The Voracity Effect Puzzle: Comment1
by Yong J. Yoon
In an earlier article in American Economic Review (March 1999), Tornell and Lane provide a formal model to explain their "Voracity Effect," a puzzle in economic development. The puzzle is, in countries with inefficient legal protection of private property rights,
1) on average, resource-rich countries tend to grow more slowly than resource-poor countries. Brazil and Argentina are rich in resources, but they under-perform against those resource-poor countries like Korea and Taiwan;
2) in some of these countries, any gains in productity or terms of trade tend to induce worse economic performances.
Tornell and Lane raise an interesting public choice question: As the number of power groups able to exploit the productive economy increases, whether in the limit the economy will dissipate like in the commons problem. They correctly point out that standard neoclassical growth models (Solow, Cass and Koopmans, etc.) are inadequate to address these questions. Into an optimal growth model Tornell and Lane introduce multiple power groups which are able to exploit the productive economy. The most interesting feature of their model is the alternative technology available in the informal, or shadow, sector. The formal sector has superior technology, but is open to exploitation by transfer seeking power groups, while the informal sector, though less productive, is free from taxes and exploitations by other groups. The Nash equilibrium, Markov Perfect Equilibrium by Tornell and Land, consists of strategies, the appropriation rules of power groups, which depend only on the state of the economy.
In the interior equilibrium, where exploitations are within bounds, their main results are,
1) the informal sector serves as a threat to exploitative behavior, and can improve the economic performance by reducing the degree of exploitation;
2) as formal sector’s productivity increases, total exploitation increases more than the productivity gain;
3) as the numbers of power groups increase, total exploitation is reduced and economic performance improves.
I can agree with the first result, but do not see how results 2) and 3) follow from their model. In the following section I summarize the Tornell-Lane argument. I point out that their necessary condition for equilibrium is in error, which implies that all of their conclusions about observed voracity effect are non sequiturs. A counter-example is provided to illustrate the inconsistencies in the Tornell-Lane results. The paper provides a resolution that will properly analyze the equilibrium respecting their original intent and insights.
I. Necessary Conditions for Interior Equilibrium
In solving the "Voracity Puzzle" Tornell and Lane develop a necessary condition (2) for equilibrium which will preclude each group from appropriating the aggregate capital stock at once. See equation (2) at page 24 in Tornell and Lane (1999). Consider the paragraph at the left of page 29:
"If this rate is lower than the private rate of return that i gets (ß), then group-i's best response is to transfer as much as possible from the formal to the informal sector."
The paragraph implies that:
the post-appropriation rate of return perceived by group-i isa p - (n - 1)x;
the return in the informal sector is ß,: and
To avoid this happen in equilibrium, Tornell and Lane continue as follows.
Page 30, paragraph at the left part:
"For i to find it optimal to set ri(t) within admissible bounds given by (6), it is necessary that the rate of return on i's closed-access capital ß be equal to its rate of return on the open-access capital after redistribution to other groups has taken place. This implies that the following condition must hold for any group i in an interior equilibrium: ß = a p - S xj' The unique solution of this system of n simultaneous linear equations is that all xj's be equal to (a p - ß)/(n - 1) as shown in (18)." (Underline added. The summation runs j = 1 through n except i.)
The thrust of the argument is this -- the "necessary condition" for an interior equilibrium requires that the appropriation rates (xi) prevent capital flight from the formal to the informal sector. To prevent capital flight, the necessary condition must negate situations that induce the inequality,
(1)a p - S xi < ß, for any j. (i assumes 1 through n except j)
which is the capital flight situation because the return in the informal sector higher than in the formal sector.
The necessary condition, the correct negation of condition (1), is
(2) For each power group i,a p - S xj "is greaer than" (>) or "equal to" (=) ß,
(the summation runs over j, 1 through n, except I)
As indicated in the quotations above, Tornell and Lane reason erroneously that the necessary condition is the equality
(3) For each i,a p - S xi "is equal to" (=) ß.
In logics a necessary condition sets a boundary that will secure the desired solution within that boundary. My proposed condition (2) sets a boundary more inclusive than the condition (3) proposed by Tornell and Lane. Thus, the suggested necessary condition (2) will include the solution proposed by Tornell and Lane; if the two conditions, (2) and (3), give different solutions, then Tornell-Lane solution is incorrect.
We demonstrate by an example that necessary condition (3) is not consistnet with the Nash solution to the problem. This means that the Tornell-Lane argument is inconsistent.
