Per-Capita Income as a Determinant of International Trade andEnvironmental Policies
James R. Markusen
University of Colorado, Boulder
Hitotsubashi University, Tokyo, Japan

Abstract

International trade policy analysis has tended to focus on the production side of generalequilibrium, with policies such as a tariff or carbon tax affecting international and internal
income distributions through a Heckscher-Ohlin nexus of factor intensities and factor
endowments. Here I move away from this structure to focus on demand, preferences, andendogenous policy in a trade/environment setting by assuming a high income elasticity ofdemand for environmental quality. I show how both non-cooperative and cooperative
abatement policies in a two-country (rich and poor) setting are affected by nonhomotheticity. I examine “issue linking” in international bargaining, in which one country isboth large and rich, and hence has both a high tariff and a high abatement effort in a noncooperative equilibrium. Several cooperative bargaining agreements are computed underalternative assumptions about linking or separating trade and environment negotiations. A
final exercise considers “policy leakage”, in which one country has an incentive to reduce its
optimal abatement effort when the other country increases its effort. The paper will also
introduce many readers to a new solver in GAMS for a class of problems referred to as
MPECs: mathematical programming with equilibrium constraints. This problem class has
wide applications in economics, including solving for multiple optimal tax rates to provide
public goods, redistribute income, internalize externalities, exploit monopoly power in tradeand so forth, when the underlying general-equilibrium model is a set of constraints on the
optimization problem.

Draft 0, October 19, 2012
I have not written a literature review for this draft, and am currently seeking and reading related
papers. I am especially interested in finding empirical work that estimates something like an income
elasticity of demand for environmental quality. Please send references with attached notes explaining
their relevance.
This paper originated following an invitation to present a paper at the conference Environmental
Protection and Sustainability Forum, held at the University of Exeter April 13-15, 2011, which
reignited a long-dormant interest in trade policy, more specifically trade and the environment. I thankMichael Finus and Christos Kotsogiannis of the University of Exeter for their invitation and their hard
work in organizing an excellent conference. The first draft was finally written while a visitor at
Hitotsubashi University, and I thank my colleagues there for a delightful and productive three months.

1.

Introduction

Much of the literature on trade policy concentrates on the production side of generalequilibrium. In the trade/environment literature for example, a typical model might involve
the nexus among factor intensities in production, factor endowments of countries, andpollution intensities across goods. A common question might be how changes in trade orenvironmental policy of one country then change total world pollution and the distribution ofpollution emissions across countries (e.g., the carbon leakage and pollution-haven
literatures). Considerations of the role and determinants of the demand for environmental
quality across countries and hence endogenous policy are less featured, with the exception ofa few papers on the environmental Kuznets curve.
This paper develops a model of non-homothetic preferences in which environmental
quality has a high income elasticity of demand. At the same time, the usual production side
factors just mentioned are neutralized via the use of a novel constant elasticity of substitution
transformation function. Global, trans-boundary pollution depends only on both countries’
total production and resources withdrawn from production are used in an abatement activity,
financed by a non-distortionary tax. Comparative advantage in two consumption goods links
the countries through trade, such that more abatement by either country shifts worldcommodity prices in that country’s favor by withdrawing resources from production.
The improved terms of trade partly compensate the environmentally conscious
country by shifting part or even half of the burden of abatement in terms of reduced
commodity consumption onto the other country. This burden sharing is increasing in aparameter determining the comparative advantage spread between the countries: full
specialization in the two goods generates full consumption reduction sharing when only one
country increases abatement. No comparative advantage and hence no goods trade means acountry with no intervention is a complete free rider on the country increasing abatement.
This production setup allows us to concentrate on the role of per-capita income anddifferences in per-capita income across countries in determining optimal cooperative andnon-cooperative policy outcomes. I develop the basic intuition for later results by showingthat the model produces (as intuitively expected) a non-monotonic Kuznets curve: as acountry’s productivity rises, environmental quality at first deteriorates, then begins toincrease as it begins an abatement effort, and eventually passes the quality level originallyprevailing when the country was poor.
Next, we examine non-cooperative and cooperative outcomes with respect toabatement efforts by two countries of equal total income. Allowing the per-capita income of
the countries to differ holding total income constant (e.g., one country has fewer but moreproductive households), the Nash non-cooperative policy outcome has a higher abatement
effort by the high-income country and a lower effort by the poor country relative to whenboth countries have identical per-capita incomes. The same is true for a cooperative Nashbargaining outcome using the non-cooperative outcome as the disagreement outcome. This
cooperative outcome is contrasted to an outcome that maximizes the product of the twocountries utilities and to a Rawlsian outcome which maximizes the minimum of the twocountries’ utility gains over no intervention by either country.
1

2
Two further experiments are conducted on this question. In the first alternative to thebase case, comparative advantage is eliminated so that there is no trade and no effect of one
country’s abatement on the terms of trade. This reduces the non-cooperative abatement effort
of both countries, since a small increase in abatement no longer shifts the terms of trade inthat country’s favor. The cooperative outcome, on the other hand, is close to that in thecomparative-advantage case, since cooperation helps internalize the terms-of-trade
externality in the comparative-advantage case. However, I show that it can now be the poor
country that gains more at both the cooperative and non-cooperative outcome relative to zeroabatement by both countries (it does not have a terms-of-trade loss).
In the second alternative case, the difference in the per-capita income of the twocountries is widened considerably (returning to base-case comparative advantage). Obvious
results follow about equilibrium cooperative and non-cooperative abatement efforts. But one
interesting result is that the poor country may be made worse off by even a modest abatement
effort by the rich country, since the term-of-trade deteriorating for the poor country may
outweigh the smaller benefits of any spillover improvement to its environment.
Suppose that countries agree on an “equal sharing rule”, by which I mean that eachcountry agrees that they will each cut their emissions by the same proportion. Having
adopted this rule, they bargain on the level of the proportional cut. I illustrate that the
maximum cut that the poor country will agree on (the one that leave it no worse off than no
cut at all) falls as its per-capita income falls. At some point, no joint cut can make it betteroff so that the cut that is adopted is zero.
The paper then turns to “issue linking” in international bargaining. Specifically, thereis the standard motive for protection in that an import tariff improves a country’s terms oftrade. Is there a role for linking together bargaining on abatement efforts with tradeliberalization? A clear possibility occurs when the rich country is also large, so that it will
have a high tariff in a Nash equilibrium in tariff rates, as well as a high abatement tax. It is
possible that the countries can benefit if the large country trades a lower tariff for the poor
country increasing its abatement effort.
Using a relatively novel optimization solver in GAMS, I model the generalequilibrium policy problem as an MPEC (mathematical programming with equilibrium
constraints). In a non-cooperative outcome, each country chooses a tariff and pollution tax tomaximizing its utility given the tariff and tax of the other country, with the two-country
general-equilibrium model as a constraint set. In a cooperative outcome, the Nash bargaining
function is maximized with respect to the four policy instruments, again with the generalequilibrium model as the constraint set.
The base case has country h having five times both the total income and per-capita
income of country s. The non-cooperative Nash equilibrium has a high tax in tariff for n, a
low tariff and a zero abatement tax for country s. From this equilibrium, I calculate three
two-instrument bargains: (1) an isolated (unlinked) agreement on tariffs, (2) an isolatedagreement on abatement taxes, and (3) a bargain in which n lowers its tariff in exchange for a
an abatement effort by country s. I then calculate a cooperative outcome in which all four
instruments are negotiated together. These two experiments produce outcomes which havequite different values for the four instruments. We will also see that the disagreement

3
outcome matters a lot; specifically, whether the Nash equilibrium is the disagreement
outcome for bargaining or whether the no-intervention equilibrium is used as thedisagreement outcome (country bargaining starting from scratch).
The paper concludes with an example of “policy leakage”: the incentive for one
country to free ride on the abatement effort of its trading partner. Again formulating the
problem as an MPEC, I solve for the level of the poor country’s optimal abatement tax foreach (exogenous) level of the rich country’s tax. Not surprisingly, the former is falling in thelatter, and the leakage rate, the shortfall in emissions relative to those that occur holding thepoor country’s tax constant, is around 50-60% for the specific parameter values used.
I will not attempt a literature review in this first draft, though I have attached anumber of references I have come across to date. Don’t get mad, get even: send me
references and describe how they fit in. Clearly, Copeland and Taylor’s (2004) review article
is an important starting point. But very few papers seem to directly ask questions about theincome elasticity of demand for environmental quality. Although their focus is different
from mine, Broner, Bustos and Carvalho (2012) have shown a relationship between
environmental standards and per-capita income (again, not their focus however), using thenational standards on the maximum lead content of gasoline (petrol) across countries. I havegone to the source of this data (Lovei 1998) and show it in a plot against per-capita income
for the same year as the lead-content data (1996) from World Bank statistics in Plot 1. I am
not passing this off as sophisticated econometrics, but the negative relationship (meaning
higher standards in rich countries) is economically and statistically strong.

