University of Victoria. Economics 325 Public Economics SOLUTIONS

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1 University of Victoria Economics 325 Public Economics SOLUTIONS Martin Farnham Problem Set #5 Note: Answer each question as clearly and concisely as possible. Use of diagrams, where appropriate, is strongly encouraged. Problem sets are ungraded. However, developing a careful understanding of the problems will, on average, dramatically improve your exam performance. True/False/Uncertain For each question, state clearly whether you find the statement to be true, false, or uncertain. Then provide a clear explanation. Answers without explanation will be given zero points. 1) The income-leisure model discussed in class generates an upward-sloping labour supply curve. False. It generates a backward-bending labour supply curve. Make sure you understand why this is the case. It has to do with when substitution effects of a wage increase dominate, vs. when income effects of a wage increase dominate. At higher levels of hours worked (lower quantities of leisure) income effects of a wage increase are likely to be larger hence the labour supply curve is more likely to bend backwards. 2) An increase in the income tax rate will lead to higher government revenues. Uncertain. If the income tax rate goes up and income remains constant, then government revenues will rise. But it s possible that income will fall in response to the higher tax rates (either due to people working less or due to people hiding income from taxation). Then whether income rises or falls will depend on the relative magnitude of the changes in the tax rate and the income level. The fact that tax revenues may rise or fall in response to a tax rate increase is captured in the Laffer Curve. 3) A reduction in the income tax would cause an increase in national saving. Uncertain. There are two reasons this is uncertain. First, even if personal saving rose as a result of the decrease in the income tax rate, the government would be collecting lower revenues and therefore government saving would fall. It is not certain (without applying additional assumptions) which of these effects would dominate, and so we can t know that national saving (or social saving as your textbook calls it) would rise.

2 Second, it s not even clear theoretically that personal saving would rise in response to the lower income tax rate. Using the intertemporal consumption model discussed in class we can model the effect of a decrease in the income tax by treating it as an increase in the after-tax interest rate (since interest income is now taxed at a lower rate). The increase in the after-tax interest rate has two effects, an income effect and a substitution effect. The income effect suggests consuming more today (which implies a reduction in saving); the substitution effect suggests consuming more tomorrow (i.e. saving more today). Only if the substitution effect dominates would we expect the decrease in the income tax rate to lead to an increase in personal saving. C 2 c 2 c 2 w 2 c 1 c 1 In the diagram above, I assume tax deductibility of interest payments is not permitted. The point (w 1,w 2 ) indicates the income earned in each period. The point (c 1,c 2 ) denotes the original consumption choice. Therefore saving was originally w 1 -c 1. After the tax rate is reduced saving is w 1 -c 1. The way I ve drawn it, the income effect dominates, so personal saving actually falls. This doesn t have to be the case, however. It just illustrates that personal saving could rise or fall. 4) A tax that can be perfectly avoided results in no deadweight loss, because if the person pays no tax, she will not change her behavior. False. In taking action to avoid the tax, the person is already altering her behavior. If she can costlessly avoid the tax, then there will be no deadweight loss. So if the tax were purely voluntary, then we would say it caused no DWL. But in general, it takes costly effort in order to avoid paying a tax, so even if the tax can be fully avoided, it will incur deadweight loss. 5) Lump-sum taxes are efficient and equitable. False. They are efficient, because they induce no change in relative prices and therefore no substitution effect. But they are highly inequitable. Making everyone pay taxes of w 1 C 1

3 $10,000 a year would be very unfair to those who make very little income. It would be highly regressive (i.e. the poor would pay a much larger fraction of their income in taxes than the rich). Short Answers 1. Using a supply-demand diagram, illustrate a tax on widgets. Assume that demand slopes down and supply slopes up. Assume that for purposes of the law, the tax is paid to government by producers. a) Describe the tax incidence. Who pays more of the tax, consumers or producers? What determines who pays more? Your answer will depend on how you draw the picture. In the first case, I draw a relative steep demand curve and relatively flat supply curve. This will yield a tax incidence that falls mostly on consumers. The intuition for this is that the relatively flat supply curve (steep demand curve) means producers can wiggle away from the tax more effectively than consumers.

