Chapter 19
Distortionary Taxes and Subsidies
In Chapter 18, we began our discussion of how policies that alter or “distort” market prices in
a competitive market can create dead weight losses.1 But such policies are not limited to those
that explicitly set price floors above the equilibrium price or price ceilings below the equilibrium
price. In fact, the most common government policies that distort market prices involve tax and
subsidy policies rather than explicit regulatory policies aimed at setting prices directly. With
federal, state and local governments funded primarily through taxes, and with all government
spending combined making up more than 40 percent of most economies, tax policy then becomes
a particularly important area for understanding how price distortions impact welfare.
Because of the important role taxes play in most economies, we have already developed many
of the concepts that are crucial to understanding tax policy in earlier chapters, particularly in the
chapters leading up to and including Chapter 10. We already understand from this development
that, on the consumer (or worker or saver) side of markets, taxes result in dead weight losses or
inefficiencies to the extent to which they give rise to substitution effects. Now that we have added
producers to the model, however, we are able to talk much more explicitly about how taxes affect
economic behavior in equilibrium when all sides of the market respond to changes in incentives.
This makes it possible to now become explicit about who is affected most by particular taxes —
who ends up paying taxes in equilibrium, and how this translates to welfare changes for consumers,
producers and workers as well as society overall.
Again, it is worth noting that, in pointing out the logic behind the emergence of dead weight
losses from taxation, the economist is not voicing opposition to taxes per se. Rather, the economist
is in the business of identifying costs and benefits — leaving it up to others to judge whether
particular policies with particular costs and benefits are good or bad. Taxes have hidden costs that
policy makers should understand, and some taxes have greater hidden costs than others. Similarly,
some taxes may appear to affect one group on the surface while in fact economic analysis suggests
that they will actually affect a different group much more. Understanding issues of this kind is the
point of this chapter, with later chapters identifying more clearly why we might indeed need to use
taxes despite their hidden costs. As in the previous chapter, it is also important to note that the
1 In addition to the usual consumer theory material, this chapter includes material on labor and capital markets
– material which draws on our development of models in Chapters 3 and 8 as well as the later sections of Chapter 9.
Students who have not read this material can skip Sections 19A.2.2, 19A.2.3 and 19B.3. The chapter also presumes
a basic understanding of producers theory from Chapter 11, partial equilibrium as developed in Chapters 14 and 15,
and elasticity as developed in the first part of Chapter 18.
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inefficiencies from taxes (and subsidies) are identified here in a competitive setting in which there
are no other distortions. We will see in upcoming chapters that, in non-competitive settings or in
the presence of other distortions, taxes and subsidies may become efficiency-enhancing. Finally, we
will develop the main ideas in this chapter within our partial equilibrium framework (focusing on a
single market), but at the end we will offer an example to illustrate how general equilibrium effects
may also be important in many settings. These theme will then carry forward into Chapter 20.
19A
Taxes and Subsidies in Competitive Markets
As we have pointed out before, almost all taxes change some opportunity costs in the economy.
Put differently, almost all taxes distort some of the market prices that, at least under certain
circumstances, coordinate all sides of a market to an efficient outcome. As a result, almost all taxes
result in deadweight losses and are thus, to one degree or another, inefficient. But not all taxes
are equally inefficient, nor do all taxes impact all groups in the same way. We therefore begin our
intuitive analysis of taxes with an analysis of who actually ends up paying taxes in equilibrium
before we revisit the issue of deadweight loss and the potential for real world taxes that might
actually be efficient.
One note before we start: Taxes and subsidies are very similar in that both change the prices
individuals face in an economy. In fact, we can think of subsidies as simply negative taxes. For
instance, a government might impose a 10% tax on every good that is sold in a market, or it
might impose a 10% subsidy. The 10% tax will cause an increase in the price of the good sold
in the market, while the 10% subsidy will cause a decrease in the price. Taxes raise revenues for
the government, while “negative taxes” (or subsidies) cause increases in government expenditures.
Thus, even when we don’t explicitly treat taxes and subsidies separately in this chapter, you should
always be able to conduct a particular economic analysis for both positive and negative taxes.
19A.1
Who Pays Taxes and Receives Subsidies?
Since there are always two sides to a market — buyers and sellers, a government that wants to
tax the good sold in a market can in principle do so by writing many different types of tax laws.
In particular, the government might write the law in such a way as to make the buyers be the
ones that pay the tax and send a check to the government whenever they purchase the taxed good.
Alternatively, the government might write the law so as to make sellers send the tax payment to the
government. Or the government might do some combination of the two. For instance, in the case
of U.S. payroll taxes that fund expenditures in the social security system, the government requires
workers to pay half of the overall tax and employers to pay the other half. Thus, on every pay stub
that accompanies your paycheck, you will notice that your employer has deducted some payroll
taxes and sent that amount to the government on your behalf. What you do not see on your pay
stub is that the employer sent a separate check for her share of your payroll taxes.
19A.1.1
Statutory versus Economic Incidence
It turns out that it ends up not mattering at all which way the government writes tax laws —
whether it requires the bulk of the tax to be paid by buyers or sellers. Economists use the term
statutory incidence of a tax to refer to the way in which the legal (or “statutory”) obligation to pay
a tax is phrased in tax laws. In the case of U.S. payroll taxes, for instance, the statutory incidence
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689
of the tax falls equally on employers and employees. We will distinguish this from the economic
incidence of a tax by which we will mean how the tax burden is actually divided among buyers and
sellers when a new equilibrium under the tax has emerged.
Consider, for instance, a tax law that imposes a statutory incidence of a per-unit tax t on the
producers of good x. In other words, for every unit of x that is produced, the firm producing it
owes a tax of amount t. This raises the marginal cost of production by t, shifting up the M C
(and AC) curves for each firm in the market. Since market supply in the short run is simply the
combination of all M C curves (above AC), this implies that the short run market supply curve
will shift up by t. Similarly, since the long run market supply curve is determined by the lowest
points of (long-run) AC curves, the long run market supply curve will shift up by t. This shift in
the market supply curve is illustrated in Graph 19.1a by the upward shift (equal to the vertical
distance of the green arrow) of the market supply curve from the initial (blue) supply curve S to
the new (magenta) supply curve S ′ . This causes an increase in the market price from p∗ to p′ , and
it reduces the quantity of x transacted in the market from x∗ to x′ .
Graph 19.1: Statutory versus Economic Incidence of Taxes
Exercise 19A.1 Will the increase in price from the tax be larger or smaller in the long run? (Hint: How
is the price elasticity of supply in the long run usually related to the price elasticity of supply in the short
run?)
Now suppose that instead the government imposed the statutory incidence of an equally sized
per-unit tax on consumers of x. In this case, costs would remain unchanged for producers but each
consumer who was previously willing to pay a price p will now only be willing to pay (p − t) given
that she knows she must still send t per unit to the government. Thus, the demand curve will
shift down by t, a distance indicated by the size of the green arrow in Graph 19.1b, causing a new
equilibrium to emerge at price p′′ . At first glance, it certainly appears that panels (a) and (b) look
quite different due to the different statutory incidence of the same per-unit tax.
If you think about what information is contained in panels (a) and (b), however, you will notice
that the two graphs actually end up being identical in the underlying predicted impact of the two
taxes on buyers and sellers. In panel (a), good x is traded at price p′ — but sellers do not get to
keep this price for each unit they sell. Rather, they still have to pay the government a tax t for
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Chapter 19. Distortionary Taxes and Subsidies
each unit they sell, leaving them with a net-of-tax price (p′ − t) while buyers pay price p′ . In panel
(b), on the other hand, good x is traded at the lower price p′′ , but buyers still need to pay the tax
t. Thus, buyers in panel (b) actually pay a price (p′′ + t) while sellers receive the lower price p′′ .
In both cases, sellers end up receiving a price that is exactly t below the price buyers pay, with the
difference going to the government.
Once we recognize this, we can graph the economic incidence of a tax regardless of the statutory
incidence in a less complicated graph depicted in panel (c). Here we simply insert a vertical green
line that is equal to the per-unit tax to the left of the pre-tax equilibrium and label the price read off
the demand curve as pd and the price read off the supply curve as ps . Since the green line segment
in panel (c) has exactly the same height as the green arrows in panels (a) and (b), it logically follows
that p′ in panel (a) is equal to pd in panel (c) and p′′ in panel (b) is equal to ps in panel (c). The
price pd is then the price paid by buyers after the tax is imposed, and ps is the price received by
sellers, with the difference t going to the government. Notice further that xt in panel (c) is logically
equal to x′ and x′′ in panels (a) and (b).
Exercise 19A.2 Using a pencil, redraw the graphs in panel (a) and (b) but this time label clearly which
price buyers end up paying and sellers end up receiving, taking into account that sellers have to pay the tax
in panel (a) and buyers have to pay the tax in panel (b). Then erase the shifted curves in your two graphs.
Do the two graphs now look identical to each other and to the graph in panel (c)? (The answer should be
yes.)
Regardless of which way tax laws are phrased and who is legally responsible for paying a tax,
the economic analysis of Graph 19.1 therefore suggests that the economic incidence of the tax will
always be exactly the same: Buyers and sellers will share the burden of the tax, with buyers paying
higher prices and sellers receiving lower prices than they did before the imposition of the tax. The
exact same is true for negative taxes known as subsidies. Graph 19.2 illustrates the impact of a
per-unit subsidy s, with the new price received by sellers ps now higher than the new price paid by
buyers pd .
Graph 19.2: The Economic Incidence of a Subsidy
Exercise 19A.3 Illustrate how the equilibrium changes when the subsidy is paid to sellers (thus reducing
their M C). Compare this to how the equilibrium changes when the subsidy is paid to buyers (thus shifting
19A. Taxes and Subsidies in Competitive Markets
691
the demand curve). Can you see how both of these types of subsidies will result in an economic outcome
summarized in Graph 19.2?
19A.1.2
Economic Incidence and Price Elasticity
Our analysis so far may lead one to incorrectly conclude that the economic burden of taxes (and
the economic benefit of subsidies) is shared equally between buyers and sellers. This has been true
so far only because of the way we happened to graph demand and supply curves in Graphs 19.1
and 19.2. The actual economic incidence of taxes and subsidies, however, depends on the relative
responsiveness of buyers and sellers to price changes.
Consider, for instance, a tax on cigarettes. The evidence suggests that most smokers are relatively unresponsive to changes in the price of cigarettes and will continue to smoke roughly as
much at higher prices as they do at lower prices. A tax imposed on cigarettes will therefore tend to
primarily be passed onto consumers regardless of who is legally responsible for paying the tax. This
is depicted in Graph 19.3a where demand is relatively inelastic. A tax t will then raise the price
paid by buyers by a lot while lowering the price received by cigarette companies relatively little.
Graph 19.3: Price Elasticities and the Relative Burden of Taxes on Buyers and Sellers
Now consider a tax on the sale of oil. Oil is, at least in the short run, in relatively fixed supply,
leaving the oil market with a relatively steep supply curve. Panel (b) of Graph 19.3 then illustrates
that a tax will cause a sharp decline in the price received by sellers while causing only a small
increase in the price paid by buyers. Thus, the economic incidence of a tax falls disproportionately
on those who are less responsive to price changes — i.e. those whose behavioral response to price
is more inelastic.
Exercise 19A.4 During the 2008 Presidential campaign in the U.S., oil prices increased sharply. Some
candidates advocated a “tax holiday” on gasoline taxes to help consumers. Others argued that this would
have little effect on gasoline prices in the short run. Assuming each side was honest, how must they have
disagreed on their estimates of underlying price elasticities?
Exercise 19A.5 In graphs with demand and supply curves similar to those in Graph 19.3, illustrate the
economic impact on buyers and sellers of subsidies. How does the benefit of a subsidy relate to relative price
elasticities?
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19A.1.3
Chapter 19. Distortionary Taxes and Subsidies
The Impact of Taxes on Market Output and Tax Revenue
Just as price elasticities determine who bears disproportionately more of the burden of a tax (or
gains a disproportionate share of the benefit of a subsidy), price elasticities determine how much
market output will respond to changes in taxes and consequently how much tax revenue will be
raised. This is illustrated in Graph 19.4 where the impact on market output is illustrated for
three different scenarios. In each panel of the graph, buyers and sellers are assumed to be similarly
responsive to price changes, and the relative burden of a tax is therefore similar for both sides of the
market. The size of the tax imposed in each of the panels is exactly the same. However, panel (a) of
the graph begins with relatively elastic market demand and supply curves that become increasingly
more inelastic in panels (b) and (c). As a result, market output drops a lot in panel (a), less in
panel (b) and even less in panel (c). Thus, as buyers and sellers become more unresponsive to price
changes, taxes have a smaller impact on market output.
Graph 19.4: Taxes and Market Output as Economic Agents become more Price-Responsive
Exercise 19A.6 Does the impact of subsidies on market output also rise with the price-responsiveness of
buyers and sellers?
In addition, each panel of Graph 19.4 illustrates the total tax revenue collected by the government (using the same per unit tax t) as the shaded green area. These areas are simply the
vertical distance (which represents the per unit tax rate) multiplied by the horizontal distance that
represents output after the tax is imposed. Note how tax revenue changes as demand and supply
become more inelastic. This should, of course make intuitive sense: If consumers and producers
are very responsive to price changes, their large response to a tax will undermine efforts to raise
revenue.
Exercise 19A.7 Suppose the government has already imposed the taxes graphed in Graph 19.4 and is now
considering raising this tax. Can you see in these graphs under what circumstances this would result in a
decrease in overall tax revenues?
