Taxes and Resource Prices: A North-South Game

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NOT FOR QUOTATION WITHOUT PERMISSlON OF THE AUTHOR

TAXES AND

RESOURCE PRICES:

A NORTHMUTH GAME

Graciela Chichilnisky

December 1983 WP-83-124

Working h p e r s are interim reports on work of the International Institute for Applied Systems Analysis and have received only limited review. Views or opinions expressed herein do not necessarily represent those of the Institute or of its National Member Organiza- tions.

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS 2361 Laxenburg, Austria

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This is one of t h r e e papers derived from research on North-South resource trade performed in t h e System and Decision Sciences Area during the s u m m e r of 1982. The aim of this research was, first, t o construct a model of N o r t h C o u t h trade, and t h e n t o use it as a framework for further work in gaming, negotiations and interactive decision making.

In t h i s paper, two models already developed by t h e author a r e modified and combined t o produce a simple model of N o r t h C o u t h trade. -This model is then reformulated in such a way t h a t it can be used as t h e basis of an experimental game for t h e development of resource pricing policies.

ANDRZE J WIERZBICKI Chairman

System and Decision Sciences

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The purpose of this note is to construct and analyze a model of North- South t r a d e in resources, which could then be used as the basis of an experimental game for the formulation of resource pricing policies. The South trades oil for industrial goods with t h e North. Each region has one policy parameter: the North, a tax on oil, and the South, the price of oil. Each region h a s two economic objectives: the North, higher output and higher real wages, a n d t h e South, higher revenues from exports and higher output levels.

The policy instruments are used t o define strategies: the payoffs of these strategies are given by t h e general equilibrium solutions of t h e model.

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TAXES

AND

RESOURCE

PRICES:

A NORTHSOUTH GAME

Graciela Chichilnisky*

1. IrnODUCIION

The economic model of N o r t h S o u t h trade presented here is a modifica- tion, and synthesis, of two models already described by Chichilnisky (1983a,b).

The model is given in t h e form most suitable for experiments on North-South trade policies, following an approach developed by Fortuna (1983) and Wierzbicki (1983).

We consider a two-region model, in which one region (the South) exports oil t o t h e other (the North) in exchange for industrial goods. Each region has two outputs, basic goods and industrial goods. Each good is produced using t h r e e inputs: capital, labor and oil.

This model differs from t h a t presented in Chichilnisky (1983a) in t h a t here we have a full macroeconomic description of both regions, rather than of

'Professor of Economics, Columbia University. I thank 2. Fortuna and A. Wierzbicki for helpful comments and suggestions. This research ^wassupported by the System and Decision Sciences Area at USA.

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one region only. This model also differs from t h e two-region model given in Chichilnisky (1983b) in t h a t (i) t h e policy variable for the South is h e r e t h e price of oil, r a t h e r t h a n t h e level of oil exports, and (ii) we also consider a policy variable for t h e North, namely a tax on oil.

Finally, we p r e s e n t t h e two-region mociel in a so-called "reduced form", i.e., a s one equation t h a t can be used t o solve t h e whole model; this is t h e most suitable form for experimental work on N o r t h S o u t h t r a d e policies.

2.

THE

_MODELFOR THE NORl'H: TAX POLICIES

The economy of t h e North is described by six behavioral equations, and seven equilibrium conditions. Considering first t h e behavioral equations, we have

~ ~ = m i n ( ~ ~ / a ~ , d ~ / b ~ . ~ ~ / c ~ ) , (1)

where B is t h e basic good. The superscript S indicates supply,

L B

i s labor employed in producing B, zpB a n d

K~

y e respectively t h e oil and capital employed in producing B, and al,b ],c a r e positive real numbers. This is a typical production function with fixed factor proportions.