4 In a second case, I reverse the relative steepness of the curves and apply the same per unit tax. Now the relative steepness of the supply curve means that producers have a harder time wiggling away from the tax than consumers, who are more price (and hence tax) sensitive. In this case the tax incidence falls more on producers than consumers. I could draw a third case, which would have supply and demand curves with slopes of equal magnitude. In this case, the tax incidence would be equally split (i.e., the drop in price received by producers as a result of the tax would be equal to the rise in price faced by consumers as a result of the tax. b) Illustrate the deadweight loss of the tax. I ll use the second case, from (a), to illustrate:

5 The DWL (shaded) is just the total benefit in excess of total cost that is lost as a result of the reduced quantity of widgets produced and consumed. c) Now assume that for purposes of the law, the tax is paid to government by consumers. Redraw the picture above, leaving everything else the same. Again, I ll use the second case from (a):

6 When the law says taxes are to be paid by producers, we show the tax as increasing the minimum willingness to accept of producers. When the law says taxes are to be paid by consumers, we can show the tax as decreasing the maximum willingness to pay of consumers. Either way, we get exactly the same result. The tax here is exactly the same size as the tax in (b), we just show a different curve shifting by an identical vertical distance. As a result, the change in quantity and prices is exactly the same. The DWL is also the same. This illustrates the point that it doesn t matter who hands the tax payment over to the government. Instead, what matters is the relative elasticity (or simply slope in this case) of supply and demand. d) Describe the tax incidence. Who pays more of the tax, consumers or producers? This is showed in the diagram in the answer to (c). e) Illustrate the DWL of the tax. This is showed in the diagram in the answer to (c).

7 Now assume that we re in the long run. Supply in a competitive market (due to free entry) is generally thought to be perfectly elastic. Go back to assuming the tax is paid to government by producers. f) Describe the tax incidence. Why has the story changed from the short-run story? Thus far, we ve faced an upward sloping supply curve. That generally means we re in the short run (where some factor of production capital, in the usual story is fixed). In the long run, we assume all factors of production are variable. Hence there s no story we can tell in the long run about diminishing returns to a variable input (e.g., labour), due to fixedness of another input (e.g., capital). Unless there are some external economies of scale present (what are called agglomeration externalities in urban economics) economists generally assume a flat long-run marginal cost curve. In the long run, because the supply curve is perfectly elastic, the entire tax incidence falls on consumers. Because producers are infinitely price responsive, they are able to pass the entire burden of the tax onto consumers, in the long run. Note that in the short run,

8 the relevant supply curve is the short-run supply curve. So the short-run tax analysis that we usually do holds in the short run. But in the long run, firms can respond to lower prices by exiting the industry, and such exit leads to yet higher prices for consumers. Below is a graphic analysis that illustrates what happens in both the short run and long run. We start at Q,P with the industry in long run (and short run) equilibrium. In longrun equilibrium all firms are earning zero economic profits. There is initially no tax. Then a per unit tax is imposed. I show the tax causing marginal willingness to pay to fall by the amount of the tax (represented by a downward shift of the demand curve). We instantly move to a new short run equilibrium at (P s,sr,q t,sr ). Consumers pay the tax so they pay a total of P c,sr per unit in this short run equilibrium. Thus far this is the usual short run tax story. So far, the incidence of the tax is shared between consumers and producers. But now we let time pass and see what happens. We were initially in a long-run equilibrium, meaning producers earned zero profits. In the short run, the price producers receive has fallen from P to P s,sr. This means all producers are earning negative profits (assuming identical producers). Over time, firms will exit the industry in response to the negative profits. Exit can be shown as a gradual shift of the short-run supply curve to the left. This might take months or even over a year, and is represented by the dotted arrows in the diagram. Eventually exit brings profits back up to zero and then exit stops. This is a new long-run equilibrium at Q t,lr. Note that we are now on both our LR MC curve and a new SR MC curve that reflects the shrunken industry. Note that the price producers receive has risen back to what it was originally, P. But the tax hasn t changed. This means the process of exit has driven the price paid by consumers up to P c,lr as supply has contracted. Thus, consumers in the long run end up paying 100% of the tax, and producers pay none of it.

9 g) Illustrate the DWL of the tax. How does this differ from the short-run story? (hint: It will probably help to draw both the short-run and long-run supply curve in the same picture and impose the tax). What lesson do we learn about about elasticity and DWL by doing this comparison? This is shown in the diagram above. In the long run, the DWL is the area in red. In the short run it is the bluish area. DWL is greater in the long run, because there is a larger quantity adjustment in the long run, after firms have responded to the tax by exiting. Now move back to considering the short run (i.e., supply is not perfectly elastic) h) Consider a case where supply is upward sloping but demand is perfectly inelastic. Illustrate tax incidence and DWL.

10 In this case, demand is vertical at Q. The untaxed equilibrium is (P,Q). Because consumers are inflexible (have inelastic demand) producers are able to pass all of the tax onto consumers. So the producer price does not increase when the tax is imposed. Just the consumer price increases, and by the full amount of the tax. Because there is no change in equilibrium quantity (Q=Q ) due to the tax, there is no DWL. i) Think about the consumer choice model where we consider income and substitution effects of a price change. Would you find the same deadweight loss (say using Equivalent Variation to measure harm to consumers) in the consumer choice model as you found in h? Why or why not? No. We would find positive DWL. As long as consumers have preferences represented by smooth indifference curves there will be substitution effects of taxation. When consumers substitute away from a taxed good, they reduce revenue collected. So long as there s no substitution effect, EV=Govt Revenue and there s no DWL. But with a substitution effect, EV>Govt Revenue. EV-Govt Rev = DWL. That is, if the dollar harm caused to the consumer exceeds the revenue generated, society is worse off from the tax.