19A. Taxes and Subsidies in Competitive Markets
19A.1.4
693
Differential Impact of Taxes on Other Markets
Whenever we use the partial equilibrium model that focuses on a single market in isolation, we are
implicitly assuming that all other prices in the economy are moving in lock-step and thus all other
goods in the economy can be modeled as one big composite good. We are also treating our analysis
of a tax change as if it occurred in an environment where other goods are not taxed. However, it
is often the case that taxes imposed in one market cause differential effects in other markets and
that taxes already exist and may be impacted in these other markets. For instance, suppose the
government imposes a large tax on gasoline. Then it is likely that markets for more fuel efficient
cars are affected differently from markets for less fuel efficient cars, while markets for paperclips
may not be impacted very much at all.
This then creates further complications for tax policy analysts. Suppose, for instance, that the
government is already taxing car sales when it contemplates the imposition of a new tax on gasoline.
Tax revenue in the market for fuel efficient cars is likely to increase as a result of an increase in
the tax on gasoline as demand for such cars increases, while tax revenue is likely to decrease in
markets for less fuel efficient cars where demand drops. When new taxes or increases in existing
taxes are contemplated in economies that already have many pre-existing taxes, a full treatment of
the economic impact of the new tax thus involves tracing the effect of the new tax through other
markets that are affected. The secondary effects in these other markets may, in some cases, end up
being of larger significance than the primary effect in the market for the taxed goods, which in turn
can mean that a tax that looks “good” when analyzed in isolation looks “bad” in a fuller economic
analysis. The reverse is, of course also possible.2
Exercise 19A.8 Suppose the tax on fuel efficient cars is low and the tax on gas-guzzling cars is high. Is it
likely that our partial equilibrium estimate of a tax on gasoline will cause us to over- or under-estimate the
full impact on government revenues?
19A.2
Dead Weight Loss from Taxation Revisited
Market demand and supply curves are full descriptions of predicted behavioral changes induced by
price changes. As such, they are the appropriate tools with which to predict the economic incidence
of taxes and subsidies — i.e. how much prices paid by buyers and received by sellers will change,
as well as the impact of such policies on market output. However, as we already began to discuss
in Chapter 10, these are not necessarily the appropriate curves to use for an analysis of changes in
welfare.
In particular, we know that consumer surplus (and changes in consumer surplus) can be measured as areas underneath marginal willingness to pay (or compensated demand) curves. Only
when these curves are the same as regular (uncompensated) demand curves can the market demand curves be used to measure consumer surplus. And, we furthermore know from our work in
Chapter 10 that compensated and uncompensated demand curves are the same only when tastes
for the underlying good are quasilinear. It is for this reason that we assumed quasilinear tastes
in the previous chapter where we identified consumer surplus along (uncompensated) market demand curves. If we know that tastes for the underlying good are either normal or inferior, we have
2 An analysis of the types of effects hinted at above is often referred to as “second best” analysis. While we are
implicitly assuming that our analysis starts in a “first best” world of full efficiency, a “second best” analysis starts
with a model in which new taxes are introduced into a “second best” world where there already exist tax distortions
elsewhere.
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Chapter 19. Distortionary Taxes and Subsidies
already demonstrated in Graph 10.9 how dead weight loss on the consumer side will be over or
underestimated if welfare changes are measured on uncompensated demand curves.
We will then begin our analysis of the full welfare impact on both buyers and sellers by initially
once again assuming that tastes are quasilinear — and thus the market demand curve can be used
to calculate consumer surplus in goods markets. We will then proceed to demonstrate cases in
which quasilinearity is clearly the wrong assumption — and how an analysis of welfare changes
from taxation will necessarily lead to large policy mistakes if conducted as if tastes were indeed
quasilinear.
19A.2.1
Dead Weight Loss from Taxes and Subsidies when Tastes are Quasilinear
Graph 19.5 illustrates the economic effect of a tax t in panel (a) and of a subsidy s in panel (b)
along the lines discussed in the previous section. Assuming for now that tastes are quasilinear and
demand curves can therefore be interpreted as compensated demand curves, changes in consumer
and producer surplus are then easily identified much as we identified such changes in the previous
chapter.
Graph 19.5: Dead Weight Loss when Tastes are Quasilinear
In panel (a), an initial consumer surplus of (a + b + c) shrinks to (a) as consumers face the higher
after-tax price pd while producer surplus shrinks from (d + e + f ) to (f ) as producers face the lower
after-tax price ps . The government earns no tax revenue before the tax but gets areas (b + d) after
it is imposed. The overall surplus in society therefore falls from the initial (a + b + c + d + e + f ) to
an after-tax (a + b + d + f ), leaving us with a dead weight loss of (c + e) represented by the shaded
blue area in Graph 19.5a.
For the subsidy in panel (b), on the other hand, both consumers and producers are better off
after the subsidy but the government incurs a cost that we also have to take into account. Consumer
surplus rises from the initial (g + h) to the final (g + h + i + n + m) as consumers now face a lower
price, while producer surplus rises from the initial (i + j) to the final (h + i + j + k) as producers now
sell goods at a higher price. The cost of the subsidy, however, is the per-unit subsidy rate times
the number of units transacted, or (sxs ) which is represented by the area (h + i + k + l + m + n).
Adding consumer and producer surpluses and subtracting the cost of the subsidy (which is zero
before its imposition), we then get an overall surplus that falls from an initial (g + h + i + j) to a
final (g + h + i + j − l), giving a dead weight loss of area (l) represented by the magenta triangle
19A. Taxes and Subsidies in Competitive Markets
695
in Graph 19.5b. Once we know we can identify dead weight loss from taxation or subsidies as
triangles to the left and right of the pure market equilibrium, we can then easily see how the size
of the deadweight loss is impacted by price elasticities of demand and supply. For instance, looking
across the three panels of Graph 19.4, the dead weight loss triangle represented as the triangle next
to the shaded rectangles clearly shrinks as demand and supply become more price inelastic. In
fact, were one of the two market curves completely inelastic, the deadweight loss triangle would
disappear entirely and the tax would be efficient. The same is true for deadweight losses from
subsidies.
If you have covered dead weight losses from taxation in a previous economics course, chances are
that you learned to read dead weight loss exactly as we just described. Remember, however, that
the analysis we have just done is valid only if the tastes for the underlying goods are quasilinear
because only then are compensated and uncompensated demand curves the same. As we already
demonstrated in Chapter 10, we will either over or underestimate deadweight loss if we use uncompensated demand curves when goods are either inferior or normal. Awareness of the difference
between compensated and uncompensated curves becomes even more important, however, as we
analyze taxes in labor and capital markets where the way we have just illustrated dead weight loss
is almost certainly quite incorrect.
19A.2.2
Dead Weight Loss from Taxes in Labor and Capital Markets
Most of the tax revenue raised by governments comes from taxation of income derived either from
labor or from investments (i.e. savings). Such taxes alter the opportunity cost of leisure (in the case
of taxes on labor income) or the opportunity cost of consuming now or in the future (in the case
of taxation of savings). And we have demonstrated before that such taxes (typically) give rise to
opposing wealth and substitution effects for labor or capital supply – thus causing (uncompensated)
labor and capital supply curves to at least partially hide the substitution effects that give rise to
dead weight losses from taxation.
Suppose, for instance, that you were told that all workers are unresponsive to changes in their
wage — i.e. as wage changes, they continue to work the same number of hours. This would imply
that the market supply curve for labor is perfectly inelastic as depicted by the vertical supply curve
S in Graph 19.6a. Inserting the market labor demand curve D then yields an equilibrium wage w∗ .
If the government now imposes a per labor hour tax of t in this market, there would be no impact
on the number of hours workers worked, nor would there be any change in the wage that employers
had to pay workers per hour. However, because of the inelasticity of labor supply, workers would
end up bearing the entire burden of the tax and would receive an after-tax wage (w∗ − t). And
the government would raise revenues equal to the blue shaded area in panel (a) of the graph. All
of this is sound economic analysis using exactly the right market curves to predict the economic
impact of the tax.
But notice that there is no triangle next to the box that indicates tax revenue — which would
lead many to conclude that there is no deadweight loss. It is at this point, however, that the market
demand and supply curves become misleading because there is almost certainly a deadweight loss
that is obscured in Graph 19.6a. And, if there is a deadweight loss, it has to lie on the worker (or
supply) side of the labor market since the wage rate paid by producers is unaffected by the tax.
Consider, then, the underlying consumer choice picture that gives rise to individual labor supply
curves and, when aggregated across all workers, to market labor supply. This picture is depicted in
Graph 19.6b where leisure hours are on the horizontal and dollars of consumption on the vertical
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Chapter 19. Distortionary Taxes and Subsidies
Graph 19.6: Dead Weight Loss from Wage Taxes when Labor Supply is Perfectly Inelastic
axis. The fact that the entire economic incidence of the tax falls on workers in this case (as seen
from panel (a)) implies that worker budget constraints shrink from the initial blue budget with
slope −w∗ to the new magenta budget with slope -(w∗ − t).
If workers are indeed unresponsive to changes in wages, then the worker depicted in panel (b)
of the graph will make the same leisure choice on the initial and the final budget. Panel (b) labels
the after-tax choice as A and the before tax choice as C, with C lying exactly above A due to
the inelasticity of the worker’s behavior. When we then add the indifference curve uA (that makes
point A optimal after the imposition of the tax) and introduce the green compensated budget (that
keeps utility at uA but leaves the wage at the before-tax level), we see that the lack of change in
worker behavior is due to fully offsetting substitution and wealth effects.
Now notice the following: The green compensated budget is equivalent to a lump sum tax that
makes the worker just as well off at B as the wage tax does at point A. However, the lump sum tax
raises revenue equal to the distance between the parallel blue and green budgets, while the wage
tax raises revenue equal to the vertical distance between A and C. Put differently, the lump sum
tax raises revenue L while the wage tax raises only revenue T , implying a deadweight loss equal to
the difference between the two quantities illustrated as DW L in the graph.
Why is there a deadweight loss in panel (b) for the individual worker we are modeling but no
deadweight loss triangle in panel (a)? It is because deadweight losses on the worker side of the
market arise only from substitution effects that are obscured by the counteracting wealth effect
when (uncompensated) labor supply is derived as perfectly inelastic. Panel (c) then presents the
uncompensated and compensated labor supply curves for these workers within the same graph,
illustrating a perfectly inelastic (uncompensated) labor supply curve but an upward sloping compensated labor supply (that represents the change in labor choices for workers whose utility is kept
at uA ). The former includes both the wealth and substitution effect while the latter includes only
the substitution effect (much as marginal willingness to pay – or compensated demand – curves for
consumers only contain substitution effects).
Under the wage tax, the workers settle at point A in Graph 19.6c and receive a wage of (w∗ − t).
19A. Taxes and Subsidies in Competitive Markets
697
With worker surplus measured along the compensated supply curve (just as consumer surplus is
measured along compensated demand curves), this gives an after-tax surplus of (b). Under the
lump sum tax that leaves the workers just as well off, they would end up at point B earning a wage
w∗ . This would give them a worker surplus of (a + b + c), (a + c) greater than the surplus at point
A. However, at point A the workers had already paid the wage tax (a) (equal to the shaded blue
area), while at point B we have not yet taken into account the fact that the workers have paid a
lump sum tax that makes them just as happy as they would be at A. Since workers are equally
happy at the two points (as seen in panel (b)) but have a surplus (a + c) greater at B than at A,
it must be that the lump sum tax raises (a + c) in revenue. This leaves a difference (c) between
the wage tax revenue and the lump sum tax revenue that both leave the workers equally well off –
implying DW L = c.
Of course labor supply is not always perfectly inelastic and may even be downward sloping
for some workers, but notice that the direction of the substitution effect always implies that the
compensated labor supply curve is upward sloping. As a result, whether one can see it or not in
a picture of market equilibrium in the labor market, wage taxes will have dead weight losses so
long as there is any substitutability at all between leisure and consumption (which there almost
certainly is.)
Exercise 19A.9 * Illustrate, using an analogous set of steps we just used as we worked our way through
Graph 19.6, how wage subsidies are inefficient even when workers are completely unresponsive to changes
in wages. (Hint: If you get stuck, read the next section and come back.)
19A.2.3
Dead Weight Losses from Subsidies in Labor or Capital Markets
We can show a similar error that may arise when we use the (uncompensated) savings-interest rate
relationship — which represents the supply curve for financial capital — to predict the welfare effect
of savings subsidies. Consider, for instance, the case where individuals are completely unresponsive
to changes in the rate of return to savings — they always put the same amount into the savings
account. This gives a perfectly inelastic supply curve for capital as presented in panel (a) of
Graph 19.7. When a subsidy for saving is now introduced, the entire benefit of the subsidy accrues
to savers as their rate of return jumps from the initial equilibrium interest rate r∗ to the new interest
rate (r∗ + s) that includes the per unit subsidy s.3 The shaded blue area is then the cost incurred
by the government, with no change in the capital saved given the inelastic response by savers.
Once again it appears as if there is no dead weight loss triangle and the subsidy is therefore
efficient. But again this is not true because the capital supply curve obscures the very substitution
effects that are responsible for the inefficiencies of subsidies. We can show this most easily by
illustrating the case of a single saver who exhibits inelastic saving behavior in panel (b) of the
graph. The subsidy changes the budget from the original blue to the final magenta, with A optimal
after the subsidy and C optimal before (and with both bundles exhibiting exactly the same level
of savings.) Next we can put in the indifference curve uA that makes point A optimal after the
subsidy raises the rate of return to savings, and we can put in the compensated green budget that
results in the same utility as this saver gets at A but at the before-tax interest rate. The reason
for the inelastic behavioral response for this worker is that the substitution effect is fully covered
up by an equally large and opposite wealth effect.