Similarly, for t h e industrial good I, t h e production function is

while for oil I9 we have

* s = o

,

i.e., t h e North produces n o oil (we could equally well put dS

=

C, ^whereC ^{is a n} a r b i t r a r y constant). Labor supply is positively linked t o real wages:

where L' r e p r e s e n t s labor supply, w wages, a n d % t h e price of B. Similarly,

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where KS represents capital supply and T the r a t e of r e t u r n on capital. We postulate t h e following demand behavior a t equilibrium:

where the superscript

D

denotes demand, i.e., wage income is spent on

B,

so t h a t B is a "wage g o o d .

The equilibrium conditions are:

KS

=

K D ,

BD

=

BS ,

i.e., B is n o t traded internationally;

q ' =

I S - I D ¹

where ~ F d e n o t e s exports of I;

xg

^{= g D}^{= z p D}^-3' ^,

where X$ denotes oil imports;

b , ~ ~ + b ~ I ' = d ~ ,

PIX?=

psx: .

⁽¹³⁾

which i s a balance of payments condition, where pI a n d p d denote the price of industrial goods and of oil, respectively.

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The national income identity (or Walras' law) is always satisfied a t equilibrium:

which (at equilibrium) is equal t o

The model of the northern economy therefore consists of 13 equations in

s

^D

15 variables: I , I ,

x?,

l p S , l p D ,

xi,

BS. B D , r , PB. pl. p d , W , L , and K. It follows t h a t for a given price of oil, p,,, the model is "closed" in relative prices.

Since the price of oil is given, there is thus no room for policy in t h e model as formulated above.

We now introduce a new parameter, a tax on oil, into the model of t h e North. The taz ate is denoted by the variable t . The above 13 equations must now be modified t o take this into account.

First note t h a t the production functions (1) and (2) are equivalent to t h e following price equations (assuming competitive behavior), before trrz:

From now on we shall use price equations (14) and (15) instead of production functions (1) a n d (2). After taz, these price equations become:

The total tax revenue is thus t g D . If a fraction A, 0

<

A

<

1, of this revenue is allocated to wage income, and the complement (1 - A ) t o capital income, equation (6) becomes:

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and Yialrase law now becomes ( a t equilibrium):

so t h a t a t equilibrium we have

The tax on oil having been introduced, t h e northern economy is now described by a system of 13 equations in 16 variables, where t h e original equations (14), (15) and (6) a r e replaced by their equivalents under tax, (14'), (15') and ( 8 ' ) . Since t h e tax r a t e represents a new variable, t h e equilibrium posi- tions of t h i s model now depend o n two p a r a m e t e r s r a t h e r than one working in relative prices. We take t h e s e parameters t o be t h e price of oil pd, which is the policy variable of t h e South. a n d t h e tax r a t e t , which is the policy variable of t h e North. In t h e next section we present a model of the southern economy, and show t h a t given t h e tax r a t e t a n d t h e price of oil pd, t h e r e exists a (locally) unique equilibrium for the N o r t h S o u t h trade model. This m e a n s t h a t when t h e North chooses its policy p a r a m e t e r t and t h e South its policy p a r a m e t e r p4, t h e equilibria of both regions, a n d of t h e world economy, a r e determined. In particular, t h e output of goods in both regions, t h e N o r t h S o u t h t e r m s of trade p4/pl, t h e volume of industrial goods exported by t h e North, and the real wages and r a t e s of profits in both regions a r e all determined by t h e choice of t and p4. Therefore, our N o r t h S o u t h model can actu- ally be interpreted a s a game l o r n

G,

i.e., a m a p from strategies to outcomes:

N o r t h S o u t h Outcomes strategies

( t ,pd>

G

+ N o r t h C o u t h trade equilibrium

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The game form G is a map t h a t assigns an equilibrium t o each pair of strategies t and p+. If in addition we define t h e objectives of t h e two regions, i.e., t h e s e t of variables t h a t each region wishes t o maximize, we would then have obtained a well-defined game from our model.

The model of t h e s o u t h e r n economy i s defined in t h e n e x t section, we then give t h e "reduced form" of t h e N o r t h S o u t h model, and t h e g a m e G is described in more detail in t h e final section.

