Common and private ownership of exhaustible resources:theoretical implications for economic growth

PET10 – 11th Annual Conference of the Association for Public Economic Theory, Istanbul, Turkey, June 25-27, 2010

European University at St. Petersburg, RussiaSt. Petersburg Institute for Economics & Mathematics RAS

Kirill Borissov & Alexander Surkov

Financial support of Exxon Mobil Corp. is acknowledged.

2

Outlines• Consumers are heterogeneous in their inter-

temporal preferences– Their discount factors are different

• Model 1: private ownership of resource stocks– Owners invest their savings in resource stocks

and earn a rent• Model 2: collective ownership of resource

stock– The resource rent is equally divided among all

the consumers – Resource utilization rate is chosen by voting

3

Model 1: private resources• Consumers

( ), , 11i t i t t t tS r S WC −+ ≤ + ⋅ +

( ) 1 2 1 211 ttt tY K L Eα α α αλ − −= + ⋅ ⋅ ⋅

• Production

( ),

,0

maxi t

ti t ct

i u Cβ∞

=

⋅ →∑

• Resources

( )1 1tt t t tER R Rρ+ = − = −

4

Equilibrium path (1)

• Consumers maximize utility• Capital and labor are paid their

marginal product• Resource price is equal to its marginal

product

( )* * * * * * * *, , 1, , 0,1,

, ,1 , , , , ,t t t t t t i t i t i L tK R r W P E C S

= … = …+

( ),0 ,0 1, ,,i i i L

K R= …

starting from the initial state

5

Equilibrium path (2)

• Consumer savings are equal to investments

* * *1t t tR R E+ = −

* * * *, 1 1

1

L

i t t t ti

S P R K+ +=

= ⋅ +∑• Natural balance of exhaustible resources is

fulfilled

• Hotelling’s rule holds true

( )* * *1 11t t tP r P+ += +

6

Steady-state equilibrium

( )* * ** * * * * *

1 ,

*

,, , , ,1 , , , , ,i i i L

K R r W P E C Sργ= …

+

( )( ) ( )( ) ( )( ) ( )

* * *

* *

*

* * * * *

* * * *

* * * *, ,

*

*

* *

1 , 1 1 ,

1 , 1 ,

1 , 1 ,

1 , 1

t

t t

t t

t t

t t

t t

t t

i t i i t i

K K r r

W W P r P

R R E E

C C S S

ρ ρ

γ

γ

γ γ

= + + = +

= + = +

= − = −

= + = +

7

In a steady-state equilibrium

• The equilibrium resource utilization rate

• The equilibrium growth rate

x*

ma1ρ β= −

( ) 21 1

11

x* 1

ma1 1 α α αγ βλ −− −⎡ ⎤+ = +⎣ ⎦

8

Model 2: common resources

• Budget constraints

( ), , , 11i t i t t t ti tC S r S W−+ ≤ + +Ω+

• Equilibrium path

( )* * * * * * * * *, , 1, , 0,1,

, ,1 , , , , , ,t t t t t t t i t i t i L tK R r W P E C S

= … = …+ Ω

starting from the initial state ( ), 1 01, ,,i i L

S R− = …

9

Equilibrium path (1)

• Consumers maximize utility• Capital is paid its marginal product• Consumer income consists of marginal

product of labor and resource income* *

* t tt

P EL

Ω =

• Resource price is equal to marginal product of the resource

10

Equilibrium path (2)• Aggregate savings are equal to the

capital stock

( )* * * *1 1t t t t tR R E Rρ+ = − = −

* *, 1

1

L

i t ti

S K +=

=∑• Natural balance of exhaustible

resources is fulfilled

11

Steady-state equilibrium

( )* * * * * * * * * *

1, ,

* , , ,1 , , , , , , ,i i i LK R r W P E C Sγ π

= …+ Ω

( ) ( )( )

( ) ( )( ) ( )

( )