3 If you have trouble seeing why this economic incidence of the subsidy emerges, try first graphing the impact of
a subsidy (as we did earlier in this chapter) for the case where the supply curve is almost but not quite perfectly
inelastic. Then make the supply curve increasingly inelastic until you see Graph 19.7a emerge.
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Chapter 19. Distortionary Taxes and Subsidies
Graph 19.7: Dead Weight Loss from Subsidies for Saving when Saving Behavior is Perfectly Inelastic
But we can also see in this graph that this subsidy is inefficient. The total government cost of
paying the subsidy to this one saver can be measured as the vertical distance between A and C and
is labeled G. At the same time, we can see from the difference between the blue and green budgets
that a lump sum subsidy of L would make this individual exactly as well off as the distortionary
subsidy that cost G. The difference between G and L is the dead weight loss for this one saver.
Translating points A, B and C to a graph with capital on the horizontal axis and the rate of
return on the vertical, we can then see what goes wrong when we try to find this deadweight loss
in panel (a) of the graph. More specifically, in panel (c) of the graph we illustrate the vertical
(uncompensated) capital supply curve that is formed from points A and C in panel (b), but we
also illustrate the compensated capital supply curve (derived from A and B in panel (b)) that
corresponds to the utility level uA . Using this compensated curve, we can identify the saver surplus
as (a + b) under the distortionary subsidy and as just (b) under the lump sum subsidy (at point
B). Since this saver is equally happy at A and B, the lump sum subsidy at B must be equal to (a).
But the distortionary subsidy cost (a + c) — which is (c) more than the lump sum subsidy that
made the saver just as well off. Thus, the DW L distance in panel (b) is analogous to the magenta
area (c) in panel (c).
19A.2.4
DWL and Revenue as Tax Rates Rise
In Chapter 10, we illustrated the idea that, on the consumer side of the market, as tax rates rise by
a factor of k, dead weight loss increases by approximately k 2 . Now that we are familiar with the
process by which taxes affect both the consumer and producer sides of the market, we can extend
this intuition more generally.
Consider, for instance, the market demand and supply curves in Graph 19.8a, and, to keep the
analysis as simple as possible, suppose that tastes for good x are quasilinear — thus allowing us to
assume that the market demand curve is equivalent to the aggregate marginal willingness to pay
curve. The market price in the absence of taxes is then p∗ . If a per-unit tax of t is imposed, the
market output drops from x∗ to xt , with prices for consumers rising and prices for producers falling.
19A. Taxes and Subsidies in Competitive Markets
699
The deadweight loss from this tax would then be equivalent to the blue triangle, with half of the
deadweight loss falling on the consumer side of the market and half falling on the producer side.
Then suppose that the tax is doubled to 2t — thus raising the price for consumers, decreasing the
price for producers and leading to the output x2t . Now the deadweight loss increases by the shaded
magenta area.
Graph 19.8: Dead Weight Loss and Tax Revenue when Tastes are Quasilinear
Suppose that each triangle, such as the two triangles that form the initial blue deadweight loss
when the tax is t, is equal to $500. This implies that each square, such as the squares contained
in the magenta area, is equal to $1000. Adding these up, we then get that the deadweight loss
associated with the initial tax t is $1000 while the deadweight loss associated with the tax 2t is
$4000. A doubling of the tax leads to a quadrupling of the deadweight loss. We can then keep
increasing the tax, to 3t, 4t going all the way to 7t, and by adding up the relevant deadweight loss
areas, we can derive the relationship between the tax rate and the deadweight loss in panel (b) of
Graph 19.8. The graph illustrates that it is indeed still the case that, with the linear demand and
supply curves graphed in panel (a), deadweight loss rises by a factor of k 2 whenever the tax rate
increases by a factor of k.
Exercise 19A.10 How large does deadweight loss get if the tax rate rises to 3t? What if it rises to 4t?
We can similarly trace out the tax revenue collected by the government as the tax increases.
When the tax rate is set at t, the tax revenue is txt which is equal to 12 squares in panel (a) or
equivalent to $12,000 when each square represents $1000. Similarly, when the tax rate is 2t, the
tax revenue is 2tx2t or $20,000. The relationship between the tax rate and tax revenue that then
emerges in panel (c) of the graph has an inverse U-shape, with tax revenue equal to zero when
there is no tax and equal to zero once again when the tax becomes sufficiently high. This is another
version of what we previously called the Laffer Curve that suggests governments will ultimately
lose revenue if tax rates get too high.
While we are illustrating this in a stylized graph of linear demand and supply curves which lead
to an equal sharing of economic tax incidence between consumers and producers, the intuitions are
applicable more generally (even if the precise relationship between tax rates, deadweight loss and
700
Chapter 19. Distortionary Taxes and Subsidies
tax revenue will differ somewhat.) As a result, the simple intuition (first discussed in Chapter 10)
emerging from these graphs has often led to the general advice from economists to governments
that it is more efficient to levy low rates on large tax bases rather then high tax rates on small tax
bases.
Exercise 19A.11 Illustrate the relationship between subsidy rates, the deadweight loss from a subsidy and
the cost of the subsidy using the same initial graph of supply and demand as in Graph 19.8a in graphs
analogous to panels (b) and (c) of Graph 19.8.
19A.3
Taxing Land: An Efficient Real World Tax
At this point, you may have given up your search for a fully efficient tax. As we have demonstrated,
it is not sufficient for demand or supply relationships in general to be fully inelastic for a tax to be
efficient, because inelastic behavioral relationships with respect to price may well mask underlying
substitution effects that make taxes inefficient. There is, however, a tax that economists have
identified as an efficient tax because of the existence of a price-inelastic relationship that does not
mask such substitution effects. This tax is a tax on land value or on land rents.
Land value is simply the market price of land, while a land rent is the income (or utility) one
can derive from a particular quantity of land over a particular time period. While land value and
land rents are therefore different concepts, they are closely connected. After all, the reason land
has value is that the owner of the land can derive land rents every year. It is easiest to discuss this
in terms of farm land, but the general lesson applies more generally to all forms of land whether
the land is used for farming, production of non-farm goods or housing. What makes land special is
that it is not itself something that is produced — and it therefore exists in essentially fixed supply.4
19A.3.1
The Relationship between Land Value and Land Rents
Sometimes I get sick of talking about economics all the time and yearn to reconnect with the land
in ways that my wife does not fully appreciate. Suppose, however, that I convince her to move to
Iowa and buy 100 acres of farm land. I can now derive annual income from this land by either
producing potatoes directly or by renting it out to someone else who will produce potatoes. For it
to be worth it to farm the land myself, I have to receive compensation that covers the opportunity
cost of my time and the opportunity cost of the land that I will be using. The opportunity cost of
my time is determined by what other market opportunities I have — perhaps my other alternative
is teaching economics which carries with it a certain level of compensation. The opportunity cost
of using the land, on the other hand, is the income I could derive from the land by just renting it
to someone else. How much I can rent the land for in the market of course depends on the quality
of the land and on how much someone else would be able to make with it.
Suppose, for instance, I could rent the 100 acres in the market for $10,000 per year and my time
is worth $100,000 per year. Then in order for me to farm the land myself, I will have to be able
to generate at least $110,000 in income by farming the land — otherwise, I am better off making
$100,000 elsewhere and collecting $10,000 in rent. In equilibrium, only those who are relatively
good at farming will end up making the choice to be farmers and the rest of us will do something
4 There are instances when land is actually “produced” — such as in the Netherlands where significant amounts
of land have been “reclaimed” from the sea in a complex system of dams and levies, or in the Florida Everglades
where marsh land is converted to usable land for housing. Even in these instances, however, the land itself simply
existed before “improvements” of that land made it usable for production or consumption.
19A. Taxes and Subsidies in Competitive Markets
701
else — and if farming is a competitive industry, those who engage in farming will make zero profits
and thus exactly an amount equal to their opportunity cost of time plus the rent they have to pay
for the land (whether they are paying it explicitly or whether they simply forego collecting rents
from others if they own the land themselves).
Put differently, the land itself produces an income stream of $10,000 per year — the annual land
rent, which we will assume that we collect at the end of each year. But the value of the land — i.e.
how much I could sell it for in the market, is based on not only this year’s income stream but also
all future income streams that can be produced from this land. In Chapter 3, we already discussed
how such future income streams are evaluated in the presence of interest rates. If, for instance,
the annual interest rate is r (expressed in decimal form), then $10,000 one year from now is worth
($10,000/(1+r)) and $10,000 n years from now is worth ($10,000/(1 + r)n ). The value of land is
then the present discounted value of all future land rents, or simply ($10,000/(1+r)) (from the rent
derived a year from now for this year’s rent) plus ($10,000/(1 + r)2 ) (from the rent derived two
years from now) and so forth. It turns out that, when all land rents into the future are added up
in this way, the resulting land value is equal to ($10,000/r). Or, put more generally, land value LV
is related to land rents LR according to the formula
LV =
19A.3.2
LR
LR
LR
... =
+
.
2
(1 + r) (1 + r)
r
(19.1)
Taxation of Land Rents
Now suppose that the government requires landowners to pay 50% of their land rents as a tax. We
will call this here a 50% land rent tax. Graph 19.9 illustrates the market for renting a particular
type of land — say land of a particular quality in Iowa. Such land is in fixed supply, which implies
that the supply curve is completely inelastic. As a result, the economic incidence of the tax is
fully on land owners who are renting the land to farmers (or to themselves if they themselves are
farming), with the annual rental value that land owners get to keep dropping by 50% (while the
rent paid by renters remains unchanged).
Graph 19.9: A 50% Tax on Land Rents
702
Chapter 19. Distortionary Taxes and Subsidies
As the current owner of the land, I will have no choice but to accept a lower rental price for my
land (once I pay the tax). I may get very upset at this, and I may try to instead sell the land, but
remember that the value of land is simply equal to the present discounted value of all future land
rents. Since all future (after-tax) land rents have just fallen by 50%, this implies that the value of
my land has just fallen from (LR∗ /r) to (0.5LR∗ /r). In other words, the 50% tax on land rents
has caused the value of an asset that I own to decline by 50% — and I have no way to substitute
to anything else and avoid the tax. If I continue to hold on to the land, I will make 50% less on it
every year, and if I decide to sell it I will make 50% less now and forego any future rents. In present
value terms, I am equally well off whether I hold on to the land, whether I sell it, or whether I hold
on to it for a little while and then sell it.
You might think that perhaps I can make myself better off by turning around and using the
land for something else, but if the tax is truly on (unimproved) land rents, it is independent of
what exactly is being done with the land – because only the value of the unimproved land is taxed.
Whether I use it for farming or for producing paperclips or for housing, a land rent tax still taxes
the rental value of the land itself. So there is literally nothing I can do to prevent paying this tax
in one form or another — and thus no possibility for a substitution effect to emerge and make the
tax inefficient. A tax on land rent is therefore a simple transfer of wealth from landowners to the
government. Land owners are worse off but the government captures all the wealth that landowners
lost. It is in part for this reason that a writer by the name of Henry George suggested over 100
years ago that all government expenditures should be financed by taxes on land rents. In fact,
Henry George went even further and suggested that all land rents should be taxed at 100%.5
Exercise 19A.12 What would be the economic impact of a 100% tax on land rents (levied on owners)?
The proposal to tax land rents is a policy option that is increasingly considered in the U.S. by
local governments who rely for much of their revenue on property taxes. Property taxes are not land
rent taxes because they tax both land rents and the improvements on land (such as housing). Thus,
to the extent that property taxes change the opportunity cost of improving land, such taxes may
give rise to substitution effects that create inefficiencies (by diverting capital away from housing
and into other uses) – and local governments can move toward more efficient taxes by lowering the
tax on improvements on land and increasing the tax on land itself.6 In developing countries where
much wealth is often concentrated in the hands of relatively few landowners, taxes on land rents
are similarly discussed for purposes of funding government expenditures and redistributing wealth.
You can learn more about land taxes and their relations to other types of taxes (such as property
taxes) in courses such as Urban Economics and Public Finance.
19A.4
General versus Partial Equilibrium Tax Incidence
Our analysis of the economic incidence of a tax has thus far focused solely on partial equilibrium
models where we have implicitly assumed that the incidence of a tax is confined solely within
the market in which the tax is imposed. As we have seen, it does not matter whether the tax is
5 Henry George (1839-97) made this argument in 1879 in his book Progress and Poverty, and the resulting Henry
George Theorem has been formalized by a number of local public finance and urban economists since then. His
proposal that all government functions be financed by a 100 percent tax on land rents was based in part on the
philosophical notion – which has come to be referred to as “Georgism” – that everyone should own what they create
but that everything found in nature – such as land – belongs to everyone equally.
6 A property tax that levies different rates on land and structures is often called a split-rate tax. You can analyze
this in more detail in end-of-chapter exercise 19.12.
19A. Taxes and Subsidies in Competitive Markets
703
statutorily imposed on one party or the the other — on buyers or sellers — because who ends up
paying the tax within this partial equilibrium framework will depend on the relative elasticities
of demand and supply curves. Put differently, we have seen that taxes are shifted from buyers to
sellers and vice versa depending on whose economic behavior is more inelastic.
Tax shifting, however, is not always confined solely to actors within a particular market. In
many instances, taxes (and tax incidence) is shifted outside the market in which a tax is imposed
and onto actors in other markets that face no legal tax obligations. When this happens, there
are general equilibrium tax incidence effects in addition to the partial equilibrium effects we have
analyzed.