3.

THE

MODEL FUR

THE SOUTH:

OIL PRICING POLICIES

The economy of t h e South is described by five behavioral equations and seven equilibrium conditions. The behavioral equations a r e similar to those of t h e North (except t h a t t h e parameters a], a2, b l , b 2 , c l , c 2 ,

a

_and/3 m a y be different):

where t h e o u t p u t of oil is n o t constrained t o zero; t h e r e is only a total endow- m e n t condition $S

r s,

which in some cases will n o t be binding. The output of oil i s therefore "passive", adjusting t o demand. We a s s u m e t h a t extraction of oil is cost-free, i.e., i t u s e s no factors of production.

The equilibrium conditions are, as in t h e North:

K S = KD

= C , B ~ + c2ls

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x f ' = r D - ~ ~

^,

where Xf' denotes imports of industrial goods;

% = a s - @

^,

where X$ denotes exports of oil;

z p D = b 2 l S + b 1 ~ S ,

which is t h e balance of payments condition.

The national income identity is always satisfied a t equilibrium:

p B ~ D

+ =

w~~

+

7 S D I ~ S

+

^{p d g S}

.

Equivalently, (30) can be written

where d D denotes the amount of oil employed in the South. Equations (30) a n d (31) a r e identical a t equilibrium, because of t h e balance of payments condition. The economy of the South is therefore represented by a system of 12

D S equations ((18) to (29)) in 15 variables, f S , f D ,

xP,

^I9', ^I9 ^,

^A&,

BS, BD.

T . p ~ , P I . p d , w , L and

K.

The equilibria of this model once again form a two-parameter family, assuming relative prices. One of these parameters is taken t o be the price of oil p d . In t h e next section we show that, when the equilibrium conditions for international trade are taken into consideration, t h e equilibrium of the N o r t h S o u t h model is fully determined u p t o two

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parameters, t h e price of o i l p * and the taz rate t in t h e North.

4. THE NORTHSOUTH MODEL

, Here we consider simultaneously t h e 1 3 equations for t h e North [ ( 3 ) - ( 5 ) , (69, (7)-(13), (14') and (15')], and t h e corresponding 12 equations for t h e South [price equations (14) and (15), and (20)-(29)]. (Recall t h a t t h e South does n o t have taxes and t h a t flS # 0 in t h e South.) We also require a s e t of world equilibrium conditions, which a r e given below.

The international market-clearing conditions for an equilibrium a r e (i) t h a t oil exports must equal oil imports:

(where the S a n d N in parentheses indicate t h e South a n d t h e North, respectively); and (ii) t h a t exports of industrial goods m u s t equal imports of industrial goods:

The following two equations s t a t e t h a t t h e prices of internationally traded goods m u s t be equal a t equilibrium:

We may now determine t h e n u m b e r of equations in t h e North--South model. First note t h a t t h e balance of payments conditions of t h e two regions (13) and (29) a r e equivalent, from (32) t o (35). We therefore have 12 equations for t h e North and 12 for the South, plus t h e four international trade conditions (32) t o (35), making 28 equations altogether.

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T h e r e a r e 16 variables in t h e North, including t h e t a x on oil, a n d 1 5 in t h e South. We t h u s have 31 variables and 28 independent equations. The North- South equilibria can therefore be d e t e r m i n e d in relative prices if we m a k e two p a r a m e t e r s exogenous; we choose t h e s e t o be t h e two policy variables, t h e oil t a x r a t e t for t h e North a n d t h e price of oil pd for t h e South.

The n e x t s e c t i o n shows this explicitly, producing one equation which gives a "reduced form" solution t o t h e model.

