Correcting Market Distortions:
Shadow Prices, Shadow Wages and
Discount Rates
Chapter 6
• Observed market prices sometimes reflect true cost to
society. In some circumstances they don’t because
there are distortions which prevent market prices from
conveying true economic values.
• When this occurs have to correct observed price to
calculate the shadow price.
– Types of distortions include taxes, subsidies & other forms
of gov’t intervention.
• In competitive markets D represents marginal benefits
to society and supply curve social costs. Social costs are
equal to private costs. Likewise private benefits equal
social benefits.
A Market with a Per Unit Tax
• Suppose have a market
for good but price
observed for the good
includes a per unit tax,
here price consumers
pay is not the price the
firms keeps.
– T – is the tax
–Pc = Pf + T
• Pc – price gross of tax
• Pf – price net of tax
Project Demand with a Per Unit Tax
• Suppose there’s a project
that requires the good as
an input.
– Demand for the good
increases
– leads to new equilibrium at
point C
– Output increases from Xe
to Xf
– price firms retain increases
from Pf to Pf ’
– Price consumers pay
increases from Pc to Pc’
• Non-project demand for the firm falls from Xe to Xc
• Note that the Government requirement of XG comes
from two sources:
– Xf - Xe – units of new supply
– Xe - Xc – units of displaced demand
• If market weren’t distorted by the tax, there would not
be a problem because consumers marginal benefit
would equal the firms marginal costs, this not the case
here because of tax (the competitive output should be
at Xf )
• The tax has driven a wedge between consumers’ and
firms’ valuation of this input.
• The tax creates a problem for someone trying to value the input
because the market outcomes are distorted by the tax.
• What the shadow price does is try to take the distorted prices and
correct them for the distortion to get a valuation/price that is
distorted.
• In this example the shadow price takes a weighted average of the
opportunity costs of the two sources of the gov’t’s input
requirement.
– For example, Suppose the gov’t needs XG units of X to complete the
project, can calculate PG the shadow price as either: 𝑃𝐺 =
𝑋𝑓 −𝑋𝑒
𝑋𝑓 −𝑋𝑒
𝑋𝑒 −𝑋𝑐
𝑋 −𝑋
𝑃𝑓
+ 𝑃𝑐
or 𝑃𝐺 = 𝑃𝑓
− 𝑃𝑐 𝑐 𝑒
𝑋𝐺
𝑋𝐺
𝑋𝐺
𝑋𝐺
• Where Pf – price net of tax and Pc – is the price gross of tax (Pc = Pf
+T)
• An alternative expression of the shadow price in the
previous example uses elasticities
𝑃𝑓 𝜖𝑠 −𝑃𝑐 Ω𝜖𝐷
𝑋
𝑃𝐺 =
,where Ω = 𝑐 , 𝜖𝑠 is the elasticity of
𝜖𝑠 −Ω𝜖𝐷
𝑋𝑓
supply and 𝜖𝐷 is the elasticity of demand
• The shadow price PG will depend critically on elasticities;
elasticities will determine how big increases are in new
demand as well as how big is displaced demand.
• Recall that the elasticity determines the slope of the
demand and supply curves.
– A more elastic demand(supply) curve will be flatter
– A more inelastic demand(supply) curve will be steeper
• → D1 is flatter than D2
• → D1 is more elastic
than D2
• Note that in general the shadow price will fall
between gross – of – tax and net – of – tax
price.
• However, there are some special cases where
the shadow price takes on specific values.
– These extreme cases occur when the demand is
prefectly elastic and inelastic and supply is
perfectly elastic and inelastic
Extreme Cases
Distortionary Sudsidies
• Analysis is basically the
same as a distortionary
tax
Shadow Wage Rates
• Governments sometimes use discretionary fiscal policy to
create jobs.
• If labour markets functioned perfectly market wages would
reflect true opportunity cost of hiring labour for a project
• However, markets usually don’t function perfectly because
of distortions so have to calculate the shadow price of
labour. i.e. the shadow wage rate.
• Can draw labour for public projects from three sources:
i.
ii.
iii.
those employed elsewhere in the economy
the voluntarily unemployed
the involuntarily unemployed
• There are implications of i) to iii)
• RE: i.) Those employed elsewhere in the
economy:
– When draw someone who already has a job into a
public job, the value of the output they produced
is forgone.
– Measure by their marginal product of labour i.e.
market wage
• Recall firm is a profit maximizer or cost minimizer so
they expand output as long MPL exceeds the wage rate.