Consider, for instance, a tax on housing. Such a tax could be considered a tax on capital invested
in housing markets — with investors bearing some of the burden of this tax as their rate of return
on housing capital declines when the tax is imposed. But owners of capital have other options of
where to invest their money. Prior to the imposition of a housing tax, it must be the case that the
equilibrium rate of return on capital is the same for all forms of capital (at least to the extent to
which other forms of capital have similar risk associated with it). If a tax on housing causes the
after-tax rate of return on housing capital to decline, rational investors will shift away from investing
in housing and toward investing in other forms of capital that now have a higher rate of return. An
inward shift in housing capital supply will then raise the after-tax rate of return on housing capital
and cause an outward supply shift in other capital markets with a corresponding decline in the rate
of return on non-housing capital. A new (general) equilibrium will then be reached when the after
tax rate of return on housing capital is equal to the rate of return on non-housing capital. Thus,
some of the incidence of the housing tax is shifted away from owners of housing capital to owners
of all capital.7
We will see other examples of this in Chapter 20 where we will investigate the role of taxes
imposed in one geographic region but not in another. As in the case of the housing tax where
some of the incidence is shifted away from the housing market and toward other capital markets,
we will see that taxes are also shifted from one region (where a tax is imposed) to another. Such
general equilibrium effects of taxes can be extremely important — and thus add a substantial layer
of complexity to tax policy. If this topic is of interest to you, you should consider taking a course
on public finance or public economics in your future studies of economics. For now, you should
merely begin to gain some intuition for the insight that, to the extent to which taxed inputs or goods
are mobile across markets, the imposition of a tax in one market will generate general equilibrium
tax incidence in other markets. This is analogous to the role of price elasticity in determining tax
incidence in a partial equilibrium model — market actors who are more “responsive” bear less of
the tax burden because they can shift that burden to actors who are less “responsive”. In the same
way, market actors who are more “mobile” across markets are able to shift tax burdens to market
actors that are less “mobile” across markets.
7 The property tax, which is a tax on both land and housing, is therefore often viewed as a tax on land (which is
efficient) and a tax on housing capital which translates in general equilibrium to a tax on all capital. An alternative
view of the property tax – known as the “benefit view”, argues that, when combined with strict zoning laws, housing
becomes much more like land — and the housing portion of the property tax therefore has some of the properties of
a land tax. Again, you can learn much more about these different views of the property tax in a local public finance
course.
704
19B
Chapter 19. Distortionary Taxes and Subsidies
The Mathematics of Taxes (and Subsidies)
In this section, we continue our exploration of tax incidence and deadweight loss from taxation
(leaving the analogous case of subsidies for end-of-chapter exercises). We begin in Section 19B.1
with a general demonstration of the relationship between tax incidence and price elasticities —
proving more formally that the degree to which market participants bear the burden of a tax is
increasing in the relative inelasticity of their response to price changes. In Sections 19B.2 and 19B.3,
we then continue by illustrating how deadweight losses are calculated — first for the quasilinear
case and then, in an application to wage taxes, more generally. While tax incidence depends on
uncompensated demand and supply curves, we will see once again that deadweight loss calculations
depend on compensated curves. Finally, we conclude with a very simple example of tax incidence
in a more general equilibrium setting where a tax on housing is shifted to other forms of capital
when capital is mobile between different sectors in the economy.
19B.1
Tax Incidence and Price Elasticities
Consider the general case where demand is given by xd (p), supply is given by xs (p) and the no-tax
equilibrium has price p∗ and quantity x∗ . Now suppose a small tax t (to be paid by consumers for
each unit of x that is purchased) is introduced. This implies that the price pd paid by buyers is t
higher than the price ps at which the good is purchased from suppliers; i.e. pd = ps + t. Taking the
differential of this, we get
dpd = dps + dt;
(19.2)
i.e. the change in the consumer price pd is equal to the change in the producer price ps plus
the change in t. In the new equilibrium, demand has to equal supply, with each evaluated at the
relevant price; i.e.
xd (pd ) = xs (ps ).
(19.3)
Taking the differential of this, we can write
dxs
dxd
dpd =
dps
dpd
dps
(19.4)
and substituting equation (19.2) into equation (19.4), this becomes
dxs
dxd
(dps + dt) =
dps .
dpd
dps
Rearranging terms in this equation, we can write it as
dxd
dxd
dxs
dps = −
−
dt.
dpd
dps
dpd
(19.5)
(19.6)
Before the tax is introduced, the equilibrium was at the intersection of supply and demand at
p∗ and x∗ — a point on both the supply and demand curve. Multiplying equation (19.6) by p∗ /x∗ ,
it becomes
dxd p∗
dxd p∗
dxs p∗
dp
=
−
−
dt,
(19.7)
s
dpd x∗
dps x∗
dpd x∗
19B. The Mathematics of Taxes (and Subsidies)
705
which you should notice contains several price elasticity terms (evaluated at the no-tax equilibrium). Rewriting the equation in terms of these price elasticities, it becomes
(εd − εs )dps = −εd dt
(19.8)
where εd is the price elasticity of demand and εs is the price elasticity of supply. Re-arranging
terms, we can also then write this as
εd
dps
.
=−
dt
εd − εs
(19.9)
What does this tell us? Suppose that supply is perfectly inelastic with εs = 0. Then the
equation says that dps /dt = −1 or dps = −dt. Put into words, the producer’s price adjusts by
exactly the change in the tax, with the producers therefore bearing the entire burden (or incidence)
of the tax. If, on the other hand, demand is perfectly inelastic (εd = 0), dps /dt = 0 or dps = 0 —
the producer’s price does not change and the producers bear none of the incidence of the tax. This
conforms entirely to the intuition we get from simple graphs. Finally, suppose that, at the initial
(pre-tax) equilibrium, consumers and producers were equally responsive to price changes with price
elasticities of demand and supply equal to each other in absolute value, or εs = −εd . Plugging this
into equation (19.9), we get dps /dt = 0.5 or dps = 0.5dt — producers bear half the incidence of the
tax. The equation therefore implies that the incidence of the tax will fall disproportionately on the
side of the market that is relatively less price elastic as we concluded intuitively in Graph 19.3.
Exercise 19B.1 * Demonstrate that, whenever εd is less in absolute value than εs , consumers will bear
more than half the incidence of the tax, and whenever the reverse is true, they will bear less than half of the
incidence of the tax.
Exercise 19B.2 * Can you show that dpd /dt = εs /(εs − εd )? (Hint: Note that equation (19.2) implies
dpd /dt = (dps /dt) + 1.)
One can derive similar conclusions regarding the economic incidence of subsidies which we leave
for end-of-chapter exercise 19.1.
19B.2
Deadweight Loss from Taxation when Tastes are Quasilinear
Tax incidence in a partial equilibrium model then depends on the relative price elasticities of uncompensated demand and supply curves. Deadweight loss calculations, however, depend on elasticities
of compensated demand and supply curves. As we know from our development of consumer theory,
the difference between uncompensated and compensated relationships disappears when income effects disappear — and income effects disappear when tastes are quasilinear. We therefore begin our
discussion of the mathematics of deadweight loss from taxation for the case when tastes are indeed
quasilinear. As we will see, you can do this by calculating areas under demand and supply curves
(as we did in part A of the chapter), but you can also employ the expenditure function we derived
in Chapter 10 and thus avoid using integral calculus.
In the previous chapter, we demonstrated that, when u(x, y) = α ln x + y, demand for the
quasilinear good x is xd (px , py ) = αpy /px . You can also verify for yourself that the demand for y
is given by yd (py , I) = (I − αpy )/py . To focus on just good x within a partial equilibrium model,
we can treat y as a composite good with py = 1, which allows us to write the demands for the two
goods as
706
Chapter 19. Distortionary Taxes and Subsidies
xd (p) =
α
and yd (I) = I − α,
p
(19.10)
where p now simply denotes the price of good x.
Exercise 19B.3 What is the price elasticity of demand for x? What is the cross-price elasticity of demand
for y?
Suppose, then, that the demand side of the market for x can be modeled as arising from the
optimization problem of a representative consumer with the above tastes and some income level I.
Suppose further that the supply side of the market can be represented by the supply curve xs = βp.
Exercise 19B.4 What is the price elasticity of supply?
Setting supply equal to demand and solving for p, we then get that the equilibrium price under
no taxation is p∗ = (α/β)1/2 and the equilibrium quantity transacted is x∗ = (αβ)1/2 .
Now suppose the government imposes a per unit tax t on producers — implying that producers
will receive a price (pd − t) when consumers pay pd . The new equilibrium then requires that supply
evaluated at the producer price equals demand evaluated at the consumer price; i.e. β(pd − t) =
α/pd . Multiplying both sides of this equation by pd and subtracting α, we get βp2d − βtpd − α = 0
which, by the quadratic formula,8 implies a new equilibrium price paid by consumers of
p
p
t + t2 + 4(α/β)
βt + (βt)2 + 4βα
=
(19.11)
pd =
2β
2
with corresponding equilibrium price for producers (net of tax obligations) of
p
−t + t2 + 4(α/β)
ps = pd − t =
.
(19.12)
2
Suppose, for instance, that α = 1, 000 and β = 10. The resulting demand and supply curves
(and their inverses) are then drawn in Graph 19.10, with before and after tax prices and quantities
calculated using the equations above and assuming t = 10. From what we have done in Section A,
we know that (since tastes for x are quasilinear), consumer surplus shrinks from the original area
(a + b + c) to just (a) while producer surplus shrinks from (d + e + f ) to just (f ) while tax revenue
grows from zero to area (b + d) — leaving a deadweight loss of (c + e).
19B.2.1
Calculating Dead Weight Loss using Integrals
If you are comfortable with basic integral calculus, we can then calculate changes in consumer and
producer surpluses using integrals to calculate the appropriate areas under the curves. (If you are
not comfortable with integral calculus, you can simply skip to Section 19B.2.2.) Using the functions
graphed in panel (b) of Graph 19.10, the change in consumer surplus (b + c) is
∆CS =
Z
pd
p∗
8 Recall
xd (p)dp =
Z
pd
p∗
α
dp = α(ln pd − ln p∗ ) = 1000(ln(16.18) − ln(10)) ≈ 481
p
that √
the quadratic formula gives two solutions to the equation ax2 + bx + c = 0: x = (−b −
and x = (−b + b2 − 4ac)/2a. It is the latter that is relevant for our particular problem.
√
(19.13)
b2 − 4ac)/2a
19B. The Mathematics of Taxes (and Subsidies)
707
Graph 19.10: Welfare Changes with Quasilinear Demand
and the change in producer surplus (d + e) as
∆P S =
Z
p∗
ps
xs (p)dp =
Z
p∗
ps
(βp)dp =
β
(p∗ )2 − p2s = 5(102 − 6.182 ) ≈ 309.
2
(19.14)
Exercise 19B.5 Can you verify that our answer for ∆P S is correct by simply calculating the area of the
rectangle (d) and the triangle e in Graph 19.10?
Summing the change in producer and consumer surplus, we then get a total loss of surplus equal
to approximately $790. The tax revenue collected by the government is equal to the $10 per unit
tax times the 61.8 units sold under the tax – or approximately $618. This gives us a deadweight
loss of approximately $172.
19B.2.2
Calculating Dead Weight Loss using the Expenditure Function
In Chapter 10, we also developed an alternative way of calculating the change in consumer surplus
using the expenditure function. In particular, we concluded that the ∆CS (area (b + c)) is equal to
the maximum lump sum tax the representative consumer would be willing to pay to avoid having
the distortionary tax imposed. Plugging the demands xd (p) and yd (I) from equation (19.10) into
the utility function u(x, y) = α ln x + y, we can derive the indirect utility function V (p, I) =
α ln(α/p) + I − α.9 Inverting this and replacing V with a utility value u, we can then get the
expenditure function
E(p, u) = u + α − α ln
α
.
p
(19.15)
Exercise 19B.6 Can you derive this expenditure function more directly through an expenditure minimization problem?
9 Because of the underlying quasilinearity in x, it does not matter in this case what income level we pick so long
as it does not result in a corner solution. In our case, there is an interior solution so long as I>α.
708
Chapter 19. Distortionary Taxes and Subsidies
The representative consumer’s utility under the distortionary tax is
ut = V (pd , I) = α ln xd (pd ) + yd (I) = α ln
α
+ (I − α)
pd
(19.16)
and the expenditure necessary to reach that utility level ut without distorting prices is
α
α
α
− ln ∗ + I
E(p , ut ) = ut + α − α ln ∗ = α ln
p
pd
p
∗
= α(ln α − ln pd − (ln α − ln p∗ )) + I = α(lnp∗ − ln pd ) + I
∗
p
= α ln
+I
(19.17)
pd
where we use the property of logarithms that ln(a/b) = ln a − ln b.
Exercise 19B.7 Can you verify that the expenditure necessary to reach the after tax utility at the pre-tax
price is always less than (or equal to) I?
Exercise 19B.8 What has to be true for E(p∗ , ut ) = I to hold?
Finally, the maximum lump sum amount our representative consumer is willing to give up to
avoid the distortionary tax (area (b + c) in Graph 19.10) is the difference between her income and
E(p∗ , ut ); i.e.
∗
∗
p
p
10
∆CS = I − E(p , ut ) = I − α ln
≈ 481. (19.18)
+ I = −α ln
= −1000 ln
pd
pd
16.18
∗
Exercise 19B.9 Can you show that in general, before substituting in specific pre- and post-tax prices,
equation (19.13) (which we derived using integral calculus) and equation (19.18) (which we derived using
the expenditure function) yield identical results?