2.

Non-homotheticity and Trade Policy: an Application to Trade and theEnvironment

In this section, I consider an international policy question that is of current interest:
the relationship between trade policy and international environmental policy with a global
pollutant such as CO2. Assume that we have two final consumption goods (X1, X2), one
environmental good (E), and two countries (n, s), where n (north) will be the higher percapita income country.
Final goods are produced by a CET (constant elasticity of transformation) technology,
with one input L, with Lz denoting the total efficiency units of labor allocation to production.
Efficiency units are the number of households times productivity per household.
(1)
where F is the elasticity of transformation along the production frontier. Z can be thought ofas the production equivalent of the “composite commodity” used in the standard DixitStiglitz model of product differentiation. Why this is useful should become clear shortly.
Equation (1) is awkward for calibrating the parameters to specific numerical values,
so general-equilibrium modelers generally write this in “calibrated share form”, where the4parameters are all easily derived from observed prices and quantities. Let superscript ‘0'denoted observed quantities and prices, and let prices of Z, X1, and X2 be denoted by p’s.

(2)

Suppose that we observe expenditures on the Xi and the total value of output, which will be
the expenditure on Z. We can choose units so that all prices equal one initially. Second,
multiply both sides through by Z0, so that (2) can be written as:(3)

so that the "i’s are the initial observed shares of the Xi’s in the value of production. With
units chosen such that prices are one initially, this simplifies to(4)This transformation function is derived in an appendix to the paper, which also shows two
further results. First, the unit revenue or national-product function (the production equivalentof the expenditure function) is given by(5)
Second, the unit supply functions are found by applying Shepard’s lemma to (5):(6)

Relative production of X1 and X2 then depend on prices and their output levels depend on
total labor allocated to produciton.
since pz = r(p) in equilibrium(7)

5
Let E* be the level or “endowment” of world environmental quality at zeroproduction. Pollution is modeled as a reduction in the endowment of good E and is
proportional to the total aggregate output of Z in both countries. Input L can also be used for
an abatement activity A. Aggregate labor in each country is then divided between production(Z) and abatement (A):where, as before, is effective labor supply.
Pollution = Reduction in E endowment =(8)

Abatement = Addition to the E endowment =For any allocation of labor between production and abatement in the two countries,
world environmental quality is then given by:(9)
Abatement is financed by a consumption taxes in n and s: tn , ts. Thus public policy,
via the consumption tax, can determine that allocation of the composite input L between
production and abatement. Specifically, the equal taxes on both goods is equivalent to a tax t
on the labor input to production. Budget balance then requires tax revenues twLz to equal
abatement expenditures wLa where w is the wage rate. Thus the tax rate (in either country)
will equal the ratio of labor in abatement to labor in production.
share of labor in abatement () (10)

The advantage of this simple model is that it implies “neutrality” in several senses.
By neutrality I mean:
=>
=>
=>
=>

no pollution-intensive sector
no comparative advantage in polluting sector
no factor-intensity, factor-endowment issues
no pollution-from-consumption-versus-production issue

The dominant model in the trade-and-environment literature has only one sector that pollutes,
This leads to policy results that are very sensitive to:
=>
=>
=>

which factor is intensive in which goodwhich good is the country’s comparative advantage goodwhether pollution is from consumption or production

Here we avoid these issues. However, at the same time, allowing the "i’s to differ
across countries generates a comparative advantage motive for trade and gains from trade in
the X’s. This will be important since world prices will change as one country withdraws
labor from production in order to increase abatement. This terms-of-trade change in therelative prices of the two X goods will always favor the environmentally conscious country, a6point I will return to shortly.
Preferences are Stone-Geary, lower case letters for per-capita quantities(11)
where e0 is a positive parameter for each household in each country which creates the nonhomotheticity: up to a critical level of income, there will be no demand for environmental
quality. It can be useful to think of e0 as an endowment good given to each household whichcannot be traded: every household can watch the sunset and that is a perfect substitute for acleaner environment. E is the (world) environmental good supply and is a pure (non-rivaled
and non-excludable) public good. So each consumer in each country gets to consume theentire world supply.
Let

denotes the demand for environment in country i,
is country i’s per-capita income
is the price (willingness to pay) by a single household for environment in i.

Consumer optimization yields:
(12)

(13)
The result is the one just noted: up to the threshold per-capita income given in (13), there isno demand for environmental quality. Once the threshold income is reached,(14)

At a constant price, the income elasticity of demand for E is greater than one once thethreshold is reached. In equilibrium, E must be the same for all consumers in all countries(perfect global public good). So for (14) to hold, it must be that the willingness to pay pe
differs across countries. This (private) willingness to pay can be found by inverting (14).
(15)

Since the environment is a public good (non-rivaled and non-excludable), optimal
policy depends on the sum of the individual benefits of the good. We can work with either(14) or (15). The right-hand side of (14) can be multiplied by hi, the number of household incountry i (the aggregate supply of the alternative “sunset” good is proportional to the number

7of households). When this is set equal to the (exogenous) quantity of E, the price that
produces equality is interpreted as the social willingness to pay, .
(16)

Alternatively, the social value of E is given by multiplying the right-hand side of (15) by hi
and the supply of the alternative endowment good (sunsets) is also multiplied by hi.
Denoting total income as, gives the same equation as found by inverting (16).
where Mi is total income of i (17)Holding total income constant, an increase in per-capita income is a reduction in thenumber of households hi, and so the social willingness to pay rises. Note that this passes one
simple check: if preferences are homothetic such that e0 = 0, then the social willingness topay depends only on total income and not per-capita income and the value of demand is aconstant share of total income:. Note also that with or without homotheticity,
the social price in (17) is greater than the private willingness to pay in (15) for h > 1.
Assume that there is no pattern of comparative advantage:= 0.5 for all i = n, s,

j = 1, 2. This assumption combined with the equal-shares on X1 and X2 in the utility function
implies that the world prices of X1 and X2 are equal, so the price of Z is also equalized acrosscountries. There is no motive to trade goods. Let Z be numeraire: its price equals one in bothcountries as do the prices of X1 and X2.
Using a consumption tax on Z, the efficiency condition for each country individuallyis that the (social) marginal rate of substitution between Z and E should equal the ratio of
marginal costs. Equations (8) above imply that the ratio of marginal costs of an added unit ofenvironment E to the marginal cost of an extra unit of Z equals one. Let
be theproducer price of Z in country i. With the consumer price of Z (and of X and Y individually)
given by, the environmental distortion from country n’s and s’s points of viewis internalized with a tax such that:(18)

A country will wish to have a positive tax if

exceeds one at the zero-tax level of E.

