1

ECMB02F -- Problem Set 3 Solutions 1. See Nicholson 2. I have attached my copy of Figure 3.1 on the next page to show my work. Since it may be hard to see exactly where the budget lines are tangent to the indifference curves, your answers may differ slightly from mine. a) If the individual takes no leisure and works all 3000 hours, earnings would be $60,000. Hence the budget line has an intercept of 60 on the income axis and 3000 on the leisure axis. This budget line is tangent to indifference curve I2 at point A on my diagram, where earnings are $35,000 per year, leisure is 1250 hours, and hence this person works 3000 - 1250 = 1750 hours. b) The proportional income tax (tax rate = 1/3) reduces the after tax wage rate to $13.33 an hour and changes the intercept for the budget line on the income axis to $40,000. This new budget line is tangent to indifference curve I1 at point B on my diagram, where earnings are $25,000 per year, leisure is 1125 hours, and hence this person works 3000 - 1125 = 1875 hours, an overall increase in hours worked of 125. To divide the impact on hours worked into an income and a substitution effect, we need to draw a budget line that is tangent to the old indifference curve at the new price ratio. That tangency occurs at point C. Hence the substitution effect shifts us from point A to point C and reduces hours worked by 125 (there are 125 more hours of leisure). The income effect shifts us from point C to point B and increases hours worked by 250. c) A lump sum tax would shift the initial budget line without changing its slope. To have the same effect on utility, it would have to shift the consumer onto the I1 indifference curve. I have drawn such a line and labelled it "lump sum tax line". Since the vertical-axis intercept is now 46, the interpretation would be that this line lowers income by $14,000 relative to the initial budget line, independent of how many hours are worked by the consumer (the vertical difference between the initial line and the lump sum tax line is $14,000 at every value for leisure). However, the 1/3 income tax shifted the consumer to point B, where the consumer worked 1875 hours, earned $37,500 before tax, and hence paid $12,500 in tax. Because the lump sum tax raises $14,000. the excess burden, as defined, is $1500. The lump sum tax only has an income effect. Because the income tax also involves a substitution effect, it will increase hours of leisure (relative to the lump sum tax), reduce hours of work, and hence reduce the tax collected. 3. This is our old friend, the Cobb-Douglas utility function, so you could probably do this problem without going through maximizing the Lagrangian, but I will do so anyway: the utility maximization problem is to maximize U = XT subject to the constraint that X + wT = 4000w. The Lagrangian is: L = XT + λ(4000w - wT - X) To maximize, we set ∂L/∂T = X - λw = 0

∂L/∂X = T - λ = 0 ∂L/∂λ = 4000w - wT - X = 0

combining the first two conditions gives us X/T = w or X = wT Plugging this into the third condition gives us T = 2000 and X = 2000w

3

a) If w = 30, then X = 60,000 T = 2000, and U = 120,000,000

b) The 50% tax on earned income effectively reduces the wage rate to 15. If w = 15, then X = 30,000 T = 2000, and U = 60,000,000 Since the individual works 2000 hours (work = 4000 - T), the government revenue is $30,000

per year (gross earnings are $60,000 and the government takes half).

If instead the government imposes a lump sum tax of LST on the consumer, this just reduces income by LST, so that the budget constraint is now X + wT = 4000w - LST. We could use some of the work in this chapter to solve this problem directly, but it may be useful to go through the Lagrangian again. The Lagrangian is now:

L = XT + λ(4000w - wT - X - LST) To maximize, we set ∂L/∂T = X - λw = 0

∂L/∂X = T - λ = 0 ∂L/∂λ = 4000w - wT - X - LST = 0

Notice that the first two condition are the same as before (this is typical of any problem with a constant in the budget constraint). Combining these two conditions yields X/T = w or X = wT Plugging this into the third condition gives us T = 2000 - (LST/2w) and X = 2000w - LST/2 Since w = 30, we know that T = 2000 - LST/60 and X = 60000 - LST/2 Thus U = (2000 - LST/60)(60000 - LST/2) = 120000000 - 2000LST + LST2/120 Since we want this utility to be the same as the utility under the income tax, and since this was 60,000,000, we set 120,000,000 - 2000LST + LST2/120 = 60,000,000 or 60,000,000 - 2000LST + LST2/120 = 0 or LST2 - 240,000LST + 7,200,000,000 = 0 This is a straight-forward quadratic equation, made difficult only by all the zeros floating around! We can solve this using the quadratic formula, which you remember is the solution to ax2 + bx + c = 0 [ x = (1/2a){-b ± (b2-4ac)1/2} ] In this case LST = (1/2){240,000 ± (240,0002 - 28,800,000,000)1/2} LST = (1/2){240,000 ± (57,600,000,000 - 28,800,000,000)1/2} LST = (1/2){240,000 ± 169,706} = 204,853 or 35,147 The correct answer is LST = 35,147, since the first answer (204,853) gives negative values for

X and T. The excess burden is thus $5147

By the way, the way to avoid these zeros is to start the problem by defining the lump sum tax in thousands of dollars. If the lump sum tax is Z thousand dollars, then LST = 1000Z, and the budget constraint is 4000w = wT+X+1000Z or 120,000 = 30T + X + 1000Z. When we solve this problem, we get X = 60,000 - 500Z and T = 2000 - 50Z/3 Thus U = (60,000-500Z)(2000-50Z/3) so U = 120,000,000 - 2,000,000Z + (25,000/3)Z2 = 60,000,000 or 60,000,000 - 2,000,000Z + (25,000/3)Z2 = 0 or Z2 - 240Z + 7200 = 0 This is the same quadratic as above, without most of the zeros, and it solves to Z = 204.853 or 35.147 Also by the way, you might observe that under the lump sum tax, the new values of X and T are 42426.5 and 1414.2

4

c) In the previous example, the tax on earnings imposed an excess burden, even though the individual worked 2000 hours both before and after the tax. This illustrates the proposition that there is an excess burden even if the income tax does not affect the amount of labour supplied by workers. d) Now assume that the individual has non-labour income equal to A (A will be 12,000 initially). The utility maximization problem is to maximize U = XT subject to the constraint that X + wT = 4000w + A. The Lagrangian is:

L = XT + λ(4000w + A - wT - X) To maximize, we set ∂L/∂T = X - λw = 0

∂L/∂X = T - λ = 0 ∂L/∂λ = 4000w + A - wT - X = 0

combining the first two conditions gives us X/T = w or X = wT Plugging this into the third condition gives us 4000w + A - wT - wT = 0 or 2wT = 4000w + A or T = 2000 + (A/2w). Of course, w = 30, so we can also write:

T = 2000 + (A/60) If A = 12,000, then T = 2000 + 200 = 2200, and the consumer takes 2200 hours of leisure and hence works 1800. The reduction in hours worked accords with our intuition that lottery winners work less. In order to reduce work to zero, we would need our consumer to consume 4000 hours of leisure, so 4000 = 2000 + A/60, or A/60 = 2000, or A = 120,000. We would need a $120,000 lottery winning to eliminate all work. 4a) The relevant budget line for this consumer in part a is the one with C1-intercept of 100 (because if the consumer only spends money in the present, setting C2=0, 100 units of C1 can be purchased) and C2-intercept of 150 (because if the consumer only spends money in the future, setting C1=0, 150 units of C2 can be purchased because of the interest earned). This budget line has the equation C2 = (100 - C1)(1.5) or C1 + C2/1.5 = 100. The budget line is tangent to indifference curve I2 at point D where the consumer spends $30 in the present (C1=30), saves $70 (100-C1=70), and thus spends $105 in the future (C2=105). b) Now the budget line shifts so that the C1 intercept is still 100, but the C2 intercept is now 120. This budget line has the equation C2 = (100 - C1)(1.2) or C1 + C2/1.2 = 100. The budget line is tangent to indifference curve I1 at point A where the consumer spends $40 in the present (C1=40), saves $60 and thus spends $72 in the future (C2=72). To divide the impact of this change in interest rates into an income and a substitution effect, we must draw a line parallel to the new budget line but tangent to the old indifference curve. This line has a C1 intercept of 115 and a C2 intercept of 138 and is tangent to I2 at point C. The substitution effect shifts us from point D to point C, while the income effect shifts us from point C to point A. From D to C, C1 increases from 30 to 50, so savings fall from $70 to $50. Thus the substitution effect decreases savings by $20. From C to A, C1 falls from 50 to 40, so savings rise from $50 to $60. Thus the income effect increases savings by $10.