– Denote market wage by w1
• RE: ii.) Voluntary unemployed, e.g., retired
people or people in school
– opportunity cost of these people is the minimum
amount required to induce them to work
– measure their opportunity cost by the wage the
project pays them w2
• RE: iii.) Involuntary unemployed people willing to
work at the market wage but unable to find a job
– On the one hand, can argue that this opportunity cost
is zero
– On the other hand, can also argue that market wage is
not sufficiently large enough to induce them to work
[should work for the market wage rate X]
– Denote this wage by w3
– Since don’t really know what this wage is you have to
make some assumptions about its values
– One assumption that is used is that the value is at
least 20% greater than the market wage.
• Let 𝛼1 , 𝛼2 , 𝛼3 denote the proportions of the
type of labour hired, i.e., i) to iii)
• The shadow wage rate for the project can be
written as 𝑤𝑠 = 𝛼1 𝑤1 + 𝛼2 𝑤2 + 𝛼3 𝑤3 , where
𝛼1 + 𝛼2 + 𝛼3 = 1
• The expression above is one potential
estimate for the shadow wage rate.
• Another approach is the Hughes formula
• The Hughes formula computes the shadow
wage rate as 𝑤𝑠 = 𝑝𝑖 𝑤𝑖 + 1 − 𝑝𝑖 𝑣𝑤𝑖 ,where
𝑤𝑖 is the wage rate paid to workers on the
project; 𝑣 is the proportion of the wage paid
that represents the opportunity cost of hiring
involuntarily unemployed labour; 𝑝𝑖 is the
probability of hiring a person who is already
employed and 1-𝑝𝑖 is the probability of
someone who is involuntarily unemployed
• You can make an assumption on 𝑝𝑖 , but
Hughes estimated it as follows 𝑝𝑖 = 1 −
0.5 𝑠𝑖𝑛 𝜋
𝑢−𝑢𝑚𝑖𝑛
𝑢𝑚𝑎𝑥 −𝑢𝑚𝑖𝑛
−
𝜋
2
+1, where
𝑠𝑖𝑛 ∙ is evaluated in radians, 𝑢 is the
unemployment rate in a region, 𝑢𝑚𝑖𝑛 is the
historical minimum of the unemployment rate
in the region, 𝑢𝑚𝑎𝑥 is the historical maximum
of the unemployment in the region
• 𝑣 can also be set by assuming values, but
Pinfold proposed the following estimate 𝑣 =
𝑤 1 − 𝑡𝑤 − UI 1 − 𝑡𝑈𝐼 , where 𝑤 is the
wage rate, UI is unemployment insurance
benefits, 𝑡𝑤 is the tax on wages and 𝑡𝑈𝐼 is the
tax rate on UI benefits.
Choosing and Computing a Discount
Rate
𝐵𝑡 −𝐶𝑡
𝑇
𝑡=1 1+𝑟 𝑡 ,
• Recall the NPV =
where r is the
discount rate and B and C represent benefits and
costs, respectively.
• The NPV will depend on r as well as benefits and
costs.
– a smaller discount rate will lead to larger values of the
NPV, large values of the discount rate lead to smaller
values of the NPV
– a discount rate of 0 means that society weights the
future equally to the present, thought to be
“altruistic” discount rate
• Marginal rate of time preference
– Consider whether someone wants a $1 today versus
tomorrow
– Whether someone picks to have the $1 today or
tomorrow reflects their time preference, or how they
trade off between these alternatives
– For example, suppose you have the choice of $1000
today or $1200 one year from today, if you pick $1000
today then your rate of time preference is 20%; you
would have a stronger preference for having
something today.
• Can formalize the idea of time preference and
choosing between today and tomorrow with
the following model.
• Suppose individuals choose between
consumption today and tomorrow, denoted
𝐶1 and 𝐶2 subject to a lifetime budget
constraint.
• Assume that individuals have preferences over
consumption today and tomorrow
• The individual’s problem can be written as
𝑐2
max 𝑢 𝑐1 , 𝑐2 𝑠𝑢𝑏𝑗 𝑡𝑜 𝑐1 +
= 𝑇,
1+𝑖
where 𝑖 is the interest rate and T is the present
value of income over the individual’s lifetime
(periods 1 and 2 in this example).
We’ll discuss the solution to this problem in
graphical terms,
• Absolute value of slope of the indifference curve
measures the rate at which individuals are
indifferent between substituting current
consumption for future consumption, i.e., the
MRS between consumption this year and
consumption next year, where 𝑀𝑅𝑆 = 1 + 𝜌, and
𝜌 is the marginal rate of time preference.