Assume that both countries have positive taxes. From (16) and (17), the ratio of the tax rates
is given by

8
(19)

from which we get the following results:
Result: holding E constant, country n will have the higher tax rate if:(a)both countries have the same total income but country n has a higher percapita income (Mn = Ms, hn < hs).

(b)

both countries have the same number of households, but n has a higher percapita income and therefore higher total income (Mn > Ms, hn = hs ).

(c)both countries have the same per-capita income, but country n has a higher
total income (hn = Dhs, Mn = DMs, D > 1).

The fact that the relative tax rates depend on country size is due to the pureinternational public-good nature of the environment. Each country gets the same E, so
willingness to pay for the fixed quantity is higher in the large country even if per-capita
incomes are the same.
Of course, the introduction of this tax will lead to an increase in E for both countries,
lowering the willingness to pay for environment: the q’s are endogenous to the tax. (18) isjust a formula in which the q’s depend on the taxes, so we have no idea what the value of thetaxes might be. Secondly, (18) does not take into account the effect of withdrawing
resources from production to use in abatement on the terms of trade in goods, an effect“neutralized” in our special case. This terms-of-trade effect can partially compensate country
n for the lower production of goods when it uses resources for abatement. Finally, the tax in(18) is not jointly optimal of course, because it does not take into consideration the benefits
to country i from the improved environmental quality by country j, a point I return to later.
In order to understand the terms-of-trade effect, I make some very special
assumptions in order to derive a clear expression for the effect of abatement by one country
on the terms of trade. First, assume that the elasticity on transformation is zero, so that goods
are produced in fixed proportions in each country, with that proportion depending on the
country’s alpha parameters. Assume that there is free trade in goods, so that the world
production ratio equals the (inverse) consumer price ratio in equilibrium (since the goods are
Cobb-Douglas symmetric substitutes in consumption). We then have:(20)

Let country n withdraw a unit of labor from production. Differentiating (20), we have:

9
(21)

where we will assume that country n has the comparative advantage in good 1:
and.
Now make some addition restrictive assumptions. Assume that the countries are
identical initially and that there are no initial abatement taxes, so that
where
is the total endowment of each country. Second, assume symmetry in the comparativeadvantage parameters such that

. Becauseand= 1 initially, (21) simplifies to (shown in the appendix)

or

(22)

The effect of country n withdrawing a unit of labor from production

is toreduce the relative price of country s’s export good X2 (country s has a deterioration in its
terms of trade) with the size of this effect being proportional to the size of comparativeadvantage spread.
.

Two extreme cases are instructive:: no comparative advantage and hence no trade in goods. Country s
has no adverse terms-of-trade effect and benefits from the reduction in admissions
from country n.

This is shown in Figure 1a, where the (right angle) production frontiers are initially the sameand country n’s frontier moves in with labor transferred to abatement. All the reduction incommodity consumption is borne by country n while both countries benefit from an increasein E.
, that is,= 0: each country is fully specialized. Country s

has the maximum possible adverse terms-of-trade effect but does benefit from thereduction in emissions from country n.
This is shown in Figure 1b: because of Cobb-Douglas preferences between X1 and X2, theworld price ratio changes so that both countries share the same reduction in commodity
consumption. Fully half the burden of withdrawing resources from production by country n

10is shifted to country s. With Cobb-Douglas demand, each country will have the same income
M:
since

which follows from (10), and the symmetry of the alphas

(23)

.

With identical Cobb-Douglas sub-utility functions over X1 and X2, the countries will
have identical (X1, X2) consumption bundles regardless of the level of country n’s abatement
effort. In other words, the change in the terms-of-trade change fully compensates country n(relative to country s) for its unilateral abatement effort. Both countries have an identicalconsumption loss and the same aggregate improvement in the environment.
If country n has a higher per-capita income (Ln of effect labor is composed fewer but
more productive households than country s) then country n gets a bigger boost in welfare
than does country s. In fact, if country s is sufficiently poor such that it places only a verysmall added value to the environmental good, then country s is likely worse off from n’s
abatement.

3.

The general-equilibrium simulation model

The model developed above seems simple, but there is a lot of simultaneity. Optimal
policy depends on income, for example, and income depends on the policy chosen, both bydetermining the domestic resources available for production of goods and through theinternational general-equilibrium terms-of-trade effect. Note, for example, that the simple
result in (22) requires severe assumptions and even then it is only locally valid in theneighborhood of zero abatement. Similarly, the simple tax formula in (18) is not very
informative in general-equilibrium. (18) is just a formula, it is not a value. So let’s turn to anumerical general-equilibrium model to see how per-capita income matters for cooperativeand non-cooperative outcomes.
The model belongs to a class of problems loosely known as MPEC: mathematicalprogramming with equilibrium constraints. In our case here, the set of equilibrium
constraints is the two-country general-equilibrium model. The latter, in turn, is known as an
MCP: mixed complementarity problem. This is a set of weak inequalities with associatednon-negative variables such as quantities and prices. When a weak inequality holds as anequality, the complementary variable is positive, zero if the inequality is strict in equilibrium.
The MPEC consists of maximizing some function such as a Nash bargaining function(cooperative) with respect to tax rates (pollution taxes and/or tariffs) subject to the economic
equilibrium constraint set. A non-cooperative Nash equilibrium is found by iteration:
maximize the welfare of i holding j’s taxes constant, then hold i’s taxes constant at thesolution values and maximize j’s welfare, repeat. This converges to a best-response, noncooperative outcome in about eight iterations. Here is a description of the model.

11
(A)

Alternative objective functions
welfare of country nwelfare of country s
joint welfare (or Nash bargaining function)(B)

The mxm economic equilibrium problem (constraint set): 28 inequalities andunknowns
Inequalities

Complementary Variables

marginal cost $ priceproduction of Z by n, s
trade in X1, X2 by n,swelfare in n and sabatement activities in n and squantities
quantities of Z
quantities tradedwelfare in n and squantity of abatement

Numbermarket clearing: supply $ demand
pricessupply / demand for Zn, Zs
prices of Z in n, s
prices of Xi in n, s
supply / demand for Xi in n, s
supply / demand for L in n and s
prices of L in n, s
supply / demand for welfare
price index in n,s
supply / demand for abatement in n, s
price of abatement in n, s
supply / demand for environment goodwillingness to pay in n,s

2
4
2
2

2
4
2
2
2
2

income balance: income $ expenditure
income balance for n,sauxiliary equations
pollution reduction = abatement
pollution = emissions(C)

income in n,s2pollution abatement
pollution

1
1

Additional unmatched variables chosen to optimize welfare:pollution abatement effort (abatement tax) in countries n, s
tariffs imposed by countries n, s

(D)

2
2

The MPEC (mathamtical programming with equilibrium constraints)
Maximize objective function, subject to:pollution abatement taxes and tariffs in n, s
mxm economic equilibrium problem constraint set

428

A sample program, written and solved in GAMS, is provided in appendix two to the paper.

12

4.