The interpretation is as follows. The fall in the interest rate has two separate effects. The first is to make spending in the future relatively more expensive (compared with spending today), since the consumer must give up more consumption today to pay for the same expenditures in the future, and thus makes spending today relatively cheaper than spending in the future. This fall in the relative price of present spending increases our demand for it, so we spend more today and save less. We call this the substitution effect. The second effect of the

5

fall in the interest rate is to reduce our real income, since we cannot buy as much C1 and C2 overall. Since we feel poorer, we buy less of all normal goods. Since present spending is normal, we spend less in the present. This leaves us with more savings. We call this the income effect. The two effects move in opposite directions, so the total impact on savings is unclear.

c) As discussed in part b, the reduction in the interest rate moves us to point A, where the consumer spends $40 today, saves $60 and earns after-tax interest of $12 (so spending in the future is $72). Now, when savings are $60, the gross (before-tax) interest earned by the consumer is $30 (50% on $60). The 60% tax on interest yields $18 to the government in the future. The present value of that tax revenue is $18/1.5 = $12. The lump sum tax which leaves the consumer no worse off corresponds to a budget line parallel to the old budget line (the one tangent to point D) and tangent to the new indifference curve I1. This budget line is tangent to I1 at point B and has C1 intercept of 86. Because this line has reduces the C1 intercept from 100 to 86, it amounts to a lump sum tax of $14. Clearly $14 > $12, so extra revenue is raised for the government by a lump sum tax. The lump sum tax corresponds to a pure income effect. Because the substitution effect of a tax on interest reduces savings, it reduces the interest which is taxed by the government. Since the lump sum tax avoids this substitution effect, it has a greater ability to generate revenue. Since the lump sum tax raises more revenue than the tax on interest, it suggests that the government ought to tax income in the present instead of taxing interest income. The government does not entirely agree with this, since it does tax some interest income. However, interest in RRSP's and in pension plans is protected, suggesting that the government buys part of this argument. 5. The diagram relevant to this problem has been attached to the next page. a) The relevant budget line for this consumer in part a is the one with C1-intercept of 120 (because if the consumer only spends money in the present, setting C2=0, 120 units of C1 can be purchased) and C2-intercept of 360 (because if the consumer only spends money in the future, setting C1=0, 360 units of C2 can be purchased because of the 200% interest earned). This budget line has the equation C2 = (120 - C1)(3) or C1 + C2/3 = 120. The budget line is tangent to indifference curve I2 at point D where the consumer spends $50 in the present (C1=50), saves $70 (120-C1=70), and thus spends $210 in the future (C2=210). b) Now the budget line shifts so that the C1 intercept is still 120, but the C2 intercept is now 180. This budget line has the equation C2 = (120 - C1)(1.5) or C1 + C2/1.5 = 120. The budget line is tangent to indifference curve I1 at point A where the consumer spends $40 in the present (C1=40), saves $80 and thus spends $120 in the future (C2=120).

To divide the impact of this change in interest rates into an income and a substitution effect, we must draw a line parallel to the new budget line but tangent to the old indifference curve. This line has a C1 intercept out beyond the graph and a C2 intercept of 270 and is tangent to I2 at point C. The substitution effect shifts us from point D to point C, while the income effect shifts us from point C to point A. From D to C, C1 increases from 50 to 80, so savings fall from $70 to $40. Thus the substitution effect decreases savings by $30. From C to A, C1 falls from 80 to 40, so savings rise from $40 to $80. Thus the income effect increases savings by $40.