The representative consumer is therefore willing to pay $481 in a lump sum amount in order
to avoid the tax. Her share of tax revenue, however, is only $6.18(61.8) ≈ $382 – implying a
deadweight loss of approximately $99 on the consumer side of the market. On the producer side,
we could similarly calculate profit before and after the tax – and then compare the change in profit
to the tax actually paid by producers. In our example, however, the supply curve is linear and we
can see in Graph 19.10 that the deadweight loss on the producer side is simply the triangle (e) –
which is (100 − 61.8)(10 − 6.18)/2 ≈ $73. Summing the deadweight losses from the two sides of the
market, we then get an overall deadweight loss of approximately $172 (just as we did when we used
integrals in the previous section).
Table 19.1 then illustrates the impact of different levels of per unit taxes for this example. Notice
that, as we have noted numerous times before, deadweight loss increases at a significantly faster
rate than the tax rate. However, because the price elasticity of demand is -1 everywhere (as you
should have concluded in exercise 19B.3), no tax rate is ever high enough to fully shut down the
market. In fact, given what we learned about the relationship between price elasticity of demand
and consumer spending, we know that a price elasticity of -1 implies that consumers will always
spend the same amount on their consumption of x regardless of price — which further implies that
tax revenue always increases with higher tax rates.
19B. The Mathematics of Taxes (and Subsidies)
t
0
1
2
3
4
5
10
25
50
100
1000
pd
$10.00
$10.51
$11.05
$11.61
$12.20
$12.81
$16.18
$28.51
$51.93
$100.99
$1,000.10
Welfare Changes from Per-Unit Tax
ps xd = xs
∆CS
∆P S Revenue
$10.00
100.00
$0.00
$0.00
$0.00
$9.51
95.12
$49.98
$47.56
$95.12
$9.05
90.50
$99.83
$90.50
$181.00
$8.61
86.12
$149.44 $129.18
$258.36
$8.20
81.98
$198.69 $163.96
$327.92
$7.81
78.08
$247.47 $195.19
$390.39
$6.18
61.80
$481.21 $309.02
$618.08
$3.51
35.08 $1,047.59 $438.48
$876.95
$1.93
19.26 $1,647.23 $481.46
$962.91
$0.99
9.90 $2,312.44 $495.10
$990.20
$0.10
1.00 $4,605.27 $499.95
$999.90
709
DW L
$0.00
$2.42
$9.34
$20.27
$34.73
$52.27
$172.19
$609.12
$1,165.76
$1,817.34
$4,105.32
Table 19.1: xd (p) = 1000/p, xs (p) = 10p
Exercise 19B.10 Does the Laffer curve in this example have a peak? Why or why not?
When the tastes are not quasilinear, substitution effects will cause the compensated demand
curve to differ from the uncompensated demand — implying that welfare changes (and deadweight
loss) cannot be measured along the market demand curve. We will encounter this in the next section
in our example of labor markets, and we treat it in the context of goods markets in end-of-chapter
exercise 19.2. In cases like this, you can, however, use the same expenditure function method we
developed here to calculate the change in consumer surplus.
19B.3
Deadweight Loss from Taxes in Labor (and Capital) Markets
Now suppose we return to the example of workers with Cobb-Douglas tastes over leisure and
consumption (as introduced in Chapter 9) and represented by the utility function u(c, ℓ) = cα ℓ(1−α) .
Given leisure endowment L, this implies demand for leisure and consumption of
ℓ = (1 − α)L and c = αwL.
(19.19)
Since labor supply is simply leisure endowment minus leisure consumption, this then implies a
perfectly inelastic labor supply function
ls = L − (1 − α)L = αL.
(19.20)
Exercise 19B.11 Verify that this labor supply function has zero wage elasticity of supply.
Suppose, for instance, that a worker has 60 leisure hours per week (L = 60) and that α = 2/3.
Then the labor supply function implies that the worker will work 40 hours per week regardless of
wage. If there are 1,000 workers in this labor market, with each having the same leisure endowment
and the same tastes, this further implies a vertical market supply of labor at 60,000 hours per week.
Suppose further that the market demand for labor is given by ld (w) = 25, 000, 000/w2. Setting this
equal to the inelastic labor supply of 60,000, we can derive an equilibrium wage of w∗ = 25.
Exercise 19B.12 What is the wage elasticity of labor demand?
710
19B.3.1
Chapter 19. Distortionary Taxes and Subsidies
Calculating Dead Weight Loss in the Labor Market
Now suppose a wage tax of $10 per labor hour is imposed as an additional cost on producers. Given
the perfectly inelastic labor supply in this market, this drives the equilibrium wage down to $15,
leaving producers entirely unaffected (given that they now pay a wage of $15 plus a $10 tax for a
total of $25 per worker hour as before). We can then focus entirely on the worker side of the market
to determine deadweight loss from the tax.
Consider an individual worker who continues to work 40 hours per week under the lower wage.
To determine the deadweight loss from the tax for this particular worker, we can ask the question
(as we have throughout this book): How much could we have taken from this worker in a lump sum
way and left him just as well off as he is when his wage drops from $25 to $15? Or, more generally,
how much could we have taken in a lump sum way to make the worker just as well off as he is when
his wage declines from w∗ to (w∗ − t)?
To answer this question, we first have to determine how happy the worker is under the tax t.
Since the worker will always consume (1 − α)L in leisure, his consumption is given by α(w∗ − t)L.
Plugging these values into his utility function, we get utility ut under a tax t of
α
ut = (α(w∗ − t)L) ((1 − α)L)
(1−α)
= αα (1 − α)(1−α) (w∗ − t)α L.
(19.21)
Next, we have to determine how much expenditure would be necessary to achieve this utility
level ut if the wage were still w∗ . The expenditure function emerges from the worker’s expenditure
minimization problem
min E = wℓ + c subject to ut = cα ℓ(1−α) .
(19.22)
c,ℓ
Solving this in the usual way, we first get the compensated leisure and consumption demands
c
ℓ (w) =
1−α
αw
α
c
ut and c (w) =
αw
1−α
(1−α)
ut ,
(19.23)
and, plugging these back into E = wℓ + c, the expenditure function
E(w, ut ) =
w(1−α) ut
.
αα (1 − α)(1−α)
(19.24)
Exercise 19B.13 Verify this.
For instance, in our example of a worker with α = 2/3 and L = 60 facing a tax that decreases
his wage from w∗ = 25 to (w∗ − t) = 15, we can use equation (19.21) to calculate his after-tax
utility as ut ≈ 193.1. Plugging this into equation (19.24), we get that the expenditure necessary
to achieve this utility level in the absence of taxes is E(w∗ , ut ) ≈ 1, 067.07. Since the value of the
worker’s leisure endowment is $1,500 (i.e. his leisure endowment of 60 hours times the wage of $25),
this implies we could have raised approximately $432.93 from the worker in a lump sum way and
kept him just as happy as he was under the $10 tax. But under the $10 wage tax, we raised only
$400 from him — implying a deadweight loss of approximately $32.93. With 1,000 workers in this
market, the overall deadweight loss is therefore approximately $32,930.
More generally, we can then write the expression for deadweight loss per worker as
DW L(t) = [w∗ L − E(w∗ , ut )] − (tls (w − t)))
(19.25)
19B. The Mathematics of Taxes (and Subsidies)
t
0
1
2
3
4
5
10
15
20
25
∗
(w − t)
$25.00
$24.00
$23.00
$22.00
$21.00
$20.00
$15.00
$10.00
$5.00
$0.00
711
Per Worker Welfare Changes from Per-Hour Wage Tax
ls (w∗ − t) lsc (w∗ )
ut E(w∗ , ut ) ∆Surplus Revenue
40.00
40.00 271.44 $1,500.00
$0.00
$0.00
40.00
40.53 264.15 $1,459.73
$40.27
$40.00
40.00
41.08 256.76 $1,418.89
$81.11
$80.00
40.00
41.63 249.27 $1,377.46
$122.54
$120.00
40.00
42.19 241.66 $1,335.40
$164.60
$160.00
40.00
42.76 233.92 $1,292.66
$207.34
$200.00
40.00
45.77 193.10 $1,067.07
$432.93
$400.00
40.00
49.14 147.36
$814.33
$685.67
$600.00
40.00
53.16
92.83
$512.99
$987.01
$800.00
40.00
60.00
0.00
$0.00
$1,500 $1,000.00
DW L
$0.00
$0.27
$1.11
$2.54
$4.60
$7.34
$32.93
$85.67
$187.01
$500.00
Table 19.2: Welfare Effects of Wage Tax
where the term in brackets is the amount we could have raised in a lump sum way without
making the worker worse off than he is under the tax and the term outside the brackets is the
actual tax revenue from the wage tax.
Exercise 19B.14 Can you find in a graph such as panel (b) of Graph 19.6 the various numbers calculated
above?
Table 19.2 then illustrates the welfare and revenue effects of different levels of wage taxes for
this example.
19B.3.2
Using Compensated Labor Supply to Calculate Dead Weight Loss
In panel (c) of Graph 19.6 we argued that there was an alternative way of identifying deadweight
loss as an area on the compensated labor supply curve. This will, however, once again involve the
use of integral calculus, and if you are not comfortable with this approach, you can once again skip
to the next section (since we already found a way to calculate deadweight loss by simply using the
expenditure function.)
Just as the uncompensated labor supply curve is simply the uncompensated leisure demand
subtracted from the leisure endowment, the compensated labor supply curve lsc is the compensated
leisure demand (from equation (19.23)) subtracted from leisure endowment L; i.e.
α
1−α
lsc (w, ut ) = L − ℓc (w) = L −
ut .
(19.26)
αw
In panel (b) of Graph 19.11, this function is graphed (for ut = 193.1, L = 60 and α = 2/3)
together with the inelastic uncompensated labor supply curve, and panel (a) graphs the inverses of
these functions to facilitate comparison to Graph 19.6 where we first argued that deadweight loss
can be measured on the compensated labor supply curve.
Areas under the compensated labor supply curve are defined by the integral
Z
lsc (w, ut )dw
=
w(1−α) ut
Lw − α
α (1 − α)(1−α)
(19.27)
712
Chapter 19. Distortionary Taxes and Subsidies
Graph 19.11: Deadweight Loss from a Wage Tax
which, when evaluated from w of 15 to 25 (with ut = 193.1, L = 60, and α = 2/3) gives area
(a + c) as
Area(a + c) =
Z
25
15
lsc (w, ut )dw ≈ 432.93.
(19.28)
Note that this is exactly equal to the lump sum tax that would get the worker to the same
utility level as the wage tax t = 10. Subtracting from that the actual tax revenue collected (area
(a) in the graph), we once again get deadweight loss of approximately $32.93 per worker — equal
to area (c).
19B.3.3
Taxation of Capital
A similar example analogous to Graph 19.7 involving savings decisions and deadweight loss from
taxation of interest is explored in end-of-chapter exercises 19.4 and 19.5, and the case of subsidies
is further considered in end-of-chapter exercise 19.3.
19B.3.4
DWL and Revenue as Tax Rates Rise
In Graph 19.8, we illustrated for linear demand and supply curves the impact of raising tax rates
on tax revenue and deadweight loss (under the assumption that uncompensated and compensated
demand are equivalent). For tax revenue, we derived an inverted U-shape for the “Laffer curve” —
indicating the existence of a tax rate that maximizes revenue. For deadweight loss, we argued that,
as in earlier chapters, increasing a tax by a factor of k will often increase the deadweight loss by a
factor of approximately k 2 .
Consider, for instance, the demand and supply curves given by xd (p) = (A − p)/α and xs (p) =
(p−B)/β (and assume that there are no income effects). You should be able to derive the equilibrium
consumer price pd and the equilibrium producer price ps = pd − t as
19B. The Mathematics of Taxes (and Subsidies)
pd =
713
βA + αB − βt
βA + αB + αt
and ps =
,
α+β
α+β
(19.29)
and the equilibrium quantity xt as
xt =
A−B−t
.
α+β
(19.30)
Exercise 19B.15 Verify these.
Tax revenue is then simply the per unit tax rate t times the quantity transacted xt which reduces
to
TR =
(A − B)t − t2
.
α+β
(19.31)
This is the functional form graphed in panel (b) of Graph 19.8, and it attains its peak when its
derivative with respect to the tax rate is zero. You can verify for yourself that this occurs when
t = (A − B)/2.
It is somewhat more tedious to derive the equation for deadweight loss, but if you are careful
in the various algebra steps involved, you can verify that
DW L(t) = ∆CS + ∆P S − T R =
t2
.
2(α + β)
(19.32)
Exercise 19B.16 ** Verify the expression for deadweight loss. (Hint: There are two ways of doing this: You
can either take the appropriate integrals of the supply and demand functions evaluated over the appropriate
ranges of prices, or you can add rectangles and triangles in a graph.)
Thus, if a tax rate t is increased by a factor k, the resulting deadweight loss will be k 2 the
original deadweight loss; i.e.
DW L(kt) =
(kt)2
t2
= k2
= k 2 DW L(t).
2(α + β)
2(α + β)
(19.33)
Both the Laffer curve and the result about increases in deadweight loss with increases in tax
rates therefore arise straightforwardly in a partial equilibrium model with linear demand and supply
curves — and these results form the basis for much intuition that guides tax policy. As we can
see from our example in Table 19.1 of the previous section, however, these are only rules of thumb
and they do not necessarily arise the same way in all models. With unitary price elastic demand in
Table 19.1, for instance, the Laffer curve does not attain a peak but only converges to a maximum
tax revenue as the tax rate rises. This is a direct consequence of the unitary price elasticity of
demand which implies consumer spending on the taxed good never declines. In the real world, of
course, it is unlikely that any demand curve truly has price elasticity of -1 regardless of how high
the price goes — and we would therefore expect an eventual downward slope to the Laffer curve.