• An equilibrium for this problem is where the rate
at which people are willing to trade consumption
today and tomorrow equals the price of moving
consumption allocations, i.e., the interest rate
• An equilibrium, will occur when the indifference
curve is tangent to the budget line, i.e., where
1+𝜌 =1+𝑖
• If you can freely borrow then you can shift
consumption to the future until the MRTP falls to
the interest rate you must pay
– If 𝑖 > 𝜌 then save and reduce consumption today
– If 𝑖 < 𝜌 then borrow and increase consumption today
• In a prefect capital market 𝑖 = 𝜌
• Investment demand
- Looks at firms making
investment decisions
- Assumes perfect capital
markets
- A firm has a variety of
investment projects to
select from which have
different rates of return
associated with them.
• supply of funds for
investment is provided
by individual saving
• if rate of interest > rate
of time preference then
save
• represented by
Aggregate savings
schedule
• Market equilibrium
occurs where supply of
savings schedule equals
the demand for
investment funds,
where rate of return
equals the rate of time
preference; the
equilibrium point is the
market interest rate
• The previous equilibrium is based on the
assumption of prefect capital markets.
• Generally, the real world is not comprised of
perfect capital markets since there are
distortions, e.g., taxes, risk, gov’t borrowing,
which all drives wedges between market and
social outcomes, and, consequently, society
can end up with under investment.
Market Equilibrium with Distortions
• On previous slide 𝐷0 and 𝑆0 represent investment
demand and supply of funds without taxes
• Introduction of taxes (both corporate and
personal) shifts back the investment demand and
supply of funds curves, denoted by 𝐷𝐼 and 𝑆𝑆
• With taxes the market clearing interest rate
would be 𝑖
– The marginal return on investment before taxes would
be 𝑟𝑧 , the opportunity cost of forgone investment
– The marginal rate of return on savings after taxes
would be 𝑝𝑧
• Suppose the government undertakes a new
project/program that it funds by borrowing.
– This would shift out the demand for funds, 𝐷𝐼
shifts out to𝐷𝐼 ’
– Private sector investment falls by ∆𝐶, crowding
out effect
• Arnold Harberger using this framework suggests
the following estimate of the social discount rate:
𝑠 = 𝑎𝑟𝑧 + 1 − 𝑎 𝑝𝑧
∆𝐼
∆𝐶
𝑤ℎ𝑒𝑟𝑒 𝑎 =
𝑎𝑛𝑑 1 − 𝑎 =
∆𝐶 + ∆𝐼
∆𝐶 + ∆𝐼
• Some empirical evidence suggests that savings is
not very sensitive to interest rates, which implies
that the savings schedule would be relatively
inelastic (i.e., vertical), so that ∆𝐶 ≈ 0 and 1 −
𝑎 ≈ 0 and a ≈ 1 , which implies that 𝑠 = 𝑟𝑧
• Another approximation to social discount rate would
be 𝑝𝑧
• Some argue in favour of 𝑝𝑧 as an approximation to
social discount rate because social discount rate
should be rate at which individuals should be willing
to postpone a small amount of consumption for
future consumption.
• As with shadow prices, the marginal rate of
time preference and the rate of return on
capital can be distorted.
• The distortions can include taxes, inflation and
risk (default or bankruptcy)
• Like shadow prices, we can take observed
interested rates and correct them for the
various distortions.
Computing 𝑟𝑧
• 𝑟𝑧 proxies for a rate of return on low risk private
sector investments before taxes but after
correcting for inflation
– Suggests that we can take an observed interest and
correct/adjust it to get an estimate of 𝑟𝑧
– Want to use a low risk corporate bond, so it would
have a lower default risk and adjust it for taxes and
inflation
• Three steps in computation, assume that
corporate bond rate is 6.86%, corporate tax rate
is 35% and inflation rate is 3.92%:
Computing 𝑟𝑧 : An Example
1. Figure out before return
0.0686
= 0.1055
1 − 0.35
2. Adjust for inflation
0.1055 − 0.0392
= 0.0638
1 + 0.0392
3. Adjust for bias in CPI
0.0638 + 0.01 = 0.0738
Computing 𝑝𝑧
• 𝑝𝑧 proxies for a rate of time preference after
correcting for inflation and taxes
– Suggests that we can take an observed interest and
correct/adjust it to get an estimate of 𝑝𝑧
– Want to use a government bond, and a higher level of
government, e.g., Federal first, provincial second, and
lastly local, so it would have a lower default risk and
adjust it for taxes and inflation
• Three steps in computation assume that interest
on government bond is 6.77%, personal tax rate
is 30% and inflation rate is 3.92%
Computing 𝑝𝑧 : An Example
1. Figure out after tax return
1 − 0.3 0.0677 =0.0474
2. Adjust for inflation
0.0474 − 0.0392
= 0.0073
1 + 0.0392
3. Adjust for bias in CPI
0.0073 + 0.01 = 0.0173
Criticisms
• 𝑟𝑧 tends to produce large discount rate
estimates; computations are based on using
corporate bond, which may have a risk
premium (e.g. firm may go bankrupt, investors
want a higher return to cover this)
• 𝑝𝑧 produces discount rate that are too low;
individuals may not properly account for the
long run effects of infrastructure programs on
future generations
Weighted Social Opportunity Cost of
Capital (WSOC)
• An alternative approach for computing the
social discount rate.
