and new budget lines are labeled A0 , AH, and AN respectively. The SE is the movement from A0 to AH, and the IE is the movement from AH to AN.
To determine whether the goods are normal or inferior we compare the optima for the hypothetical and new budget lines. The shift from the hypothetical to the new budget line is exactly the type of parallel shift that would be caused by an increase in income. With the shift from AH to AN , the individual selects to consume more of both goods. Both goods are normal goods.
6. A household consumes two goods: educational quality and good Y. The household’s preferences satisfy the usual assumptions. The household has income I; educational quality and good Y can be purchased at market prices. Public education is not available. The household decides how much educational quality and how much of good Y to purchase. We’ll assume that the household has only one child so that we do not have to consider the possibility of different quality for different children.
a. Construct a budget line and indifference graph map. Use the standard consumer theory model to describe the household’s choice problem.
Answer: We have the usual negatively sloped BL. The household selects the best bundle in the budget set.
b. The government now offers free, public education of quality level Qg. The government selects the quality of the public education. (We are ignoring taxes used to support the schools.) A household’s choices are to (i) purchase private educational quality as in part a or (ii) send the child to the public school of quality Qg for free.
An indifference curve and budget lines for three different households are shown below. The dark straight line is the household’s budget line if it purchases private education and does not send the child to the public schools. The public schools offer quality Qg for free. Each graph also shows one indifference curve.
b1. What is the household’s budget set? [Hint: What bundles can the household afford if it purchases private education? What bundle can if have if it selects the free public education? ]
Answer: The budget set consists of the same bundles as were in the budget set when only private education was available plus the one additional bundle (Qg, I/py). If the household sends the child to public school, then the educational quality is the public school quality and the family spends all of its income on good Y.
b2. For each family, determine the optimum choice with free public schools of quality Qg available (in addition to private education).
b3. For each family, compare the quality of education chosen (i) when no free education is available and-(ii) when free public education of quality Qg is available.
Household of Type 1: A household which selects public education and for which the public education is of higher quality than the private education chosen previouslyHousehold of Type 2: A household which continues to select private education when public education becomes available and for which the educational quality is higher than for public education
Household of Type 3: A household which selects to send the child to public school when it becomes available and for which the public education is of lower quality than the private education chosen when public education was not available.
Answer: A household of type 1 selects to send the child to the public school when it becomes available. Introducing the public school option allows the family to achieve a higher IC and increases the child’s educational quality.
A family of type 2 continues to select the same bundle as before free, public education was available. Selecting free public education would put the family on a lower IC.
A household of type 3 selects to send the child to public school. Before free public education was available, the family selected the bundle at the tangency of the IC and the budget line. With the introduction of free public education, the family achieves a higher IC and the child’s educational quality is lower.
We have found examples in which offering free, public education lowers a child’s educational quality, has no effect on the child’s educational quality, and increases the child’s educational quality. All three possibilities (an increase in educational quality, a decrease in educational quality, or no change) are consistent with the consumer theory model
c. Suppose now that the government offers households an option. They may select the public education for free or they may have a voucher. The voucher may be used only to purchase education. The value of the voucher equals the expenditure required to purchase education of quality Qg. For each of the households, answer the following questions.
c1. Construct the household’s new budget line.
Answer: For each case, the budget line with the voucher is the dotted line. The dot at the upper end of the dotted budget line is the bundle with the public school quality and all of the household’s income spent on good Y. For each graph, Qn* denotes the optimal educational quality when the government does not offer public school, Qg* denotes the quality of the public school, and Qv* is the optimal educational quality with the voucher and the public school available.
c2., c3. Answer: The first type of household switched to the public school when it became available and the public education quality was higher than the previously selected private education quality. Adding the voucher will either (i) have no effect on the household’s school quality choice or (ii) lead the family to select a private school of the higher quality than the public school. The first graph below shows an example in which the family continues to select educational quality Qg* with the voucher program. The second graph shows an example in which the educational quality selected with the voucher program is higher than Qg*.
The household of type 2 selects private education of type 2 even when free public education of quality Qg* is available. Offering the voucher to such a family has the same effect as an increase in income. If both educational quality and good Y are normal goods, the household will select higher educational quality and more of good Y than with no voucher program. The free public education is not chosen.
The third type house sent the child to public school when this option became available. If both goods are normal goods, then the family will select a private school once the voucher is available. The educational quality will be higher than initially chosen when there was no free public education and no voucher program.
This model addresses some issues related to family choice of educational quality and how vouchers (which cost the taxpayer the same as the public education of the children) affect the educational quality selected by a child’s family. This very simple model does not deal with other issues related to voucher programs, including whether parents (some or most) can identify educational quality and whether voucher programs might lead to separation of children by ability, income, ethnic background, or race.