5. SOLVING

THE

NORTH--SOUTH MODEL: A mDUCED FDRM

In o r d e r t o solve t h e model in relative prices, we choose B in t h e North a s t h e ' n u m e r a i r e ' , i.e., pB(N)

=

1. F r o m now on, all p r i c e s a r e given relative t o t h e price of B in t h e North.

The production functions (1) a n d (2) yield equations for t h e d e m a n d for f a c t o r s K, L a n d IJ. At e a c h level of o u t p u t , when f a c t o r s a r e u s e d efficiently, we h a v e

L~ = ~t ISaz ~ a ~ (36)

Equations (36) a n d (37) imply

where D i s a l c

-

^a2c^I.

The (taxed) price equations for t h e North (14') a n d (15') c a n be regarded as a s y s t e m of two equations in two variables (taking p,, a s a constant), yield- ing

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Then, substituting L and K from (4) and (5), and w and ^Tfrom (41) and (42) into (39) we obtain an equation relating the supply of basic goods B t o the price of industrial g o o d s p ~ and the price of o i l p d , i.e.,

where

M =

^{a 1 b 2}

-

^{a 2 b 1}and

N =

^{c 1 b 2}

-

b 1 c 2 . Similarly, we obtain

- ^L] ⁺

- c 2 - m d ( 1

+

t ) ] ^a (44) PI

Consider now the demand relation

Since BS

= B ~ ,

we obtain, using (41), (42), (44) and (12)

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Equation ( 4 7 ) above is the r e d u c e d f o r m of the N o r t h S o u t h model, link- ing t h e price of oil p d and t h e tax r a t e t with t h e price of industrial goods a t equilibrium, p;. I t is now a simple m a t t e r t o check t h a t when p,, and t a r e known, a n d therefore an equilibrium value of p i can be obtained from ( 4 7 ) , t h e N o r t h S o u t h model is "closed", i.e., t h e equilibrium values of all other variables c a n be computed. We s t a r t with t h e North.

Given pd a n d t . we obtain t h e equilibrium price p; from ( 4 7 ) . F'rom (4'1) and ( 4 2 ) we obtain the equilibrium wage level w and profit r a t e r

'.

We can t h e n obtain t h e equilibrium capital supply K ~ * and labor supply L ~ * from ( 4 ) a n d (5). Equations (39) and ( 4 0 ) yield and lS*, a n d t h u s we obtain

@* =

A$*(N). Since

**p f i ~ i * ( ~ ) ⁼ p ; ~ r ( ~ ) ,**

i t follows t h a t

XY

can be found, so t h a t lD*

=

lS*

**- q S * ( N )**

can also be found. The model for t h e North is t h u s closed.

We show next t h a t t h e model for t h e South is also closed. Consider t h e two price equations for t h e South:

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or equivalently,

and

These equations lead t o

Note t h a t pd is an exogenously given policy variable, and pi is given by ( 4 7 ) as the equilibrium solution corresponding to p4. P;

=

P:(pd). Therefore (49) yields a relation connecting w / p B and r . There is an implicit assumption here t h a t the price of industrial goods is determined both by the price of oil (the policy parameter of the South) and by t h e market activity of the North.

Next, consider t h e equilibrium condition BD

=

B S . From BD

=

w L / p B and g S

=

( c 2 L

-

a z K ) / D , since L

=

a w / p B and K

=

/3r, we obtain

which is another relation between w / p B and r . Equating ( 4 9 ) a n d ( 5 0 ) , we obtain the equilibrium values of real wages ( w / P B ) ' and the r a t e of profit r in the South. From ( w / p B ) * and r

'

we obtain L" and K", a n d hence B'* and

**rS*.**

^using ^{( 3 9 )}^{a n d}( 4 0 ) . Finally, since

x ~ ' ( s ) =

X?*(N) a t equilibrium, we obtain t h e equilibrium demand for I in t h e South,

**rD*(s),**

which equals

I'*(s)

+ @'(s).

This completes our calculation of t h e equilibrium for the South.

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6. THE NORTHSOUTH MODEL AS A GAME

We have previously mentioned t h a t the model can be used t o c o n s t r u c t a g a m e f o m , which is t h e generalization of a payoff matrix,

G : ( t ,p+) Equilibrium outcomes:

pd* ,pI*, r (N)* , r(S)* , Afs*,...