Similarly, in Table 19.2, tax revenue for a wage tax continues to rise with the tax rate because of
the perfectly inelastic labor supply curve.
You might also have noticed that deadweight loss in Table 19.1, while increasing at an increasing
rate, does not increase in the same way as it does in the linear case. The rule of thumb that an
increase in a tax rate by a factor k will lead to an increase in deadweight loss by a factor k 2 is
714
Chapter 19. Distortionary Taxes and Subsidies
therefore just that: a rule of thumb derived from the linear case. In Table 19.2, on the other hand,
deadweight loss from multiplying the wage tax by a factor k increases by more than k 2 . Even
though the rule of thumb about the relationship between increases in tax rates and increases in
deadweight losses does not hold precisely in all cases, it is typically the case that deadweight loss
increases at an increasing rate as tax rates rise — leading to the common policy recommendation
that it is more efficient to raise tax revenues through low tax rates on large tax bases rather than
high tax rates on small tax bases.
19B.4
Taxing Land
We argued in Section A that a tax on land rents is one real world tax that does not give rise to
deadweight losses and is therefore efficient. The mathematics behind this was already explored
somewhat in Section A, and you can practice it further in the context of end-of-chapter exercises
19.7 and 19.12.
19B.5
A Simple Example of General Equilibrium Tax Incidence
In Section A, we also briefly introduced the notion that tax burdens may not only be shifted between
buyers and sellers within the taxed market (as in the partial equilibrium models of this chapter)
but may also be shifted to agents outside the taxed market through general equilibrium effects. We
mentioned in particular a tax on housing that leads to a reallocation of capital away from housing
and into other uses, thereby reducing the rate of return to non-housing capital and thus shifting a
portion of the tax burden to owners of non-housing capital.
We can illustrate the basic intuition behind this in a very simple setting. Suppose we modeled
owners of capital as a “representative investor” who chooses to allocate K units of capital between
the housing sector and all other sectors that make use of capital. Letting capital invested in housing
be denoted by k1 and capital invested in other uses by k2 , let’s assume that the before-tax rate of
1/2
return in the housing sector is determined by the production function f1 (k1 ) = αk1 , and the rate
1/2
of return in the untaxed remaining sector is determined by the production function f2 (k2 ) = βk2 .
But suppose the government imposes a tax of t% on returns from housing. Then the after-tax
1/2
return on k1 is (1 − t)f1 (k1 ) = (1 − t)αk1 .
Our representative investor then wants to maximize her total after-tax return by optimally
choosing the allocation of her capital K across the two sectors. Put differently, she wants to solve
the maximization problem
max (1 − t)f1 (k1 ) + f2 (k2 ) subject to k1 + k2 = K.
k1 ,k2
(19.34)
The solution to this problem is
k1∗ =
β2K
(1 − t)2 α2 K
∗
and
k
=
.
2
(1 − t)2 α2 + β 2
(1 − t)2 α2 + β 2
(19.35)
Table 19.3 then demonstrates how the tax t on housing is partially shifted to other forms of
capital when 1,000 units of capital are available to the representative investor and when α = β = 100
(which implies that equal amounts are invested in housing and other forms of capital in the absence
of taxes). The last column of the table represents the marginal product of a unit of capital in the
untaxed sector — and in equilibrium this has to be equal to the after-tax marginal product of a
19B. The Mathematics of Taxes (and Subsidies)
t
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Shifting
k1∗
500.00
447.51
390.24
328.86
264.71
200.00
137.93
82.57
38.46
9.90
715
of Housing Tax to Other Forms of Capital
k2∗ M P1 (k1∗ ) (1 − t)M P1 (k1∗ ) M P2 (k2∗ )
500.00
2.24
2.24
2.24
552.49
2.36
2.13
2.13
609.76
2.53
2.02
2.02
671.14
2.76
1.93
1.93
735.29
3.07
1.84
1.84
800.00
3.54
1.77
1.77
862.07
4.26
1.70
1.70
917.43
5.50
1.65
1.65
961.54
8.06
1.61
1.61
990.01
15.89
1.59
1.59
Table 19.3: K = 1000, α = β = 100
unit of capital in the taxed sector (which is reported in the second to last column). In the absence
of taxes (first row), these marginal products are equal to 2.24. As the tax on housing is increased
(going down in the table), this marginal product declines as capital is shifted out of the taxed sector
(where its after-tax return is falling) and into the untaxed sector. Thus, even though the tax is
imposed on housing, the burden of the tax falls equally on all capital. Implicitly we are assuming
that capital is perfectly mobile between sectors.
Exercise 19B.17 For t = 0.5, verify that the marginal product columns of the table report the correct
results.
Exercise 19B.18 If capital is “sector-specific” and cannot move from one use to another, would you still
expect the housing tax to be shifted? Explain.
In addition to the degree of capital mobility between sectors, the degree to which owners of
capital in other sectors are affected by a tax on housing also depends on the pre-tax size of the
housing sector relative to the non-housing sector. In Table 19.3, we set values for the example so
that the two sectors are initially of equal size. In Table 19.4, on the other hand, we keep α at 100
but increase β — which has the effect of reducing the housing sector relative to the non-housing
sector. The final column of this table then reports the percentage drop in the marginal product of
capital that results from a 50% tax on housing.
Exercise 19B.19 Why is the relative size of the housing sector relevant for determining how much owners
of capital in other sectors are affected by a tax on housing capital?
Conclusion
The first welfare theorem guarantees efficiency of market outcomes so long as a number of conditions
are satisfied. In this (and the previous) chapter, we have explored how inefficiencies are introduced
(and the first welfare theorem fails) when prices are distorted — in Chapter 18 because of explicit
price ceilings and floors and now because of taxes (and subsidies). In both cases, we have seen that
the group who bears the burden of the price distortion is not necessarily the one we might think
716
Chapter 19. Distortionary Taxes and Subsidies
β
100
200
300
400
500
1000
5000
Tax Shifting Depends on Relative Size of Housing Sector
(k1 /k2 )bef ore (k1 /k2 )af ter M Pbef ore M Paf ter %Change
0.5000
0.2500
2.236
1.768
-26.471%
0.2500
0.0625
3.536
3.260
-8.466%
0.1111
0.2778
5.000
4.809
-3.972%
0.0625
0.0156
6.519
6.374
-2.275%
0.0400
0.0100
8.062
7.945
-1.473%
0.0100
0.0025
15.890
15.831
-0.373%
0.0004
0.0001
79.073
79.061
-0.015%
Table 19.4: α = 100, K = 1000, t = 0.5
of first – that the economic incidence of price distortions is determined in equilibrium and often
depends critically on the relative price responsiveness of different actors in the market. The cost
of price regulations as well as the cost of taxes can thus be passed from one side of the market
to the other in ways that our partial equilibrium framework can clarify. We have similarly seen
that not all price regulation or tax/subsidy policies are equally inefficient – and that the relative
inefficiency of different price-distorting policies once again often depends on the price responsiveness
of those in the market. And we re-discovered in this chapter the fact that inefficiencies arise from
substitution effects which may be masked by income or wealth effects that prohibit us from relying
solely on (uncompensated) market demand and supply curves for purposes of welfare (and efficiency)
analysis. Finally, we have at least briefly pointed out that the impact of price regulations or taxes
and subsidies can extend beyond a particular market through general equilibrium effects that cross
from one market to another. We will explore this latter notion more explicitly in Chapter 20 where
we investigate the consequences of taxation or regulation in one of multiple connected markets
where such policies erect barriers to unfettered trade.
As before, we should caution again to not misinterpret the lessons of these chapters: While
economists unapologetically point out that there is an efficiency cost to using distortionary taxes to
raise revenue, this does not imply that economists are uniformly opposed to the use of such taxes
to raise revenues for expenditures considered to be socially valuable. Similarly, we make no apology
for pointing out the efficiency cost of using distortionary subsidies while not necessarily judging all
such subsidies to lack social value if they foster activities that are important to policy-makers. The
economist’s role is to clarify how taxes and subsidies affect individuals in an economy, how costly
they are and how they might be re-designed to become less costly.
End of Chapter Exercises
19.1 In our discussion of economic versus statutory incidence, the text has focused primarily on the incidence of
taxes. This exercise explores analogous issues related to the incidence of benefits from subsidies.
A: Consider a price subsidy for x in a partial equilibrium model of demand and supply in the market for x.
(a) Explain why it does not matter whether the government gives the per-unit subsidy s to consumers or
producers.
(b) Consider the case where the slopes of demand and supply curves are roughly equal in absolute value at
the no-subsidy equilibrium. What does this imply for the way in which the benefits of the subsidy are
divided between consumers and producers?
19B. The Mathematics of Taxes (and Subsidies)
717
(c) How does your answer change if the demand curve is steeper than the supply curve at the no-subsidy
equilibrium?
(d) How does your answer change if the demand curve is shallower than the supply curve at the no-subsidy
equilibrium?
(e) Can you state your general conclusion – using the language of price elasticities – on how much consumers
will benefit relative to producers when price subsidies are introduced. How is this similar to our conclusions
on tax incidence?
(f) Do any of your answers depend on whether the tastes for x are quasilinear?
B: * In Section 19B.1, we derived the impact of a marginal per-unit tax on the price received by producers –
i.e. dps /dt.
(a) Repeat the analysis for the case of a per-unit subsidy and derive dps /ds where s is the per-unit subsidy.
(b) What is dpd /ds?
(c) What do your results in (a) and (b) tell you about the economic incidence of a per-unit subsidy when the
price elasticity of demand is zero? What about when the price elasticity of supply is zero?
(d) What does your analysis suggest about the economic incidence of the subsidy when the price elasticities
of demand and supply are equal (in absolute value) at the no-subsidy equilibrium?
(e) More generally, can you show which side of the market gets the greater benefit when the absolute value
of the price elasticity of demand is less than the price elasticity of supply?
19.2 In the chapter, we discussed the deadweight loss from taxes on consumption goods when tastes are quasilinear
in the taxed good, and we treated deadweight loss when tastes are not quasilinear for the case of wage taxes. In this
exercise, we will consider deadweight losses from taxation on consumption goods when tastes are not quasilinear.
A: Suppose that x is a normal good for consumers.
(a) Draw the market demand and supply graph for x and illustrate the impact on prices (for consumers and
producers) and output levels when a per-unit tax t on x is introduced.
(b) Would your answer to (a) have been any different had we assumed that all consumers’ tastes were quasilinear in x?
(c) On a consumer diagram with x on the horizontal and “all other goods” (denominated in dollars) on the
vertical axes, illustrate the impact of the tax on a consumer’s budget.
(d) In your graph from (c), illustrate the portion of deadweight loss that is due to this particular consumer.
(e) On a third graph, depict the demand curve for x for the consumer whose consumer diagram you graphed
in (d). Then illustrate on this graph the same deadweight loss that you first illustrated in (d).
(f) Now return to your graph from (a). Illustrate where deadweight loss lies in this graph. How does it
compare to the case where the original market demand curve arises from quasilinear tastes rather than
the tastes we are analyzing in this exercise?
(g) True or False: We will overestimate the deadweight loss if we use market demand curves to measure
changes in consumer surplus from taxation of normal goods.
B: Suppose that consumers all have Cobb-Douglas tastes that can be represented by the utility function u(x, y) =
xα y (1−α) and each consumer has income I. Assume throughout that the price of y is normalized to 1.
(a) Derive the uncompensated demand for x by a consumer.
(b) Suppose income is expressed in thousands of dollars and each consumer has income I = 2.5 (i.e. income
of $2,500). There are 1000 consumers in the market. What is the market demand function?
(c) Suppose market supply is given by xs = βp. Derive the market equilibrium price and output level.
(d) Suppose α = 0.4 and β = 10. Determine the equilibrium pd , ps and xt when t = 10. How do these
compare to what we calculated for the quasilinear tastes in Section 19B.2.1 (where we assumed α = 1000
and β = 10) graphed in Graph 19.10?
(e) What is the before-tax and after-tax quantity transacted?
(f) If you used the market demand and supply curves to estimate deadweight loss, what would it be?
(g) Calculate the real deadweight loss in this case – and explain why it is different than in Section 19B.2.1
where market demand and supply curves were the same as here.
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Chapter 19. Distortionary Taxes and Subsidies
19.3 In the text, we discussed deadweight losses that arise from wage taxes even when labor supply is perfectly
inelastic. We now consider wage subsidies.
A: Suppose that the current market wage is w ∗ and that labor supply for all workers is perfectly inelastic. Then
the government agrees to pay employers a per-hour wage subsidy of $s for every worker hour they employ.
(a) Will employers get any benefit from this subsidy? Will employees?
(b) In a consumer diagram with leisure ℓ on the horizontal and consumption c on the vertical axes, illustrate
the impact of the subsidy on worker budget constraints.
(c) Choose a bundle A that is optimal before the subsidy goes into effect. Locate the bundle that is optimal
after the subsidy.
(d) Illustrate the size of the subsidy payment S as a vertical distance in the graph.
(e) Illustrate how much P we could have paid the worker in a lump sum way (without distorting wages) to
make him just as well off as he is under the wage subsidy. Then locate the deadweight loss of the wage
subsidy as a vertical distance in your graph.