Policy experiments

The first policy experiment is shown in Figure 2. This considers environmental
quality as a function of per-capita income, where I make the countries identical (in total andin per-capita incomes) for simplicity. Productivity or “effective” labor units per household
are increased holding the number of households constant. The MPEC solves for the optimal
taxes or abatement effort at each level of income (productivity).
This produces a non-monotonic Kuznets curve in Figure 2 as we expect. At very low
levels of per-capita income, there is no demand for abatement or additional environmental
quality and the latter falls with increases in productivity. At a critical level around 0.9 in thisexperiment, there is a positive demand for additional environmental quality and the tax kicks
in. The tax rate rises steadily thereafter due to the non-homotheticity, and is equal to 0.25 on
the right-hand boundary where environmental quality is now higher than in the very poor
county.
What is perhaps not so obvious is that there is still some non-monotonicity in the
Kuznets curve with homothetic demand. This is due to the fact that the initial fixed
endowment of E = E* is to high or rather the demand price too low to justify abatement at
low income levels. Intuitively, a really poor country would want to sell off some of theenvironmental good if it could under either homothetic or non-homothetic demand. Adding
the non-homotheticity assumption shifts the minimum point to the right in Figure 2 (notshown). If there was no initial endowment of E, E* = 0, then the homothetic case would
produce a constant tax rate at all levels of income such that environmental quality is a linear
function of income passing through the origin, while the non-homothetic case continues tolook like Figure 2 (from (2), E can go negative provided that -E < e0). This possibly
expected result occurs only when E* = 0.
The next exercise is to examine how cooperative and non-cooperative policy
outcomes depend on per-capita income. Some result are shown in Figure 3. In all cases, thetotal incomes of the countries are equal. There is a strong pattern of comparative advantage
in the case considered,, and F = 1. Thus when per-capita incomes and taxes
are unequal, there will be a fairly strong terms-of-trade effect that favors country n.
The solid boxes in Figure 3 show an outcome when the two countries have equal percapita incomes (or same number of equally productive households). Each country’s(unilateral) optimal tax is shown when the other country’s tax is zero as are the noncooperative Nash equilibrium rates. There is no difference in the values of the taxes acrosscountries in this latter equilibrium as we expect to be the case. The (equal) cooperative tax
rates are Nash bargaining outcomes where the disagreement outcome is the non-cooperative
Nash equilibrium. The cooperative rates are considerably higher than the non-cooperative
ones as shown in Figure 3. The terms-of-trade in both outcomes is one and there is no tradein goods.
The second set of outcomes, shown with a slash through the boxes in Figure 3,
assumes that country n has a per-capita income 1.5 times that of country s. Total incomes are
the same as in the first case (solid boxes), so country n now has fewer, more productive

13households while s has more, less productive households. Figure 3 shows that the noncooperative equilibrium shifts to a higher tax for country n and to a lower tax for country s(as we would expect). The cooperative Nash bargaining outcome using the non-cooperative
outcomes as the disagreement point shifts from the equal-per-capita-income scenario in about
the same way.
As suggested earlier, country n will be partially, perhaps significantly, compensatedfor its higher abatement effort by an improvement in its terms of trade. Specifically, therelative price of good X1, country n’s export good, is 1.12 at the non-cooperative Nashoutcome and 1.14 at the cooperative outcome.
Table 1 presents numerical values for these results and compares them to several
alternative scenarios. The first three columns of number are the case where country n’s percapita income is 1.5 times that of country s as in Figure 3. The first row gives the welfare
values when there is no intervention by either country, where these welfare values arenormalized at one. The second and third row of Table 1 give the welfare values for the noncooperative and cooperative outcomes, and the lower panel the corresponding tax rates(PTAXN, PTAXS), which are those in Figure 3. Note that the non-cooperative outcome
results in a substantial welfare gain over non-intervention: unlike a non-cooperative tariff“war” for example, here the non-cooperation is a failure to internalize a positive rather than anegative externality.
The tax rates that maximize the Cobb-Douglas world welfare function and a Rawlsian
welfare function are also shown (world welfare is the minimum of the two countries’ welfare
levels.). The former is equivalent to a Nash bargaining outcome when the disagreement
outcome for both countries is zero. Curiously, the Rawlsian outcome is not very different
from the cooperative Nash outcome, but does require a higher tax for country n and a slightly
lower one for country s relative to the cooperative outcome. I do not have any intuition totell me if this is just a “coincidence” or if there is some deeper truth here. Also curious, thetax rates that maximize the Cobb-Douglas index of world welfare are equal (0.35). Here I do
think there is a deeper truth. World production efficiency requires equal tax rates, but it isunclear to me why this should be a characteristic of maximizing the Cobb-Douglas index.
An interesting feature of these results is that country n is actually the relative gainer
over the no-intervention outcomes (except the Rawlsian one). How much of this is simply
due to the fact that country n places a much higher value on the environment at the nointervention point, and how much might be due to the this terms-of-trade effect? In order toexamine this question, I compute an scenario in which there is no comparative advantage:

Results are shown in columns 4 and 5 of Table 1. Here we see some significant
differences. First, the relative gainer is reversed in the cooperative and non-cooperative Nashoutcomes. Now country s is the relative gainer. This verifies the conjecture that the termsof-trade effect that favors the higher-tax country n in columns 1 and 2 is indeed important indetermining the relatively larger gains for country n in those columns. Second, note in thelower part of the Table that the non-cooperative rates for both countries are lower when there
is no comparative advantage. This is clearly due to the fact that raising your tax rate has nocompensating beneficial effect on the terms of trade with no comparative advantage. Third,

14
the cooperative Nash is almost unchanged from the comparative-advantage case, presumably
because this case internalizes the terms-of-trade externality. The CD maximum rates are alsounchanged, because there is no terms-of-trade effect when the countries have the same rates.
Fourth, the Rawlsian maximum rate is now lower for country n and higher for country s. I
think that the intuition here is that, starting from the Rawlsian taxes with comparativeadvantage, removing comparative advantage reduces the welfare of high-tax country n andraises it for country s. The Rawlsian outcome then adjusts n’s tax down and s’s tax up.
A second alternative case is presented in columns six and seven of Table 1.
Comparative advantage is reinstated, but country n now has nine times the per-capita incomeof country s. Results are now that country s has a zero tax rate at both the cooperative andnon-cooperative Nash outcomes. The non-cooperative outcome is in fact a Pareto optimum:
there are no gains from cooperative. Second, note that country s is almost indifferent
between the no-intervention outcome and the cooperative or non-cooperative equilibrium. If
I push the size difference a little higher, then country s is actually worse off than with nointervention. Country s places little value on improved environmental quality and suffers anegative terms-of-trade effect when country n imposes its abatement tax. The relative priceof country n’s export good is 1.22 at the cooperative and non-cooperative tax rate of 0.41.
Third, there is now a large difference between CD world welfare maximum tax rates
and the Rawlsian rates. The CD rates are again equal to one another, while country n bears
all the abatement effort in the Rawlsian equilibrium. As in alternative case 1, the welfare
levels of the two countries are not equal in the Rawlsian equilibrium in alternative 2; in fact,
they are quite different (there is some rounding error that make the Rawlsian outcome look
Pareto inferior to the cooperative one - it isn’t). At the Rawlsian equilibrium, any increase ineither country’s tax makes country s worse off.
The model is then used to consider an experiment in which countries can agreed tobargain over a common level of abatement taxes or, given (10) above, a common share ofresources devoted to abatement. The countries are initially identical in total and per-capita
incomes, and then the per-capita income of country s is reduced holding its total income
constant (more but less productive households). Results are shown in Figure 4, where thecommon tax rate is on the horizontal axis. Both countries’ welfare levels are shown by the
curve “country n’s welfare” when they have the same per-capita income. Now lower country
s’s per-capita income to one-fifth and then one-tenth of country n’s per-capita income. The
curve for country n does not change, while that for country s is flatter and has its maximum ata lower level of per-capita income. When the difference between the countries’ per-capita
incomes is a factor of ten, country s cannot gain at any common tax rate. Thus having agreed
on an “equal sharing rule” and then bargaining on the rate, the outcome is a zero rate when
the per-capita-income difference is large. This has some clear applications to situations suchas the Kyoto protocol, where there was considerable controversy over whether or not thepoor and rich countries should suffer the same proportional cuts in carbon emissions.
Now I turn to issue linking and introduce two additional policy instruments: importtariffs for countries n and s. We will assume that country n has five times both the total
income and the per-capita income of country s. Comparative advantage is the same as we
used earlier:and F = 1. Some results are shown in Figure 5 and more
complete results in Table 2. A shorted version of the GAMS program (it doesn’t solve for all