Household of Type 1 which consumes the same quality of education with the voucher and public school available as with only the public school availableHousehold of Type 1 which consumes higher quality education with the voucher and public school available than with only the public school available
Household of Type 2 with the voucher program
Household of Type 3 with the voucher program
Consider the standard labor-leisure choice model and assume that Jo’s preferences satisfy the usual assumptions. The standard graph for this model shows leisure per time period on the horizontal axis and the consumption of good Y on the vertical axis. For simplicity, assume that Jo has no unearned income. She may select the number of hours to work at the market wage.
a. Construct a graph and show an example for which Jo chooses to work fewer hours (consume more leisure) at a high wage than a low wage. Determine the income and substitution effects of the change in the wage.
b. Construct another example for which she chooses to work more hours at a higher wage.
Answer: Figure 5.24 of Besanko and Braeutigan show an example in which the individual works more hours as the wage increases from $5 per hour to $10/hour to $15/hours. As the wage increase from $15/hour to $20/hour to $25/hour, the hours worked fall. Figure 5.26 shows the income and substitution effects for the case in which hours worked fall with an increase in the wage.
The graph printed here shows the IE and SE for the case in which the hours worked increase (or equivalently, leisure hours fall,) with an increase in the wage.
The initial and new budget lines are labeled BL0 and BLN. The initial optimum is at (16,48) and the new at (12,216). The hypothetical BL is the dotted line. The optimum for the hypothetical budget line is approx. (10,115). The substitution effect is a reduction in the number of hours of leisure from 16 to 10. The income effect is an increase in the number of hours of leisure from 10 to 12.
Suppose that the government imposes a tax of 50% on wage income. Do individuals necessarily work fewer hours? Explain
Answer: The tax has the same effect upon the BL as a wage decrease. As reasoned above, some individuals select to work more at a lower wage and some to work less.
Consider the standard labor-leisure choice model and assume that Jo’s preferences satisfy the usual assumptions. The standard graph for this model shows leisure per time period on the horizontal axis and the consumption of good Y on the vertical axis. For simplicity, assume that Jo has no unearned income. She may select the number of hours to work at the market wage.
c. Construct a graph and show an example for which Jo chooses to work fewer hours (consume more leisure) at a high wage than a low wage. Determine the income and substitution effects of the change in the wage.
d. Construct another example for which she chooses to work more hours at a higher wage.
Answer: Figure 5.24 of Besanko and Braeutigan show an example in which the individual works more hours as the wage increases from $5 per hour to $10/hour to $15/hours. As the wage increase from $15/hour to $20/hour to $25/hour, the hours worked fall. Figure 5.26 shows the income and substitution effects for the case in which hours worked fall with an increase in the wage.
The graph printed here shows the IE and SE for the case in which the hours worked increase (or equivalently, leisure hours fall,) with an increase in the wage.
The initial and new budget lines are labeled BL0 and BLN. The initial optimum is at (16,48) and the new at (12,216). The hypothetical BL is the dotted line. The optimum for the hypothetical budget line is approx. (10,115). The substitution effect is a reduction in the number of hours of leisure from 16 to 10. The income effect is an increase in the number of hours of leisure from 10 to 12.
Suppose that the government imposes a tax of 50% on wage income. Do individuals necessarily work fewer hours? Explain
Answer: The tax has the same effect upon the BL as a wage decrease. As reasoned above, some individuals select to work more at a lower wage and some to work less.
7. Consider the standard labor-leisure choice model and assume that Jo’s preferences satisfy the usual assumptions. The standard graph for this model shows leisure per time period on the horizontal axis and the consumption of good Y on the vertical axis. For simplicity, assume that Jo has no unearned income. She may select the number of hours to work at the market wage.
a. Construct a graph and show an example for which Jo chooses to work fewer hours (consume more leisure) at a high wage than a low wage. Determine the income and substitution effects of the change in the wage.
b. Construct another example for which she chooses to work more hours at a higher wage.
Answer: Figure 5.24 of Besanko and Braeutigan show an example in which the individual works more hours as the wage increases from $5 per hour to $10/hour to $15/hours. As the wage increase from $15/hour to $20/hour to $25/hour, the hours worked fall. Figure 5.26 shows the income and substitution effects for the case in which hours worked fall with an increase in the wage.
The graph printed here shows the IE and SE for the case in which the hours worked increase (or equivalently, leisure hours fall,) with an increase in the wage.
The initial and new budget lines are labeled BL0 and BLN. The initial optimum is at (16,48) and the new at (12,216). The hypothetical BL is the dotted line. The optimum for the hypothetical budget line is approx. (10,115). The substitution effect is a reduction in the number of hours of leisure from 16 to 10. The income effect is an increase in the number of hours of leisure from 10 to 12.
Suppose that the government imposes a tax of 50% on wage income. Do individuals necessarily work fewer hours? Explain