11 j) Now consider a case where demand is downward sloping but supply is perfectly inelastic. Illustrate tax incidence and DWL. Again, there is no change in equilibrium quantity, so no DWL. And incidence is born by those on the inelastic side of the market. This time, that is producers. The consumer prices remains the same after the tax is imposed. The producer prices falls by the full amount of the tax.

12 2. Consider a consumer who is endowed with income that they can use to buy combinations of two commodities, gas and other stuff. Assume that gas is a normal good. a) Illustrate the overall effect of a tax on gas on an individual s consumption bundle. What does the tax do to the individual s consumption of gas? Consumption of other stuff? other stuff C A Assuming gas is a normal good, the income effect and substitution effect will both lead the person to consume less gas (the optimal choice changes from A to C). If other stuff is normal, the income effect will lead to a reduction in consumption of other stuff. The substitution effect will lead to an increase in consumption of other stuff, as its relative price has fallen. Whether consumption of other stuff will rise or fall depends on the relative magnitude of the income and substitution effects. b) Decompose the overall change in consumption of gas into income and substitution effects. other stuff gas C B A gas

13 Substitution effect: The move from A to B, measured along the gas axis. Income effect: The move from B to C, measured along the gas axis. c) Explain what is different in the theoretical predictions about the commodity tax versus the wage tax. What is the source of the difference? With the commodity tax, the theoretical prediction on the effect of the tax is clear for the taxed good. Both income effects and substitution effects of the tax are negative. With the wage tax income effects and substitution effects of the tax go in opposite directions (SE says consume more leisure, income effect says consume less leisure). The source of the difference is the fact that in the commodity tax model, agents are endowed with cash that they they spend on either gas or other stuff. In the leisureincome model, people are endowed with leisure, which they can sell (by working) in exchange for other stuff (or income or consumption as we can call the y-axis good). So the tax on the wage essentially amounts to a tax on other stuff in the income-leisure model. Therefore the income and substitution effects (on leisure) go in opposite directions, just as they do for other stuff in the commodity tax case we are currently discussing. d) On a single diagram, illustrate both the Equivalent Variation of the tax and the Compensating Variation of the tax. This is most easily done if we assume that 1 unit of other stuff costs $1. That way we can interpret vertical distances as dollar amounts.

14 The equivalent variation of a tax is the maximum amount the person would be willing to pay to avoid having the tax imposed on them. In other words, sitting at point A, the person would be willing to pay as much as EV to avoid getting taxed and thus being forced to move from A to C. The compensating variation of a tax is the minimum amount the person is willing to accept in order to agree to have the tax imposed. In other words, sitting at point C and wanting to get back to point A, the person could be convinced to have the tax left in place if they were offered a cash payment of CV. Notice that EV does not generally equal CV. It is also true that neither of these is generally equal to the change in consumer surplus from the tax, which would be the equivalent measure of lost consumer welfare in the Marshallian supply-demand diagram. e) Using the diagram from (d), illustrate the deadweight loss associated with the commodity tax, using Equivalent Variation as your measure of harm to consumers. DWL=EV-GR. There is DWL because there exists some other tax (a lump-sum tax) that would generate the same utility loss but with a greater revenue gain. Therefore, this tax must be wasteful, relative to that other tax. The waste is measured by the loss in government revenue relative to the efficient tax (lump-sum) resulting from use of the commodity tax. Note, now do this analysis for a subsidy instead of a tax. Recall that a gas subsidy would have the opposite effect of a gas tax. Calculate EV and DWL from the subsidy.