15
scenarios) is attached at the end of the paper. The no-intervention case shown in the first row
of Table 2 fixes the four policy instruments at zero. The joint-welfare function is thenessentially a “dummy”: the solution to the model is equivalent to simply solving the twocountry general-equilibrium model at zero taxes and tariffs (PTAX denotes pollution(abatement) taxes and TAR denotes tariffs of n and s). The welfare numbers are normalized
at 1.0 in the non-cooperative outcome, but I should note that the difference in the welfare
levels is nowhere near a factor of five: n’s welfare is about twice that of country s. This is due
to a large relative price advantage for country s: the relative price ratio is 0.472 or inverting,the relative price of country s’s export good X2 is 2.121.
Tables 2 and Figure 5 then give the non-cooperative Nash equilibrium, where eachcountry jointly chooses its abatement tax and tariff for fixed values of the other country’s
instruments. This is solved as an iterative MPEC. Country n’s welfare is maximized withrespect to PTAXN and TARN holding PTAXS and TARS constant. Then the solution values of
PATXN and TARN are held constant and the welfare of country s is maximized with respect
to PTAXS and TARS. The iteration converges to the non-cooperative Nash equilibrium inabout six to eight iterations.
Table 2 and Figure 2 show that, not surprisingly, country n has a high abatement tax
and tariff. Country s has a small tariff and a zero abatement tax. Table 2 notes a substantial
welfare improvement over no intervention for both countries. However, I imagine a case caneasily be produced in which country s is worse off than in the no-intervention equilibrium
following the logic of Table 1.
Four cooperative bargaining outcomes are then computed using the Nash equilibrium
as a disagreement point in Table 2. The first computes an isolated environmental tax bargain.
The second computes an isolated tariff bargain. In each of these two cases, the instruments
not under negotiation are held at their non-cooperative Nash levels. One interesting result
here is that there are no gains to an isolated environment negotiation. No Pareto improving
changes in taxes can help and nothing will come of such a negotiation (sound familiar?). The
isolated tariff negotiation does produce a positive result, as country n lowers its tariff
significantly in exchange for country s eliminating its tariff.
In the third case using the non-cooperative outcome as a disagreement point (row 5 ofTable 2), country n offers a lower tariff in exchange for a positive abatement effort bycountry s. In my view, this is what some writers and politicians in high-income countries
want to do: offer trade liberalization in exchange for environmental and labor standards topoor countries. The joint welfare function WELG (global) is maximized holding PTAXN andTARS at their non-cooperative levels. Results are shown in Table 2 and in Figure 5.
In order to get a feel for the “global” properties of the linking payoff, I also computed
this by a brute-force grid search, which is the surface shown in Figure 5 (using only two
instruments allows for a nice graphical representation). The non-cooperative equilibrium is
on “Country n’s tariff” axis with “Country s’s ptax” equal to zero. The vertical axis gives thevalue of the Nash bargaining function, equation WELFAREG in the GAMS program at theend of the paper. Only points that are Pareto improving have positive values for this function
and the maximum point of the surface in Figure 5 is of course the payoffs in the Nashbargaining equilibrium. As noted in Table 2 and Figure 5, country n gives up a substantial
reduction in its tariff in exchange for an increase in country s’s pollution tax from 0.0 to 0.15.

16Note that country s would rather have an isolated negotiation about trade only, the thing it
really cares about, than negotiate over PTAXS for a lower TARN.
Row 6 of Table 2 takes the non-cooperative Nash outcome as a disagreement outcome
and computes a cooperative bargaining solution treating all four instruments as endogenous
variables. Here the outcome is a zero abatement tax and a zero tariff for country s. Country
n bears the burden of abatement and, in exchange, retain a high tariff but substantially lessthan the non-cooperative tariff. The four-instrument cooperative outcome is a “long way” inthe values of the instruments from the two-instrument outcome shown in row 5 of Table 2.
Yet the welfare gains to the four-instrument bargain are not a great deal larger than the twoinstrument case, a point I will return to in a minute.
In order to see the importance of the disagreement outcome to the cooperative
solution, row 7 of Table 2 gives the results of a computation in which no intervention is thedisagreement outcome with all four instruments endogenous. This is equivalent tomaximizing the Cobb-Douglas product of the two countries’ utilities. Note that this is very
different from row 6 which uses the non-cooperative outcome as the disagreement outcome
in bargaining. In row 7, country n gives up its tariff entirely, while country s gets a small
tariff to help raise its welfare. The difference between rows 6 and 7 in Table 2 is due to thebig gain that country n gets in the non-cooperative case by being able to turn the terms-oftrade substantially in its favor through a large tariff (1.567). This high level of welfare in the
non-cooperative case then allows it to extract more in the bargaining solution in row 6 than if
no intervention is the disagreement outcome. Country s is much better off with nointervention as the disagreement outcome, while country n is worse off even relative to thenon-cooperative Nash outcome. Quantitatively, the assumption of what is the disagreement
outcome matters a lot.
It is interesting to see, in both Table 1 and Table 2, that cooperative bargaining does
not extract much in the way of additional gains. The non-cooperative outcomes do a “goodjob” of extracting gains, since the non-internalized pollution externality is a positive spillover
between countries. But this remains true in Table 2, even though the non-cooperative
outcome has a big tariff for country h, which has a negative effect on country s. Apparently
country s gets a sufficient benefit from n’s pollution tax and from s’s own tariff to much morethan compensate. I have no reason to believe that this specific result has great generality
beyond this specific case and I do not attach too much significance to the particular
numerical values we are getting here. It clearly is the case that (a) the values of the
instruments vary widely depending on what is bargained over and (b) both welfare and the
instruments are very sensitive to the disagreement option.
The final exercise considers “policy leakage”, a term I will use to indicate one
country reducing its abatement effort in response to an increase in its trading partner’s
abatement effort. There is no particular role for differences in per-capita income here, but I
thought a couple of brief comments are worthwhile having developed the model. Most
literature that I am aware deals with leakage through changes in market prices when one
country imposes a carbon tax, for example, and the other country is (countries are) passive.
Action by the first country lowers world fossil fuel prices and so firms and consumers in thepassive country will use more.
This traditional mechanism doesn’t work here by virtue of the fact that there is no17
carbon-intensive sector and hence no relative price change lowering fossil fuel prices in
response to one country’s tax. But we can get a policy response by the second country. A
computational example is shown in Figure 6, where the horizontal axis is the (exogenous)
level of country n’s abatement tax. As we would expect, country s’s option tax is negatively
related to country n’s tax as shown: the tax rates are “strategic substitutes” even thoughcountry s’s welfare is increase in n’s tax. A higher tax in n improves the environment andlowers the willingness to pay in country s. This is nothing other than a free-rider problem inenvironmental policy.
The marginal leakage rate can be defined in a few different ways. In Figure 6, it isdefined as one minus the change in actual abatement when country s adjusts its tax optimallydivided by the change in abatement that occurs when country s’s tax is fixed, in response to a
tax increase by country n. Thus if the increase in abatement by n is perfectly offset by adecrease in abatement by country s, then the leakage rate is 1. If country s does not adjust its
tax, the leakage rate is zero. Under this definition, the leakage rate is high until country s’s
tax rate is driven to zero.

5.