II. Numerical Counter Example
My strategy of argument is to follow the construct of Tornell and Lane and show that their interior equilibrium is not consistent with the definition of Markov Perfect Equilibrium, a Nash equilibrium.
A community consists of two power groups, Group-1 and Group-2. Each group starts with endowments of capital goods, Ki, i = 1, 2, in the formal sector of the economy. But the protection of private property rights is inefficient and each group treats the productive economy as a commons for exploitation. Technically, capital goods are taxed to finance transfers. The exploitative tax can be avoided if the power group invests in the informal sector which produces by an inferior technology.
The technology in the formal sector allows one unit of capital good grow into (1 +a ) units of output in a period. The output can be consumed or invested either in the formal sector or in the informal sector. The productivity of the informal sector is (1 + r), and r < a . We can say that this is a two asset multiagent growth model. The period starts with aggregate capital K(t), which grows into (1 + a )K(t) at the end of the period. The transfer requested by each group is a function of capital K(t), which is the only variable that depends on the state of the economy.
Tornell and Lane suggest to consider only the linear forms, ri(K) = xiK, for transfer functions. Each group tries to maximize the present value of expected utilities by exploiting the productive formal sector (K), also by investing in the informal sector (b) to avoid exploitation in the formal sector.
As Tornell and Lane argue in equation (3) above, we assume that Group-2 appropriates at a rate x2 = a - r. Given this, we derive the appropriate strategy of Group-1. If this turns out to be x1 = a - r, then we can confirm the Tornell-Lane result. The budget constraint for Group-1 is
(4) c(t) = (1 +a )K(t) + (1 + r)b(t) - (a - r)K(t) - K(t+1) - b(t+1) = (1 + r)[K(t) + b(t)] - [K(t+1) + b(t+1)]
We note that each group exploits the formal sector by using the same rule period after period. This is part of the definition of Markov perfect equilibrium. In each period, throughout the entire future, Group-2 will take away the surplus, r2(K) = (a - r)K, from the formal sector. Thus, Group-1 faces a virtual technology that has productivity (1 + r) per period, because for Group-1 the perceived productivity is (1+r) in both formal and informal sectors.
Under this condition, Group-1's intertemporal problem is to choose a consumption-investment rule that will maximize its present value of utils subject to the "virtual technology."
(5) V(K) = Max U(c) + ß V(K')
(6) c(t) = (1 + r)K(t) - K(t+1)
where we let K(t) denote the virtual capital, the aggregate capital stock both in the formal sector and in his private informal sector.
The value function V(K) can be intepreted as follows: the value of capital K is the present value of utils if Group-1 behaves optimally in each future period.
A solution to the problem (5)-(6) is c*(K) = zK, where z is Group-1's appropriation rate. This is consistent with the Markov Perfect equilibrium because is c*(K) = zK is a solution to the dynamic programming in which the state variable is the aggregate capital K. Note that z is a number determined by parameters ß and r, and not bya . But according to Tornell and Lane argue that, in (symmetric) equilibrium, transfer rule for each group is,
(7) r1(K) = r2(K) = (a - r)K
which increases ina . This condition could be satisfied if z = a - r. For instance, if a = 0.2, r = 0.1, and z = 0.1, then we have z = a - r, which is consistent with equation (7). If a becomes 0.4, then a - r = 0.3 while z is still 0.1 because r1(K) = zK does not depend on a .
The equilibrium concept used by Tornell and Lane is that of Nash: Given Group-2's strategy, r2(K) = (a - r), strategy r1(K) will maximize (5) subject to (6). We have demonstrated that the strategy r1(K) = (a - r)K fails to meet this requirement, while Tornell and Lane solely depends on this relation (7) in computing the equilibrium. This error comes from confusion in the necessary condition. The correct necessary condition comes from (2), while Tornell and Lane solution depends on condition (3).
The inconsistency comes from following the wrong different equilibrium condition, necessary condition (3), rather than (2) which is the correct one. Based on this observation, an equilibrium concept, a threat equilibrium, can be formulated. In a separate paper, our equilibrium results will be proffered.
1In May 1999, James M. Buchanan and Yong J. Yoon made an announcement for a special seminar presentation, indicating that "Tornell and Lane argue that, as the number (n) of power groups able to exploit the productive economy increases, the degree of total exploitation increases at the beginning then decreases for large number of n. Something seems wrong here."
Tornell, Aaron and Lane, Philip. "The Voracity Effect." American Economic Review, March 1999, pp. 22-46.