• Takes the perspective the discount rate should
reflect social opportunity cost of the resources
required for a project, with weights based
based on the relative contributions of the
different sources of resources
• The weighted social opportunity cost of
capital can be computed as𝑊𝑆𝑂𝐶 = 𝑎𝑟𝑧 +
𝑏𝑖 + 1 − 𝑎 − 𝑏 𝑝𝑧 , where a is the proportion
of the projects resources that displace private
investment, b is the proportion of resources
that are financed by borrowing from
foreigners, (1-a-b) is the proportion of
resources displacing domestic consumption,
and 𝑖 is the government's real long-term
borrowing rate
• Since 𝑝𝑧 < 𝑖 < 𝑟𝑧 ⟹ 𝑝𝑧 < 𝑊𝑆𝑂𝐶 < 𝑟𝑧
• We already know how to compute 𝑝𝑧 and 𝑟𝑧 ,
but not 𝑖; However, 𝑖 is relatively
straightforward to compute.
• Recall that 𝑖 is the government’s real long
term borrowing rate, so all we need to do is
adjust a nominal return government bond for
inflation to obtain 𝑖
Computing 𝑖
• Only two steps are need to compute 𝑖. (Figures
continue from previous example)
1. Adjust for Inflation
0.0677−0.0398
= 0.0268
2+0.0398
2. Adjust for Bias in CPI
0.0268+0.01=0.0368
• Note: there is no adjustment for taxes because
the government doesn’t pay taxes to itself.
• 𝑝𝑧 , 𝑖, 𝑟𝑧 are relatively easy to compute based
on available interest rate data
• The weights, i.e., a, b and (1-a-b) are harder to
determine
• In a Canadian context, Jenkins suggested using
the following values: a=0.75 and b=0.20,which
suggest that
WSOC=0.75(0.0738)+0.2(0.037)+0.05(0.0173)
=0.05696 or about 5.7%
• On the other hand, Burgess suggests that for
Canada a is likely to be between 0.26 and
0.32, b is between 0.55 and 0.64 and (1-a-b) is
likely to be between 0.1 and 0.13. Picking the
smaller value of a and the bigger value of b
produces a smaller value of WSOC; e.g.,
WSOC=0.26(0.0738)+0.64(0.037)+0.05(0.0173
)=0.0437 or 4.4%
• As another example, Suppose have a project that
is financed exclusively with taxes, then b=0. The
weight should represent the proportion of taxes
that reduce investment and 1-a-b should
represent the proportion of taxes that reduce
consumption. One can obtain an estimate of a
with the ratio of gross fixed investment to real
GDP. Recently, this ratio was computed as 16.8%,
so that
WSOC=0.168(0.0738)+0.0(0.037)+0.832(0.0173)=
0.0268 or 2.7%
Rules of Thumb: United States
• What do policy makers use in practice?
– In the United States the Office of Budget Management
used a real discount rate of 10 percent during the 1970s,
but had lowered this estimate to about 7 percent by 1992.
Recently, the Congressional Budget Office and the General
Accounting Office have used the 𝑝𝑧 approach to get a
discount rate of about 2 percent.
– Municipalities in the United States tend to use discount
rates of 3 percent with sensitivity analysis between 0 and 7
percent.
Rules of Thumb: Canada
• The Federal Treasury Board Secretariat has
recommended from about 1976 to the late-1990s, a
discount rate of 10 percent, with a sensitivity analysis
at 5 and 15 percent. But they recommend much lower
discount rates (0 to 3 percent) for health or
environmental cost benefit analysis.
• More recently, the Treasury Board Secretariat
(recommends) a discount rate of about 8 percent, with
a sensitivity analysis of 3 and 13 percent.
• The Treasury Board Secretariat also estimates the
social rate of time preference of about 3 percent.