Summary and Conclusions

The purpose of the paper is to make the case that per-capita income has an interestingrole to play in cooperative and non-cooperative international trade policy. While some of theresults are not surprising, there are a number of interesting subtleties than may have goneunnoticed.
The model can inform policy and perhaps help us understand how we got to where weare. On the former, there has been much debate about linking issues in negotiations, andmuch of it has been ideological in nature: in high-income countries, the left thinks its great,
the right thinks its awful. This paper suggests that linking can extract additional gains: recall
that there are no gains to be had negotiating over environmental policy alone in our example.
In the area of understanding how we got to where we are, the results here may help
explain why the high-income countries often seem to give up more than they get ininternational negotiations (some will surely dispute this assertion), such as the Kyoto
Protocol. While this may indeed be all or in part due to simple altruism, some of oursimulation results here suggest that this is also predicted by standard economic theories ofcooperative and non-cooperative behavior. Specifically, with environmental quality a highincome-elasticity good, standard theory predicts a high abatement effort by high income
countries, no need to appeal to altruism.
Some results make provide food for further thought and discussion. The point about
the quantitative importance of the disagreement outcome is interesting. If the current statusquo point is given legitimacy, then an initially highly-protected, free-riding country is goingto do well in negotiations. If the current status quo point in not granted legitimacy then thebargaining outcome can be very different. For example, some rule that if bargaining fails
then the outlier country faces sanctions changes the disagreement outcome and therefore thebargaining outcome. That of course opens the pandora’s box of negotiating over what isgoing to be negotiated, something I’ll leave to better theorists than myself.

18
Appendix 1
First, the algebra on the CET transformation function, the national product function
and the supply functions. I derive the unit national product function: the revenue derived
from one unit of labor input into production.

(A1)

(A2)

(A3)

(A4)

(A5)

Inverting this gives unit supply functions: the optimal output of each good for one
unit of labor input into production.
(A6)

(A7)

(A8)

(A9)

This gives us the unit national product function, which could also be termed the
producer price index in an analogy to the CES consumer price index so widely used in19monopolistic competition and economic geography.
(A10)
The unit national product function or producer price index is also CET, a self-dual prorerty
familiar from the CES demand literature. Substitute this into the supply functions above.
(A11)
This gives us more compact unit and total supply functions.
(A12)
Second, the algebra for the terms-of-trade effect under the special assumptions notedin the text.
(A13)

(A14)

Assume that the countries are identical initially and that there are no initial abatement
taxes, so that
where
is the total endowment of each country. Second,
assume symmetry in the comparative advantage parameters such that
. Then (xx) can be simplified to(A15)

Becauseand= 1 initially, this further simplifies toor

The effect of country n withdrawing a unit of labor from production(A16)

is to reduce

the relative price of country s’s export good (country s has a deterioration in its terms of trade)with the size of this effect being proportional to the size of comparative advantage spread.

20
Appendix 2:
In this appendix, I briefly introduce the general idea of an MPEC problem, and show asimple example.
MPEC:

mathematical programming with equilibrium constraints
an MPEC is basically a:

NLP (non-linear programming):
constrained optimization where the constraint set is an
MCP (mixed complementarity problem):an economic equilibrium problem consisting of aa set of weak inequalities each with a matched non-negative variable.
Here is a simple example of an MCP: simple supply-demand problem with two
unknown, price (p) and quantity (x). First, solve the underlying producer and consumer
optimization problems (themselves MPECs) to getc’(x): the producer’s marginal cost function andd(p):the consumer’s demand function

The economic equilibrium problem should be specified as follow:
c’(x) $ p

matched to x (if c’ > p, then the good is not produced => x = 0)

x $ d(p)matched to p (if x > d(p), then x is a free good => p = 0)x, p $ 0
One technique is to add two non-negative “slack” variables to convert the first two
inequalities to four equalities. Let subscript m denote market and subscript p denote pricing
or profits. The four-equation system is:sm = x - d(p)sm: measure of excess supply in equilibrium

sp = c’(x) - p

sp: measure of unprofitability in equilibrium

smp = 0s px = 0

sp, sm, x, p $ 0

The importance of matching inequalities and variables can be seen by the fact that thissimple MCP has one of three possible solutions.
(1)

sm = sp = 0, x and p > 0

interior solution(2)

sm = 0, sp > 0, x = 0, p > 0technology too costly, not used(3)sm > 0, sp = 0, x > 0, p = 0

x is a free good

21
Figure 1A shows the three possible outcomes of the supply-demand problem. Case 2
comes up a lot in economic modeling. One example would be production of electricity, where
there are multiple technologies to produce electricity. Some technologies will be unprofitable
and will not be used in equilibrium. A second example occurs in international trade models,
where many trade linking are slack (not used), since they are unprofitable.
Case 3 is somewhat less common. It can occur in trade models, for example, when
there are fixed input coefficients in production (Leontief technologies). Then if one factor of
production is in sufficiently plentiful supply, its price will be zero in equilibrium.
Here is an example of an MPEC in our simple problem. Suppose that the government
wants to set a tax t in this market to maximize tax revenue. Let p be the consumer price of x,
so c’(x) = p/(1+t) is the producer price. and the producer price is the tax base.
MPEC:
maximize wrt to t:

taxrev = c’(x)tx

subject to the economic equilibrium problem (MCP):c’(x)(1+t) $ pzxx $ d(p)zpx, p $ 0
Suppose our good is electricity produced by solar cells, but they are unprofitable inequilibrium (case 2). What is the subsidy rate s, on marginal cost, that will get the industry to
produce x = 1 units of electricity with this technology?
MPEC
minimize wrt to s:

subsidy payments = c’(x)sx

subject to the economic equilibrium problem (MCP) and auxiliary constraint:c’(x)(1-s) $ pzxx $ d(p)

z

p

x $1
x, p $ 0
where x $ 1 is an additional unmatched constraint. If the technology is profitable without thesubsidy and x > 1 with no subsidy (case 1), this program will return the solution s = 0. If s is
specified as a free variable (may take a negative value) then, in this case, the program will
return an s < 0 (tax revenue) that achieves x = 1.

22
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23
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Maximum lead content of gasoline: grams per litre, 1996

Plot 1: Maximum lead content of gasoline (petrol) in relation to log
GDP per-capita at PPP: regression line: y =D+ Eln(x)
1.1

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

LEAD = 1.292 - 0.105*ln(GCAP)(37.103) t-stat

0.2

R2 = 0.246

0.1

0

-0.1
4

5

6

7

8

ln GDP per-capita at PPP exchange rates, 1996

9

10

11

TABLE 1:

Welfare and abatement taxes under alternative secnarios
Base case: n's per-capita
income 1.5 that of country s

Alternative case 1: nocomparative advantage

Alternative case 2: n's
per-capita income
9 times that of s

Welfare n

Welfare s

Welfare n

Welfare s

Welfare n

Welfare s

PTAX = 0

1.000

1.000

1.000

1.000

1.000

1.000

Non-cooperative NE

1.207

1.196

1.136

1.120

1.161

1.012

Coop NashNE outside option

1.237

1.217

1.197

1.255

1.161

1.012

CD world welfare
index max*

1.290

1.173

1.290

1.173

1.267

0.944

Rawlsian world
welfare max**

1.225

1.225

1.237

1.228

1.160

1.012

PTAXN

PTAXS

PTAXN

PTAXS

PTAXN

PTAXS

Non-cooperative NE

0.32

0.08

0.24

0.03

0.41

0

Coop NashNE outside option

0.51

0.21

0.51

0.22

0.41

0

CD world welfare
index max*

0.35

0.35

0.35

0.35

0.22

0.22

Rawlsian world
welfare max**

0.57

0.20

0.48

0.28

0.45

0

TAX RATES*Cobb-Douglas index used is (WELN)^0.5*(WELFS)^0.5; welfare n,s normalized to 1 at zero taxes**Rawlsian welfare index is MIN(WELN, WELS); welfare n,s normalized to 1 at zero taxes

TABLE 2:

Welfare and abatement taxes under alternative linking secnarios
Country n has five times the total income and per-capita income of country s
Welfare n

Welfare s

PTAXN

PTAXS

TARN

TARS

No intervention

0.731

0.788

0

0

0

0

Non-cooperative Nasheach county coordinates tax and tariff

1.000

1.000

0.386

0

1.567

0.196

Isolated environment negotiation

1.000

1.000

0.386

0

1.567

0.196

Isolated tariff negotiation

1.008

1.048

0.386

0

1.015

0

Bargain over TARN and PTAXS

1.005

1.025

0.386

0.150

0.692

0.196

Bargain over all four instuments

1.009

1.059

0.452

0.034

1.015

0

Bargain over all four instruments
1.024with transfer s to n ( = 22.0% of s's income)