6

This result is not unusual, since the effect of a change of the interest rate on savings is ambiguous. To explain this to the noneconomist, we would repeat the explanation in part b of the previous question. A more intuitive way to see this result is as follows. The fall in the interest rate makes it "harder" to save for the future, which tends to reduce savings. On the other hand, if we have something that we wish to buy in the future ("a secure retirement' for example), we now have to save more to achieve that goal. This tends to increase savings. c) As discussed in part b, the reduction in the interest rate moves us to point A, where the consumer spends $40 today, saves $80 and earns after-tax interest of $40 (so spending in the future is $120). Now, when savings are $80, the gross (before-tax) interest earned by the consumer is $160 (200% on $80). The 75% tax on interest yields $120 to the government in the future. The present value of that tax revenue is $120/3 = $40. The lump sum tax which leaves the consumer no worse off corresponds to a budget line parallel to the old budget line (the one tangent to point D) and tangent to the new indifference curve I1. This budget line is tangent to I1 at point B and has C1 intercept of 75. Because this line has reduces the C1 intercept from 120 to 75, it amounts to a lump sum tax of $45. Clearly $45 > $40, so extra revenue ($5) is raised for the government by a lump sum tax. We have called this extra revenue the excess burden, which in this case is $5. As before, the lump sum tax corresponds to a pure income effect. Because the substitution effect of a tax on interest reduces savings, it reduces the interest which is taxed by the government. Since the lump sum tax avoids this substitution effect, it has a greater ability to generate revenue. 6a) The Lagrangian is:

L = C1C2 + λ(120 - C1 - (1/3)C2) To maximize, we set ∂L/∂C1 = C2 - λ = 0

∂L/∂C2 = C1 - λ/3 = 0 ∂L/∂λ = 120 - C1 - C2/3= 0

combining the first two conditions gives us C2/C1= 3 or C2 = 3C1 Plugging this into the third condition gives us C1 = 60 and C2 = 180, and savings are 60 When the interest rate falls to 100%, we can resolve the problem, changing the Lagragian accordingly: L = C1C2 + λ(120 - C1 - (1/2)C2) The solution of this is straightforward and gives us C1 = 60 and C2 = 120, with savings once again 60 (thus, this is a case where the change in interest rate does not affect savings). (b) The Lagrangian is now:

L = C11/2 + C2

1/2 + λ(120 - C1 - (1/3)C2) To maximize, we set ∂L/∂C1 = 0.5C1

-1/2 - λ = 0 ∂L/∂C2 = 0.5C2

-1/2 - λ/3 = 0 ∂L/∂λ = 120 - C1 - C2/3= 0

combining the first two conditions gives us C1-1/2/C2

-1/2 = 3 or C2 = 9C1 Plugging this into the third condition gives us C1 = 30 and C2 = 270, and savings are 90 When the interest rate falls to 100%, we can resolve the problem, changing the Lagragian accordingly: L = C1

1/2 + C21/2 + λ(120 - C1 - (1/2)C2)

The solution of this is similar to the previous solution, and gives us C1 = 40 and C2 = 160, with savings now at 80. Here the fall in the interest rate reduces savings.

7

(c) If the consumer has $120 in the future, then at an interest rate of 200%, this present value of this income is worth ($120)/3 = $40. As such, we can proceed as we have in part (a), but using an income of $40 instead of $120.

L = C1C2 + λ(40 - C1 - (1/3)C2) To maximize, we set ∂L/∂C1 = C2 - λ = 0

∂L/∂C2 = C1 - λ/3 = 0 ∂L/∂λ = 40 - C1 - C2/3= 0

combining the first two conditions gives us C2/C1= 3 or C2 = 3C1 Plugging this into the third condition gives us C1 = 20 and C2 = 60.