15 3. Using the commodity tax in (2) as basis for comparison, explain why a lump-sum tax would be efficient. A lump sum tax is efficient because there is no other tax that can raise more revenue for the same loss in utility. To see this, try pivoting the dotted budget line in a way that holds revenue (measured by vertical distance between the dotted line and the original budget constraint) constant while improving utility (or that increases utility while holding revenue constant). It cannot be done. The commodity tax is inefficient, because at point C, we can pivot around the new indifference curve (thus holding utility loss constant) and generate a higher level of revenue by finding the dotted budget constraint above. 4. Recalling that the change in relative prices leads to deadweight loss, consider a commodity tax where the government taxes both gas and other stuff such that relative prices don t change. a) In theory, would such a tax cause deadweight loss? No, not if gas and all other stuff are truly taxed so that relative prices don t change. Recall that if relative prices don t change, we re back to what s essentially a lump-sum tax. It s only when relative prices change that we get deadweight loss. b) In practice, can you think of difficulties with implementing such a tax? Hint: Consider the goods consumed in problem 1. Is it practical to tax both of these goods? Yes. Managing to truly tax all other stuff would be tricky. For instance, leisure is commonly thought of as a good that people consume. We all enjoy leisure time and leisure is available to us to use in limited quantities. It has an opportunity cost associated with it, represented by the wage we would earn if we engaged in one less hour of leisure. A tax on all other stuff would have to be applied to leisure as well. But it s difficult to tax leisure because people don t actually go to the store and buy it. If this commodity tax was not applied to leisure, then it would be distortionary (because it would change relative prices) and would induce deadweight loss. Since leisure cannot be taxed, we could consider subsidizing the wage as an alternative. Or we could impose a higher tax on complements to leisure (airline tickets to Hawaii, etc), in order to prevent people from responding to commodity taxes by substituting toward leisure. Both of these solutions have problems as well. 5) Now let s analyze an income tax that applies to interest income. Consider the model of consumption choice from lecture, in which an individual chooses between consumption in period 1 and consumption in period 2. Assume that consumption in periods 1 and 2 is normal. Assume also that individuals earn the same nominal income in each period. a) In the absence of an income tax, what determines the slope of the budget constraint?

16 The slope of the budget constraint is determined by the interest rate. Technically, the slope is (1+r). This says that in order to get one more unit of consumption today, you must forgo (1+r) units of consumption tomorrow. This should make intuitive sense. If you consume $10 extra today, you pass up the opportunity to put that money in the bank, and earn back 10*(1+r) tomorrow. b) What effect does an increase in the interest rate have on consumption in period 1? Are any assumptions required to make an unambiguous theoretical prediction? An increase in the interest rate has two effects. It raises the relative price of consumption today (the budget constraint becomes steeper) and therefore causes a substitution effect toward consumption tomorrow. At the same time, it can have an income effect that is either positive or negative depending on whether you started off as a saver or a borrower. If you were a saver (i.e. you consumed up the budget constraint from the endowment), the increase in the interest rate makes you richer. If you were a borrower (i.e. you consumed down the budget constraint from the endowment), the increase in the interest rate makes you poorer. Thus the sign of the income effect will depend on what your original consumption choice was. C 2 c 2 w 2 B A w c 1 c 1 w 1 C 1 For someone originally choosing to consume at A, what is the effect of the increase in interest rate on saving? The substitution effect says consume less today; the income effect says consume more today and more tomorrow. The way I ve drawn it here, the overall effect has the substitution effect dominating. Therefore saving (w 1 -c 1 ) increases. c) Now consider introducing an income tax. Assume that there is no deductibility of interest payments. Illustrate the effect of this on the consumer s budget constraint. What is the intuition for the effect on the budget constraint?

17 An income tax (that is applied to interest income) effectively lowers the after-tax interest rate. This means you have to give up less consumption tomorrow to get another dollar of consumption today. This is equivalent to flattening out the budget constraint above the endowment point (because of the non-deductibility of interest payments the slope does not change below the endowment point). C 2 w d) What will be the effect of the income tax on saving? Recall that saving is w 1 -c 1 (income-consumption). Is there a clear theoretical prediction on the effect of the income tax on saving? Again, it depends what the initial consumption choice of the person is. Notice that if the person were originally borrowing, this would have no effect on their consumption choice. It will only affect people who were originally savers. For savers, the income tax lowers the after-tax interest rate. The substitution effect of this change should be to consume more today. The income effect should be to consume less today. The overall effect will depend on whether the income or substitution effect dominates. C 1 C 2 A C w saving (w/o tax) saving (w/tax) C 1

18 The way I ve drawn this, the income effect actually dominates, so saving rises. But it is just as plausible that saving would fall. e) What does the empirical literature suggest about the effect of income taxes on saving? Evidence from both Canada and the US suggests that the effect of income taxes on saving are small to none. They may slightly reduce saving (suggesting that in reality, the substitution effect may dominate). 6) Using the federal marginal tax rate schedule in your notes: a) calculate the average and marginal tax rates and the dollar tax liability of someone who makes $200,000 a year. 0.15*(40,970-10,382)+0.22*(81,941-40,970)+0.26*(127,021-81,941)+0.29*(200, ,021)=$46, b) calculate the average and marginal tax rates and the dollar tax liability of someone who makes $50,000 a year? 0.15*(40,970-10,382)+0.22*(50,000-40,970)= c) Does this system of tax rates constitute a progressive, proportional, or regressive income tax? Explain. It constitutes a progressive income tax. This is a result of the increasing marginal tax rates. Note that the average tax rate is around 23% for the person making $200,000 a year. It s around 13% for the person making $50,000 a year.