1.075

0.541

0

0

0

Cooperative Nash usin non-cooperative outcome
as disagreement outcome

Cooperative Nash using no intervention
as disagreement outcome( now normalized to nointervention = 1)

Isolated environment negotiation

0.945

1.306

0.631

0

0

0

Isolated tariff negotiation

0.731

0.788

0

0

0

0

Bargain over TARN and PTAXS

0.785

0.792

0

0.126

0

0

Bargain over all four instuments

0.943

1.310

0.628

0

0

0.048

Bargain over all four instruments
0.935
with transfer s to n ( = -2.3% of s's income)

1.333

0.621

0

0

0

Figure 1a: Effect of abatement by country n on the commodity consumptionof both countries: no comparative advantage

X2

Initial production frontiers ofboth n and s: no abatement

Composite commodity
consumption of s (and ninitially)

Production frontier of country nafter resources used forabatement
p1/p2

p1/p2

Composite commodity
consumption of n after
abatement

X1

Figure 1b: Effect of abatement by country n on the commodity consumptionof both countries: complete specialization

X2

Production point of country s
Composite commodity
consumption of s and n initiallyafter abatement by nComposite commodity
consumption of s and n initiallyp1/p2 initially
Production pointof country n after
abatement

p1/p2 after
abatement

Initial production point ofcountry n before abatement

X1

Figure 2: Welfare, environmental quality
1.75

13

9

1.5

5

1.25

Environmental quality index

Per-capita welfare level

per-capita welfare: left-hand axis

enviromental quality: right-hand axis
1

1

-3

0.75
0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Per-capita income level

Cooperative Nash bargaining:n higher per-capita
Nash eq. default option
n's optimal tax at s's tax = 0
n higher per-capita
045
Non-cooperative Nash equilibrium:n higher per-capita

n's optimal tax
at s's tax = 0
equal per-capita

Cooperative Nash bargaining:equal per-capita income

Non-cooperativeNash equilibrium:
equal per-capita income

s's optimal tax at n's tax = 0
equal per-capita

0.40

0.39

0.38

0.37

0.36

0.35

0.34

0.33

0.32

0.31

0.30

0.29

0.28

0.27

0.26

0.25

0.24

0.23

0.22

0.21

0.20

0.19

0.18

0.17

0.16

0.15

0.14

0.13

0.12

0.11

0.10

0.09

0.08

0.07

0.06

0.05

0.04

0.03

0.02

s's optimal tax at n's tax = 0
n higher per-capita
0.01

0.60
0.59
0.58
0.57
0.56
0.55
0.54
0.53
0.52
0.51
0.50
0.49
0.48
0.47
0.46
0.45
0.44
0.43
0.42
0.41
0.40
0.39
0.38
0.37
0.36
0.35
0.34
0.33
0.32
0.31
0.30
0.29
0.28
0.27
0.26
0.25
0.24
0.23
0.22
0.21
0.20
0.19
0.18
0.17
0.16
0.15
0.14
0.13
0.12
0.11
0.10
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0.00

0.00

Country h's tax

Figure 3: Effect of differing per capita income on equilibria

Country
f's tax

Figure 4: Infeasibility of an equal tax bargaining rule (ptaxn = ptaxs) whencountries differ greatly in per capita income: PCI = 10*PCI
160

150

Welfare of countries n and s

140

Country n's welfare
130

120

Country s's welfare: PCIN = 5*PCIS
110

100

90

Country s's welfare: PCIN = 10*PCIS
80
0

0.1

0.2

0.3

Equal taxes: ptax = ptax

0.4

0.5

0.6

(countries sizes identical)

Figure 5: Gains from cooperative linked bargaining
lower tarn traded for higher ptaxs

1.05

Nash eq:ptaxn: 0.39
ptaxs: 0.00
tarn: 1.57
tars: 0.20

0.39

0.33

0.3

0.27

0.24

0.21

0.18

0.15

0.12

0.09

0.06

0.03

0
0

0.2

0.1

0.6

0.4

0.3

Country n's tariff

0.5

0.8

0.7

1

Non-cooperative
Nash eq.

0.9

2
1.9
1.8
1.7
1.6
1.5
1.4
1.3
1.2
1.1

0

0.36

0.525

Country n's ptax

Nashbargain:ptaxn: 0.39
ptaxs: 0.15
tarn: 0.70
tars: 0.20

Figure 6: Illustration of "policy leakage": country s makes abest response to country n's (exogenous) tax
1.6

Abatement by countries n and s

1.6

1.4

1.4

Country n's abatement
Country s's abatement: best
response tax to n's tax

1

1

Marginal leakage rate

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0
0

0.1

0.2

0.3

0.4

0.5

0.6

Country n's tax rate

Figure A1: Three outcomes of the mcp exampleppSupply (MC):slope = B

Supplysp
Demand:slope = 1/D

pDemandxxCase 2: x is too expensive, notproduce in equilibrium (x = 0)Case 1: interior solutionpSupplyDemandCase 3: excess supply,
x is a free good in equilibrium(p = 0)x
smx

Leakage rate

1.2

1.2

C:\jim\RESEARCH\per-cap-income\environment paper\link-mpec-app.gms Tuesday, September 18, Page
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1$TITLE:LINK-MPEC.GMS

cooperation over all four policy instruments$ONTEXT INITIAL CALIBRATION (countries h (n in paper) f (s in paper))Rows = markets;
row sums zero: zero profits
Columns = sectors; column sums zero: market clearing, income balance
ZH
ZF
PXH
180
PYH
20
PXF
20
PYF
180
PWH
PWF
PLH
-200
PLF
-200
PENVH
PENVF

XHF
-80

XFF

YFF

YFH
80

WH
-100
-100

80

WF

CONSH

CONSF

-100
-100

-80
300

-300
300

-300
200
200

-100

100
-100

100

100 units of initial E endowment reduced by production (not shown)$OFFTEXTPARAMETERS
VALH
per-capita income paramater for the S-G preferences - h
VALF
per-capita income paramater for the S-G preferences - f
SIGMA
elasticity of transformation between X and Y
ENDOWLH factor endowment of country h
ENDOWLF factor endowment of country f
WELHN
disagreement outcome for country h in bargaining
WELFN
disagreement outcome for country f in bargaining;
VALH = 1;
VALF = 1;
SIGMA = 1;
ENDOWLH = 200; ENDOWLF = 200;
WELHN = 1;
WELFN = 1;
VARIABLES
WELH
welfare of country h
WELF
welfare of country f
WELG
value of Nash bargaining objective function;
POSITIVE VARIABLES
ZH
activity for composite X-Y commodity output in h
ZF
activity for composite X-Y commodity output in f
XHF
export activity: X from h to f
YHF
export activity: Y from h to f
XFH
export activity: X from f to h
YFHexport activity: Y from f to h
welfare index for h (activity for "production" of WH)
WH
WF
welfare index for h (activity for "production" of WF)
ABH
abatement activity for h
ABF
abatement activity for f
PZH
PZF
PXH
PXF
PYH
PYF
PLH

price of Z (composite X-Y) in hprice of Z (composite X-Y) in hprice of X in hprice of X in f
price of Y in hprice of Y in f
price of composite factor L in h

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PLF
PWH
PWF

price of composite factor L in f
price of welfare in h (real consumer price index)
price of welfare in f (real consumer price index)

PABH
PABF
PENVH
PENVF

price (cost) of the abatement activity in hprice (cost) of the abatement activity in f
consumer valuation of environmental quality in h
consumer valuation of environmental quality in h

CONSH
CONSF

consumer income in h
consumer income in f

POLRED
POL

pollution reduction due to combined abatement in h and fpollution generated by production of ZH and ZF