In order to fully define a game, we now only have t o specify t h e objectives which t h e two players (the North and t h e South) pursue when choosing their strategies.

We assume t h a t t h e North wishes to optimize two variables, the total v a l u e o j n e t o u t p u t

and the real w a g e

The South wishes t o optimize two variables, total n e t o u t p u t

and the l e v e l of indwtq-ial goods i m p o r t e d in ezchange f o r oil

xpcs, .

In a traditional game-theoretical approach, we would define preferences over t h e outcomes, and subsequently adopt an equilibrium concept t o determine t h e solutions of this game. The approach pioneered by Fortuna (1983) a n d Wierzbicki (1983) is different, since i t is n o t necessary t o have well- defined preferences over all possible outcomes in order t o find a (Pareto) solution.

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I t would also be of i n t e r e s t t o resolve a n u m b e r of questions r e l a t e d t o m a r k e t behavior, s u c h as:

-

What i s t h e r a n g e of oil prices for which factor p r i c e s (profits r a n d wages w ) a r e positive in both regions?

-

How do r e a l wages w / p B in t h e South depend on t h e price of oil pd?

A n u m e r i c a l solution of t h e model would provide approximate answers t o t h e s e questions.

7 . CHANGES IN TECHN0LM;IES

We shall close with a discussion of t h e possible i m p a c t of t a x policies on t h e technologies used in t h e North: t h e model of t h e North m a y b e a l t e r e d t o include t h e i m p a c t of t h e oil t a x on t h e s e technologies u n d e r a n u m b e r of d i f - f e r e n t assumptions, s u c h a s "learning by doing", o r m o r e d i r e c t g o v e r n m e n t intervention.

For i n s t a n c e , consider t h e c a s e where h

=

^0,s o t h a t all t h e proceeds f r o m t h e oil t a x a r e t r a n s f e r r e d t o capital i n c o m e (increasing t h e d e m a n d for indus- t r i a l goods) u n d e r t h e c o n d i t i o n t h a t oil/output coefficients 1/ b a n d 1/ b 2 be i n c r e a s e d (i.e., i n v e s t m e n t s h o u l d be i n technologies which are' l e s s oil- intensive). In t h i s c a s e we m a y postulate* t h a t t h e inverses of t h e oil/output coefficients b l , b 2 d e c r e a s e a s t h e t a x r e v e n u e allocated t o t h e i m p r o v e m e n t of technology in e a c h s e c t o r i n c r e a s e s , i.e.,

*This formulation was suggested by Wierzbicki.

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Equations (51) and (52) now7 become part of t h e model of t h e North. Since t h e r e a r e two m o r e variables

(Kl

and Gz) and two more equations, t h e solutions have t h e s a m e characteristics as before.

However, i t should be pointed out t h a t some of t h e computations made in t h e previous sections no longer apply, and in particular t h a t t h e reduced-form equation must be modified to include

b",

and

K2

as variables depending on

t

and gD.

This completes t h e forhulation of t h e N o r t h S o u t h trade game. The next s t e p is t o implement t h i s game experimentally, in an a t t e m p t t o discover resource pricing policies t h a t a r e consistent with t h e goals, and t h e strategic behavior, of both North and South.

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REFERENCES

Chichilnisky, G. (1983a) "Oil Prices, Industrial Prices and Outputs: A General Equilibrium Macro Analysis". Working Paper, lnternational Institute for Applied Systems Analysis (forthcoming).

Chichilnisky, G. (1983b) "Resources and North--South Trade: A Macro Analysis in Open Economies". Working Paper, International Institute for Applied Systems Analysis (forthcoming).

Fortuna, Z. (1983) Working Paper, International Institute for Applied Systems Analysis (forthcoming).

Wierzbicki, k (1983) Working Paper, International Institute for Applied Sys- t e m s Analysis (forthcoming).