(f) On a separate graph, illustrate the inelastic labor supply curve as well as the before and after-subsidy
points on that curve. Then illustrate the appropriate compensated labor supply curve on which to measure
the deadweight loss. Explain where this deadweight loss lies in your graph.
(g) True or False: As long as leisure and consumption are at least somewhat substitutable, compensated labor
supply curves always slope up and wage subsidies that increase worker wages create deadweight losses.
B: Suppose that, as in our treatment of wage taxes, tastes over consumption c and leisure ℓ can be represented
by the utility function u(c, ℓ) = cα ℓ(1−α) and that all workers have leisure endowment of L (and no other source
of income). Suppose further that, again as in the text, the equilibrium wage in the absence of distortions is
w ∗ = 25.
(a) If the government offers a $11 per hour wage subsidy for employers, how does this affect the wage costs
for employers and the wages received by employees?
(b) Assume henceforth that α = 0.5. What is the utility level us attained by workers under the subsidy (as
a function of leisure endowment L)?
(c)
*
(d)
*
(e)
**
What’s the least (in terms of leisure endowment L) we would need to give each worker in a lump sum
way to get them to agree to give up the wage subsidy program?
What is the per worker deadweight loss (in terms of leisure endowment L) of the subsidy?
Use the compensated labor supply curve to verify your answer.
19.4 This exercise reviews some concepts from earlier chapters on consumer theory in preparation for exercise
19.5.
A: Consider an individual saver who earns income now but does not expect to earn income in a future period
for which he must save.
(a) Draw a consumer diagram with current consumption c1 on the horizontal axis and future consumption c2
on the vertical. Illustrate an intertemporal budget constraint assuming an interest rate r – then draw an
indifference curve that contains the optimal bundle A.
(b) Now suppose the interest rate increases to r ′ . Illustrate the new budget constraint and indicate where the
new optimal bundle C will lie given that the individual does not change his savings decision when interest
rates change.
(c) How much, in terms of future dollars, would this person be willing to pay to get the interest rate to change
from r to r ′ ? If he pays that amount, will he end up saving more or less?
(d) Suppose instead that the interest rate starts at r ′ and then falls to r. Illustrate how much I would have
to give this individual to compensate him for the drop in the interest rate. If this is done, will he save
more or less than he did at the high interest rate?
(e) On a new graph, illustrate the individual’s inelastic savings supply curve. Then illustrate the compensated
savings supply curves that correspond to the utility levels the individual has at the interest rates r and
r′ .
(f) True or False: Compensated savings supply curves always slope up.
B: Suppose your tastes over current consumption c1 and future consumption c2 can be modeled through the
(1−α)
utility function u(c1 , c2 ) = cα
, your current income is I and you will earn no income in the future. The
1 c2
real interest rate from this period to the future is r.
19B. The Mathematics of Taxes (and Subsidies)
719
(a) Derive your demand functions c1 (r, I) and c2 (r, I) for current and for future consumption.
(b) Define “savings” as the difference between current income and current consumption. Derive your savings
– or capital supply – function ks (r, I). (Note: It turns out that this function is not actually a function of
r.)
(c) Derive the indirect utility function V (r, I) – i.e. the function that gives us your utility for any combination
of (r, I).
(d) Next, derive your compensated demand functions cc1 (r, u) and cc2 (r, u) for current and future consumption.
(e) Define the expenditure function E(r, u) – i.e. the function that tells us the current income necessary for
you to reach utility level u at interest rate r.
(f) Can you verify your answers by comparing V (r, I) to E(r, u)?
(g) Finally, suppose that we begin with an interest rate r and derive from it V (r, I). Define the compensated
savings or compensated capital supply function as ksc (r, r) = I − cc1 (r, V (r, I)).
(h) What is the interest rate elasticity of savings? Without deriving it precisely, can you tell whether the
interest rate elasticity of compensated savings is positive or negative?
19.5 (This exercise builds on exercise 19.4 which you should do before proceeding.) Through the income tax code,
governments typically tax most interest income – but, through a variety of retirement programs, they often subsidize
at least some types of interest income.
A: Suppose all capital is supplied by individuals that earn income now but don’t expect to earn income in some
future period – and therefore save some of their current income. Suppose further that these individuals do not
change their current consumption (and thus the amount they put into savings) as interest rates change.
(a) What is the economic incidence of a government subsidy of interest income? What is the economic
incidence of a tax on interest income?
(b) In the text, we illustrated the deadweight loss from a subsidy on interest income when savings behavior is
unaffected by changes in the interest rate. Now consider a tax on interest income. In a consumer diagram
with current consumption c1 on the horizontal and future consumption c2 on the vertical axis, illustrate the
deadweight loss from such a tax for a saver whose (uncompensated) savings supply is perfectly inelastic.
(c) What does the size of the deadweight loss depend on? Under what special tastes does it disappear?
(d) On a separate graph, illustrate the inelastic savings (or capital) supply curve. Then illustrate the compensated savings supply curve that allows you to measure the deadweight loss from the tax on interest
income. Explain where in the graph this deadweight loss lies.
(e) What happens to the compensated savings supply curve as consumption becomes more complementary
across time – and what happens to the deadweight loss as a result?
(f) Is the special case when there is no deadweight loss from taxing interest income compatible with a perfectly
inelastic uncompensated savings supply curve?
B: Suppose everyone’s tastes and economic circumstances are the same as those described in part B of exercise
19.4 – with α = 0.5 and I = 100, 000.10
(a) Suppose further that there are 10,000,000 consumers like this – and they are the only source of capital in
the economy. How much capital is supplied regardless of the interest rate?
(b) Suppose next that demand for capital is given by Kd = 25, 000, 000, 000/r. What is the equilibrium real
interest rate r ∗ in the absence of any price distortions?
(c) Suppose that, for any dollar of interest earned, the government provides the person who earned the interest
a 50 cent subsidy. What will be the new (subsidy-inclusive) interest rate earned by savers, and what will
be the interest rate paid by borrowers? What if the government instead taxed 50% of interest income?
(d) Consider the subsidy introduced in (c). How much utility V will each saver attain under this subsidy?
(e) How much current income would each saver have to have in order to obtain the same utility V at the
pre-subsidy interest rate r ∗ ? In terms of future dollars, how much would it therefore cost the government
to make each saver as well off in a lump sum way as it does using the interest rate subsidy?
10 Among
other functions, you should have derived uncompensated and compensated savings function as
"
#
„
«
1 + r (1−α)
c
ks (r, I) = αI and ks = 1 − α
I.
(19.36)
1+r
720
Chapter 19. Distortionary Taxes and Subsidies
(f) How much interest will the government have to pay to each saver (in the future) under the subsidy? Use
this and your previous answer to conclude the amount of deadweight loss per saver in terms of future
dollars. Given the number of savers in the economy, what is the overall deadweight loss?
(g) Derive the compensated savings function (as a function of r) given the post-subsidy utility level V .
(h)
**
Use your answer to (g) to derive the aggregate compensated capital supply function – and then find
the area that corresponds to the deadweight loss. Compare this to your answer in part (f).
(i)
**
(j)
**
Repeat parts (d) through (h) for the case of the tax on interest income described in part (c).
You have calculated deadweight losses for interest rates that are reasonable for 1-year time horizons.
If we consider distortions in people’s decisions over longer time horizons (such as when they plan for
retirement), a more reasonable time frame might be 25 years. With annual market interest rates of 0.05
in the absence of distortions, can you use your compensated savings function (given in the footnote to
the problem) to estimate again what the deadweight losses from a subsidy that raises the effective rate of
return by 50% and from a tax that lowers it by 50% would be?
19.6 Business and Policy Application: City Wage Taxes: In the U.S., very few cities tax income derived from wages
while the national government imposes considerable taxes on wages (through both payroll and income taxes) – and
then passes some of those revenues back to city governments.
A: In this exercise, we will consider the reason for this difference in local and national tax policy – and why
city governments might in fact be “employing” the national government to levy wage taxes and then have the
national government return them to cities.
(a) Consider first a national labor market. While workers and firms can move across national boundaries to
escape domestic taxes, suppose that this is prohibitively costly for the labor market that we are analyzing.
Illustrate demand and supply curves for domestic labor (assuming that supply is upward sloping). Indicate
the no-tax equilibrium wage and and employment level and then show the impact of a wage tax.
(b) Next, consider a city government that faces a revenue shortfall and considers introducing a wage tax. Why
might you think that labor demand and supply are more elastic from the city’s perspective than they are
from a national government perspective?
(c) Given your answer to (b), draw two Laffer curves – one for tax revenue raised in a city when the tax is
imposed nationally and one for tax revenues raised in the same city when it is imposing the tax on its
own. Explain where the peaks of the two Laffer curves are relative to one another.
(d) How do your answers to (b) and (c) most likely contain the answer to why cities do not typically use wage
taxes to raise revenues?
(e) Suppose you are a mayor of a city and would like to impose a wage tax but understand the problem so
far. How might it make sense for you to ask the federal government to increase the wage tax nationwide
– and then to give cities the additional revenue collected in each city?
(f) Of those cities that do have wage taxes, most are relatively large. Why do you think it is exceedingly rare
for small cities to impose local wage taxes?
(g) Does any of this analysis depend on whether there are wealth (or income) effects in the labor market?
B: Suppose that labor demand and supply are linear – with ld = (A − w)/α and ls = (w − B)/β.
(a) For a given per-unit wage tax t, calculate the employment level and tax revenue.
(b) Consider two scenarios – scenario 1 in which (A − B) is large and scenario 2 in which (A − B) is small.
What has to be true about (α+β) in scenario 1 relative to scenario 2 if the no-tax equilibrium employment
level is the same in both cases.
(c) Suppose one scenario is relevant for predicting tax revenue from your city when it is collected nationwide
and the other is relevant for predicting tax revenue when the wage tax is collected just in your city. Which
scenario belongs to which tax analysis?
(d) Find the tax rate t at which government revenue is maximized.
(e) Demonstrate that the scenario appropriate for the tax analysis when only your city imposes the wage tax
leads to a Laffer Curve that peaks earlier.
(f) As cities get small, what happens to (A − B) in the limit? What happens to the peak of the Laffer Curve
for a local city tax in the limit?
19B. The Mathematics of Taxes (and Subsidies)
721
19.7 Business and Policy Application: Land Use Policies: In most Western democracies, it is settled law that
governments cannot simply confiscate land for public purposes. Such confiscation is labeled a “taking” – and, even
when the government has compelling reasons to “take” someone’s property for public use, it must compensate the
landowner. But, while it is clear that a “taking” has occurred when the government confiscates private land without
compensation, constitutional lawyers disagree on how close the government has to come to literally confiscating
private land before the action constitutes an unconstitutional “taking”.
A: Any restriction that alters the way land would otherwise be used reduces the annual rental value of that
land and, from the owner’s perspective, can therefore be treated as a tax on rental value.
(a) Explain why the above statement is correct.
(b) Suppose a land use regulation is equivalent (from the owner’s perspective) to a tax of t% on land rents to
be statutorily paid by landowners (where 0 < t < 1). How does it affect the market value of the land?
(c) I am about to buy an acre of land from you in order to build on it. Right before we agree on a price, the
government imposes a new zoning regulation that limits what I can do on the land. Who is definitively
made worse off by this?
(d) Suppose you own 1000 acres of land that is currently zoned for residential development. Then suppose
the government determines that your land is home to a rare species of salamander – and that it is in
the public interest for no economic activity to take place on this land in order to protect this endangered
species. From your perspective, what approximate tax rate on land rents that you collect is this regulation
equivalent to? Do you think this is a “taking”?
(e) Suppose that, instead of prohibiting all economic activity on your 1000 acres, the government reduces
your ability to build residential housing on it to a single house. How does your answer change? What if
it restricts housing development to 500 acres? Do you think this would be a “taking”?
B: * Suppose that people gain utility from housing services h and other consumption x, with tastes described by
the utility function u(x, h) = ln x+ln h. Consumption is denominated in dollars (with price therefore normalized
to 1). Housing services, on the other hand, are derived from the production process h = k 0.5 Lα where k stands
for units of capital and L for acres of land. Suppose 0 < α < 1. Let the rental rate of capital be denoted by r,
and assume each person has income of 1000.
(a) Write down the utility maximization problem and solve for the demand function for land assuming a
rental rate R for land.
(b) Suppose your city consists of 100,000 individuals like this – and there are 25,000 acres of land available.
What is the equilibrium rental rate per acre of land (as a function of α)?
(c) Using your answers above, derive the amount of land each person will consume.
(d) Suppose the government imposes zoning regulations that reduce the coefficient α in the production function
from 0.5 to 0.25. What happens to the equilibrium rental value of land?
(e) Suppose that what you have calculated so far is the monthly rental value of land. What happens to the
total value of an acre of land as a result of these zoning regulations assuming that people use a monthly
interest rate of 0.5% to discount the future?
(f) Suppose that, instead of lowering α from 0.5 to 0.25 through regulation, the government imposes a tax t
on the market rental value of land and statutorily requires renters to pay. Thus, if the market land rental
rate is R per acre, those using the land must pay tR on top of the rent R for every acre they use. Set up
the renters’ utility maximization problem, derive the demand for land and aggregate it over all 100,000
individuals. Then derive the equilibrium land rent per acre as a function of t (assuming α = 0.5).
(g) Does the amount of land consumed by each household change?
(h) Suppose you own land that you rent out. What level of t makes you indifferent between the zoning
regulation that drove α from 0.5 to 0.25 and the land rent tax that does not change α?