PTAXH
PTAXF
TARH
TARF

pollution tax in h - receipts allocated to abatement in hpollution tax in f - receipts allocated to abatement in f
tariff in country htariff in country f;

EQUATIONSWELFAREH welfare of country h
WELFAREF welfare of country f
WELFAREG Nash bargaining objective function
COSTZH
COSTZF
COSTXHF
COSTYHF
COSTXFH
COSTYFH

pricing equation for ZH (cost =G= price) - comp var ZH
pricing equation for ZF (cost =G= price) - comp var ZF
pricing equation for XHF (cost =G= price) - comp var XHF
pricing equation for YHF (cost =G= price) - comp var YHF
pricing equation for XFH (cost =G= price) - comp var XFH
pricing equation for YFH (cost =G= price) - comp var YFHCOSTWHCOSTWF

pricing equation for WH
pricing equation for WH

COSTABH
COSTABF

pricing equation for ABH (cost =G= price) - comp var ABH
pricing equation for ABF (cost =G= price) - comp var ABF

MKTPZHMKTPZFMKTPXHMKTPYHMKTPXF
MKTPYF

market clearing equation for ZH - complementary variable PZH
market clearing equation for ZF - complementary variable PZF
market clearing equation for XH - complementary variable PXH
market clearing equation for YH - complementary variable PYH
market clearing equation for XF - complementary variable PXF
market clearing equation for YF - complementary variable PYF

MKTPLHMKTPLF
MKTPWHMKTPWF

market clearing equation for LH - complementary variable PLH
market clearing equation for LF - complementary variable PLF
market clearing equation for WH - complementary variable PWH
market clearing equation for WF - complementary variable PWF

MKTPABH
MKTPABF
MKTENVHMKTENVFmkt clear for abatement in h: taxes = expenditure - comp PABH
mkt clear for abatement in F: taxes = expenditure - comp PABF
mkt clear environment in h: supply = demand- comp PENVH
mkt clear environment in f: supply = demand- comp PENVFICONSHICONSF

income balance equation in h - complementary variable ICONSH
income balance equation in f - complementary variable ICONSF

APOLRED
APOL

pollution reduction = abatement by h and f - comp var POLRED
pollution generated by production by h and f - comp var POL;

(cost =G= price) - comp var WH(cost =G= price) - comp var WF

C:\jim\RESEARCH\per-cap-income\environment paper\link-mpec-app.gms Tuesday, September 18, Page
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3*specify equations and inequalities
WELFAREH.. WELH =E= WH;
WELFAREF.. WELF =E= WF;
WELFAREG.. WELG =E= ((WH - WELHN)*(WF - WELFN));
COSTZH.. 200*PLH =G= 200*PZH;COSTZF.. 200*PLF =G= 200*PZF;COSTXHF.. 80*PXH*(1+TARF) =G= 80*PXF;COSTYHF.. 80*PYH =G= (80*0.999)*PYF;COSTXFH.. 80*PXF =G= (80*0.999)*PXH;COSTYFH.. 80*PYF*(1+TARH) =G= 80*PYH;COSTWH..
COSTWF..

300*((PXH*(1+PTAXH))**(1/3) * (PYH*(1+PTAXH))**(1/3)* PENVH**(1/3)) =G= 300*PWH;
300*((PXF*(1+PTAXF))**(1/3) * (PYF*(1+PTAXF))**(1/3)* PENVF**(1/3)) =G= 300*PWF;COSTABH.. 100*PLH =G= 100*PABH;COSTABF.. 100*PLF =G= 100*PABF;MKTPZH..
MKTPZF..
MKTPXH..
MKTPYH..
MKTPXF..
MKTPYF..

MKTPLH..
MKTPLF..
MKTPWH..
MKTPWF..

200*PZH =E= 200*(0.9*PXH**(1+SIGMA)+ 0.1*PYH**(1+SIGMA))**(1/(1+SIGMA));
200*PZF =E= 200*(0.1*PXF**(1+SIGMA)+ 0.9*PYF**(1+SIGMA))**(1/(1+SIGMA));
0.9*200*ZH*(PXH/PZH)**SIGMA - 80*XHF + 80*XFH=G= 100*WH*PWH/(PXH*(1+PTAXH));
0.1*200*ZH*(PYH/PZH)**SIGMA - 80*YHF + 80*YFH=G= 100*WH*PWH/(PYH*(1+PTAXH));
0.1*200*ZF*(PXF/PZF)**SIGMA - 80*XFH + 80*XHF=G= 100*WF*PWF/(PXF*(1+PTAXF));
0.9*200*ZF*(PYF/PZF)**SIGMA - 80*YFH + 80*YHF=G= 100*WF*PWF/(PYF*(1+PTAXF));ENDOWLH =G= 100*ABH + 200*ZH;
ENDOWLF =G= 100*ABF + 200*ZF;
300*WH =G= CONSH/PWH;
300*WF =G= CONSF/PWF;

MKTPABH.. 100*PABH*ABH =G= PTAXH*200*WH*PWH/(1+PTAXH);
MKTPABF.. 100*PABF*ABF =G= PTAXF*200*WF*PWF/(1+PTAXF);
MKTENVH.. (100+100*VALH) + 100*POLRED - 100*POL =G= 100*WH*(PWH/PENVH);
MKTENVF.. (100+100*VALF) + 100*POLRED - 100*POL =G= 100*WF*(PWF/PENVF);
ICONSH..
ICONSF..

CONSH =E= ENDOWLH*PLH +((100+100*VALH)+100*POLRED-100*POL)*PENVH+ 80*YFH*PYF*TARH;CONSF =E= ENDOWLF*PLF +((100+100*VALF)+100*POLRED-100*POL)*PENVF+ 80*XHF*PXH*TARF;

APOLRED.. POLRED =G= 2*ABH + 2*ABF;
APOL..
POL =G= (ZH + ZF)/2;*declare model definition, including matching equations and variables*in the economics equilibrium constraint set (general-equilibrium model)
MODEL NHD /WELFAREGWELFAREH
WELFAREF
COSTZH.ZH
COSTZF.ZF

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COSTXHF.XHF
COSTYHF.YHF
COSTXFH.XFHCOSTYFH.YFHCOSTWH.WH
COSTWF.WF
COSTABH.ABH
COSTABF.ABF
MKTPZH.PZHMKTPZF.PZF
MKTPXH.PXH
MKTPYH.PYH
MKTPXF.PXF
MKTPYF.PYF
MKTPLH.PLH
MKTPLF.PLF
MKTPWH.PWH
MKTPWF.PWF
MKTPABH.PABH
MKTPABF.PABFMKTENVH.PENVHMKTENVF.PENVFICONSH.CONSHICONSF.CONSF
APOLRED.POLRED
APOL.POL /;*set starting values
ZH.L = 1; ZF.L = 1; XHF.L = 1; YHF.L = 0; XFH.L = 0; YFH.L = 1;
WH.L = 1; WF.L = 1; ABH.L = 1; ABF.L = 1;
PZH.L = 1; PZF.L = 1; PXH.L = 1; PXF.L = 1; PYH.L = 1; PYF.L = 1;
PLH.L = 1; PLF.L = 1; PWH.L = 1; PWF.L = 1;
PABH.L = 1; PABF.L = 1; PENVH.L = 1; PENVF.L = 1;
CONSH.L = 300; CONSF.L = 300; POLRED.L = 0; POL.L = 1;*choose numeraire
PXH.FX = 1;
VALH = 1;
VALF = 1;
ENDOWLH = 400;
ENDOWLF = 80;* simplist scenario: outside options are zero
* equivalanet to maximzing Cobb-Douglas world welfare function
WELHN = 0;
WELFN = 0;
OPTION MPEC = NLPEC;* solve command specifying maximand
SOLVE NHD USING MPEC MAXIMIZING WELG;