* * * *, ,

* * * *

*

* *

*

* * * *

* * * *

* *

*

*

1 , 1

1 , 1 1 ,

1 , 1

1 , 1 ,

,

1

,

t t

i t i i t i

t

t t

t t

t t

t tt t

t

t

C C S S

K K r r

W W

R R E E

P P

γ γ

π

ρ ρ

γ

γ γ

= + = +

= + + = +

= + Ω = + Ω

= − = −

= +

• Let tρ ρ=

12

In a steady-state equilibrium

• Hotelling’s rule may be violated

• The equilibrium growth rate

max*

*

11 1r

βρ

π+=

−+

( )( ) 1 12

11* 11 1 1 α α αργ λ − −−⎡ ⎤+ = + −⎣ ⎦

13

Voting

• Consumer i is in favor of increasing ρt if

,0

ti i t

ti uV Cβ

∞

=

= ⋅

0/ tiV ρ∂ ∂ >

• Consumer i is in favor of decreasing ρt if

0/ tiV ρ∂ ∂ <

14

Voting equilibrium path

• For each t the number of consumers who are in favor of increasing ρt and the number of those who are in favor of decreasing ρt are less than half the total number of consumers.

( )* * * * * * * * *, , 1, , 0, ,

*

1, , ,1 , , , , , ,t t t t t t t i t i t it L tK R r W P E C Sρ

= … = …+ Ω

15

Voting steady-state equilibrium

( )* * * * * * * * * *

1, ,

**, , , ,1 , , , , , , ,i i i LK R r W P E C Sρ γ π

= …+ Ω

( )( ) ( )( ) ( )( ) ( )

( )

* * * * *

* * * *

* * * *

*

* *

* ** * * *, ,

* * *

*

* *

, 1 , 1 1 ,

1 , 1 ,

1 , 1 ,

1 , 1

1

,

t

t t t

t t

t t

t t

t t

t t

i t i i t i

t

t

K K r r

W W

R R E E

C C S S

P P

γ

γ γ

γ

ρ

γ

π

ρ

ρ

ρ = = + + = +

= + Ω = + Ω

= − = −

= + = +

= +

16

Median voter theorem

• Hotelling’s rule may be violated

• The equilibrium growth rate

*

*max1

1 mr βπ β+

=+

( ) 1 2 1

11* 11 1 mα α αγ λ β −− −⎡ ⎤+ = +⎣ ⎦

• In a voting steady state equilibrium* 1 mρ β= −

17

Common vs. private• Common resources

( ) 1 2 1

11* * 11 , 1 1m mα α αρ γ λβ β − − −⎡ ⎤= − + = +⎣ ⎦

• Private resources

( ) 1 12max m

11* * 1

ax1 , 1 1 α α αρ γ λβ β − − −⎡ ⎤= − + = +⎣ ⎦

Private ownership of resource stocks may be favorable for the economic growth

18

Further speculations (1)

• Risk of nationalization may decrease patience of the most patient consumers who own the resources thus decreasing the economic growth rate

19

Further speculations (2)• Borissov, K. and S. Lambrecht. Growth and

distribution in an AK-model with endogenous impatience. Economic Theory. 2009. 39(1). 93-112

• Model of endogenous growth where discount factors of heterogeneous consumers are determined endogenously and depend on economic inequality: – positively related to individual consumer's relative

wealth– negatively affected by a possibility of social

conflict increasing with inequality

20

Further speculations (3)• Private ownership of resource stocks

increases economic inequality and thus can have an ambiguous effect on economic growth: – positive influence in the case of moderate

inequality– negative effect due to decrease in the discount

factors of capital owners induced by increased risk of social conflict at high level of inequality

• There can be a range of parameters where common ownership of resource stocks may lead to even larger economic growth rate then in the case of private resources

21

Conclusion (1)• Two models of economic growth with

heterogeneous consumers and exhaustible resources are developed– a model with private ownership of the resource

stocks– a model with collectively owned resource stocks

and the resource rent equally divided among all the consumers

• Equilibrium paths are defined and some properties of steady-state equilibria are analyzed

• In the second model the notion of voting equilibrium is introduced and properties of steady-state voting equilibria are described

22

Conclusion (2)• In the case of common resources in a voting

steady-state equilibrium the resource utilization rate as well as the steady-state rate of growth is determined by patience of the median consumer.

• In the case of private resource stocks the resource utilization rate and the steady state rate of growth are determined by the patience of the most patient consumers.

• If the median consumer is less patient than the most patient one, in the case of privateresources the resource utilization rate is lower and the economic growth rate is higherthan in the case of common resources.