(i) Suppose the government statutorily collected the land rent tax from the owner instead of from the renter.
What would the tax rate then have to be set at to make the land owner indifferent between the zoning
regulation and the tax?
19.8 Business and Policy Application: Price Floors for Corn: Is it a Tax or a Subsidy?: In exercises 18.9 and 18.10,
we investigated policies that imposed a price floor in the corn market.
A: We will now see whether some of the price regulation proposals we considered are equivalent to taxes or
subsidies. For simplicity, assume that tastes are quasilinear in corn.
722
Chapter 19. Distortionary Taxes and Subsidies
(a) In exercise 18.9, we began by considering a price floor without any additional government program.
Illustrate the equilibrium impact of such a price floor on the price of corn paid by consumers as well as
the price of corn received by producers.
(b) If you were to design a tax or subsidy policy that has the same impact as the stand-alone price floor, what
would it be?
(c) In exercise 18.10, we considered the combination of a price floor and a government purchasing program
under which the government guaranteed it would purchase any surplus corn at the price ceiling and then
sell it at a price sufficiently low for all of it to be bought. Illustrate the impact of this program – including
the deadweight loss.
(d) If you were to design a tax or subsidy policy with the aim of achieving the same outcome for the marginal
consumer and producer as the policy in (c), what would you propose?
(e) Would your proposal result in the same level of consumer and producer surplus? Would it result in the
same deadweight loss?
B: Suppose, as in exercises 18.9 and 18.10, that the domestic demand curve for bushels of corn is given by
p = 24 − 0.00000000225x while the domestic supply curve is given by p = 1 + 0.00000000025x.
(a) Suppose the government imposes a price ceiling of p = 3.5 (as in exercise 18.9). In the absence of any
other program, how much will consumers pay (per bushel) and how much will sellers keep (per bushel)
after accounting for the additional marginal costs incurred by producers to compete for consumers?
(b) If you wanted to replicate this same outcome using taxes or subsidies, what policy would you propose?
(c) Suppose next that the government supplemented its price floor from (a) with a government purchasing
program that buys all surplus corn – and then sells it at the highest possible price at which all surplus
corn is bought. What is that price?
(d) If you were to design a tax or subsidy policy that has the same impact on the marginal consumer and
producer, what would it be?
19.9 Policy Application: Rent Control: Is it a Tax or a Subsidy?: In exercise 18.11 we analyzed the impact of rent
control policies that impose a price ceiling in the housing rental market. The stated intent of such policies is often
to make housing more affordable. Before answering this question, you may wish to review your answers to exercise
18.11.
A: Begin by illustrating the impact of the rent control price ceiling on the price received by landlords and the
eventual equilibrium price paid by renters.
(a) Why is it not an equilibrium for the price ceiling to be the rent actually paid by renters?
(b) If you wanted to implement a tax or subsidy policy that achieves the same outcome as the rent control
policy, what policy would you propose?
(c) Could you credibly argue that the alternative policy you proposed in (b) was designed to make housing
more affordable?
(d) If you did actually want to make housing more affordable (rather than trying to replicate the impact of
rent control policies), would you choose a subsidy or a tax?
(e) Illustrate your proposal from (d) – and show what would happen to the rental price received by landlords
and the rents paid by renters. What happens to the number of housing units available for rent under your
new policy?
(f) True or False: Policies that make housing more affordable must invariably increase the equilibrium quantity of housing – and rent control policies fail because they reduce the equilibrium quantity of housing
while subsidies succeed for the opposite reason.
(g) True or False: Although rental subsidies succeed at the goal of making housing more affordable (while
rent control policies fail to do so), we cannot in general say that deadweight loss is greater or less under
one policy rather than the other.
B: Suppose, again as in exercise 18.11, that the aggregate monthly demand curve is p = 10000 − 0.01x while
the supply curve is p = 1000 + 0.002x. For simplicity, suppose again that there are no income effects.
(a) Calculate the equilibrium number of apartments x∗ and the equilibrium monthly rent p∗ in the absence
of any price distortions.
(b) In exercise 18.11, you were asked to consider the impact of a $1,500 price ceiling. What housing tax or
subsidy would result in the same economic impact?
19B. The Mathematics of Taxes (and Subsidies)
723
(c) Suppose that you wanted to use tax/subsidy policies to actually reduce rents to $1,500 – the stated goal
of the rent control policy. What policy would you implement?
(d) Consider the policies you derived in (b) and (c). Under which policy is the deadweight loss greater?
19.10 Policy Application: Incidence of U.S. Social Security Taxes: In the U.S., the social security system is funded
by a payroll (wage) tax of 12.4% that is split equally between employer and employee; i.e. the statutory incidence of
the social security tax falls half on employers and half on employees.
A: In this exercise we consider how this split in statutory incidents impacts the labor market. Assume throughout
that labor supply is upward sloping.
(a) Illustrate the labor supply and demand graph and indicate the market wage w ∗ and employment level l∗
in the absence of any taxes.
(b) Which curve shifts as a result of the statutory mandate that employers have to pay the government 6.2%
of their wage bill? Which curve shifts because of the statutory mandate that employees pay 6.2% of their
wages in social security tax?
(c) Suppose the wage elasticity of labor demand and supply are equal in absolute value at the pre-tax equilibrium. Can you illustrate how the market wage at the post-tax equilibrium – when both parts of the
social security tax are taken into account – might be unchanged from the initial equilibrium wage w ∗ ?
(d) In your graph, illustrate what the imposition of the 2-part social security tax means for the take home
wage ww for workers. What does it mean for the real cost of labor wf that firms incur?
(e) How would the equilibrium wage in the market change if the government imposed the entire 12.4% tax
on workers (and let employers statutorily off the hook)? How would it change if the government instead
imposed the entire tax on employers?
(f) What happens to the take-home wage for workers and the real labor cost of firms as a result of the two
statutory tax reforms raised in part (e)?
(g) Does any of this analysis depend on whether there are wealth effects in the labor market?
B: Suppose, as in exercise 19.6, that labor demand and supply in the absence of taxes are given by ld = (A−w)/α
and ls = (w − B)/β.
(a) Determine the equilibrium employment level l∗ and the equilibrium wage w ∗ .
(b) Now suppose the government imposes a per-unit tax t on workers and a second per-unit tax t on employers.
Derive the new labor demand and supply curves that incorporate these (as you would when you shift
demand and supply curves in response to statutory tax laws).
(c) Determine the new equilibrium wage and employment level. Under what condition is the new observed
equilibrium wage unchanged as a result of the two-part wage tax? Is there any way that employment will
not fall?
(d) Determine the take-home wage ww for workers and the real labor cost wf for firms.
(e) Suppose you did not know the statutory incidence of the wage tax but simply knew the total tax was
equal to 2t. How would you calculate the economic incidence – i.e. how would you calculate ww and wf ?
(f) Compare your answers to (e) to your answers to (d). Can you conclude from this whether statutory
incidence matters?
19.11 Policy Application: Mortgage Interest Deductibility, Land Values and the Equilibrium Rate of Return on
Capital: In the text, we suggested that the property tax can be thought of in part as a tax on land and in part as a
tax on capital invested in housing. In the U.S., property taxes are typically levied by local governments – while the
major piece of federal housing policy is contained in the federal income tax code which allows individuals to deduct
(from income) the interest they pay on home mortgages prior to calculating the amount of taxes owed.
A: Whereas we can think of the property tax as a tax on both land and housing structures, we can think of the
homeownership subsidy in the federal tax code as a subsidy on land and housing structures.
(a) If your marginal federal income tax rate is 25% and you are financing 100% of your home value, how much
of your housing consumption is being subsidized through the tax code? What if you are only financing
50% of the value of your home?
(b) Suppose homeowners are similar to one another in terms of their marginal tax rate and how much of their
home they are financing, and suppose that this implies a subsidy of s for every dollar of housing/land
consumption. How would you predict the value of suburban residential land (assumed to be in fixed
supply) is different as a result of this than it would have been in the absence of this policy?
724
Chapter 19. Distortionary Taxes and Subsidies
(c) When s was first introduced, who benefitted from the implicit land subsidy: current homeowners or future
homeowners?
(d) Now consider s as a subsidy on housing capital. Do you think houses are larger or smaller as a result of
the federal income tax code?
(e) Suppose that the overall amount of capital in the economy is fixed and that capital is mobile across sectors.
Thus, any given unit of capital can be invested in housing or alternatively in some other non-housing sector
where it earns some rate of return. If the overall amount of capital in the economy is fixed, what happens
to the fraction of capital invested in the housing sector?
(f) What would you predict will happen to the rate of return on capital in the non-housing sector? Explain.
(g) True or False: Even though only housing capital is statutorily subsidized, the economic incidence of this
subsidy falls equally on all forms of capital (so long as capital is mobile between sectors).
B: Suppose we model owners of capital as a “representative investor” who chooses to allocate K units of capital
between the housing sector and other sectors of the economy. With k1 representing capital invested in housing
and k2 representing capital invested in other sectors, suppose f1 (k1 ) = αk10.5 and f2 (k2 ) = βk20.5 are the
production functions of the two sectors.
(a) In the absence of any policy distortions, calculate the fraction of total capital (K) that is invested in the
housing sector.
(b) What changes as a result of the federal income tax code’s implicit housing subsidy s.
(c) What happens to the marginal product of capital in the non-housing sector?
(d) What happens to the equilibrium rate of return on capital?
(e) True or False: The general equilibrium subsidy incidence of the implicit subsidy of housing capital falls
equally on all forms of capital.
19.12 Policy Application: The Split-Rate Property Tax: As we have mentioned several times, the usual property tax
is really two taxes: one levied on land value (or on land rents) and the other levied on the value of the “improvements”
of land – or the rents from capital investments. The typical property tax simply sets the same tax rate for each part,
but in an increasing number of places, governments are reforming property taxes to levy a higher rate on land than
on improvements. Such a tax is called a split-rate property tax.
A: Suppose you are in a locality that currently taxes rental income from capital at the same rate as rental
income from land. Assume throughout that the amount of land in the community is fixed.
(a) Which portion of your local tax system is distortionary and which is non-distortionary?
(b) Next, suppose that your community lowers the tax on capital income and raises it on land rents – and
suppose that overall tax revenues are unchanged as a result of this reform. Do you think the tax reform
enhances efficiency?
(c) Your community has a fixed amount of land – but capital can move in and out of your community and
therefore changes depending on economic conditions. Do you think the land in your community will be
more or less intensively utilized as a result of the tax reform – i.e. do you think more or less capital will
be invested on it?
(d) What do you think happens to the marginal product of land in your community under this tax reform?
What must therefore happen to the rental value of land (before land rent taxes are paid)?
(e) Suppose half of your community has land that is relatively substitutable with capital in production –
and the other half of your community has land that is relatively complementary to capital in production.
Might it be the case that land values go up in part of your community and go down in another part of
your community as a result of the tax reform? If so, which part experiences the increase in land values
despite an increase in the tax on land rents?
(f) Will overall output in your community increase or decrease as a result of the tax reform? Under what
extreme assumption about the degree of substitutability of land and capital in production would local
production remain unchanged?
(g) True or False: The more substitutable land and capital are in production, the more likely it is that the tax
reform toward a split-rate property tax (that taxes land more heavily) will result in a Pareto improvement.
19B. The Mathematics of Taxes (and Subsidies)
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B: * Suppose we normalize units of land so that the entire land area of a particular locality equals one unit.
Economic activity is captured by the constant elasticity of substitution production function y = f (k, L) =
(0.5L−1 + 0.5k −1 )−1 . The government collects revenues through a property tax that taxes land rents at a rate
tL and the rental value of capital at a rate tk – resulting in total tax revenue of T R = tL R + tk rK where R is
the rental value of the 1 unit of land in the locality, r is the interest rate in the local economy and K is the total
capital employed in the locality. (Note that we have defined capital units such that the interest rate is equal to
the rental rate of capital).
(a) Suppose that this locality is sufficiently small so that nothing it does can affect the global economy’s rental
rate r – i.e. the supply of capital is perfectly elastic. If the locality taxes the rental value of capital at
rate tk , at what local interest rate r would investors be willing to invest here?
(b) Suppose that land is utilized optimally given the local tax environment – which implies that the marginal
product of capital must equal r. Define the equation that you would have to solve in order to calculate
the level of capital invested in this locality.
(c) Suppose r = 0.06. Solve for the level of capital K invested on the one unit of land of this locality (as a
function of tk ).
(d) Can you determine the rental value of land? (Hint: Derive the marginal product of land and evaluate it
at the level of capital you calculated in the previous part and the 1 unit of land that is available.)
(e) Now consider the case where the local tax system is (tL , tk ) = (0, 0.5). Derive the total capital K invested
in the locality, the land rental value R, the value of land P (assuming that future income is discounted at
the interest rate r = 0.06), the production level y and the tax revenue T R. (You may find it convenient
to set up a simple spreadsheet to do the calculations for you).
(f)
**
Repeat this for the tax system (tL , tk ) = (0.05, 0.3637), the tax system (tL , tk ) = (0.1, 0.1748) and the
tax system (tL , tk ) = (0.1353, 0). Present your results for K, R, P , y and T R in a table (and keep in mind
that r changes with tk even though r remains at 0.06.) (Hint: All three systems should give the same tax
revenue.)
(g) Use your table to discuss how the shift from a tax solely on capital (i.e. structures) toward a revenueneutral tax system that increasingly relies on taxing land rents impacts the local economy. Which of the
rows in your table could look qualitatively different under different elasticity of substitution assumptions?
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Chapter 19. Distortionary Taxes and Subsidies