A. Environment and Technology

Consider an economy comprised of a continuum of two-period lived agents distributed over the interval [0,1]. Agents are assumed to have identical preferences and abilities and differ only in their initial wealth. The preferences of each agent are described by the twice-continuously differentiable quasi-concave utility function U (c₁, c₂), where c₁ and c₂ denote consumption of the economy’s single good in each period of life. The arguments of the utility function are gross substitutes With lim c_i→ ∞ [u_i(c₁, c₂)] = ∞, and u_i, represents marginal utility in period i, i=1,2.⁶ Each agent is endowed with w > 0 units of the consumption good when young. The initial distribution of goods endowment in the economy is represented by the cumulative distribution function Γ: R₊₊→ [0,1].

There are two sectors in the economy: production and rent seeking. Each agent operates in one of these two sectors in the second period of his life; in the first period agents are not occupationally active. Each producer has access to a standard concave production technology that yields f(i) units of the consumption good in period two of his life to an investment of i units when young (period 1), where f(0) = 0 and f′(0)=∞. Producers, however, face appropriation of some share of their market production by rent seekers. There is no law enforcement and no government taxation. Let 0< γ < 1 denote the exogenously given proportion of market production that can be extracted from each producer in the form of bribes, taxes, or outright expropriation.⁷

The description of the rent seeking sector is as follows:⁸ Let the probability of being approached by a rent seeker be denoted by π^P. The second period expected return to a producer is then given by (1 – π^P γ)f(i). Let n denote the mass (fraction) of rent seekers in the economy. Each rent seeker can expropriate a share γ of the production of λ ≥ 1 producers in the economy, where λ can be regarded as the efficacy of rent seeking. However, each producer is only approached by a single rent seeker.⁹ The probability π^P can then be defined as the ratio of the total number of rent seeking cases in the economy divided by the mass of producers, (1- n). Assuming full information, π^Pcan be defined as

Entry into rent seeking requires payment of a lump-sum fee of θ units of the consumption good when young. Payment of this cost allows rent seekers to tax market production in the second period of their lives. Agents are assumed to be unable to borrow to finance entry into rent seeking.

Rent seekers save through a simple technology that returns (xs) goods in period 2 of their lives to an input of s units when young. The inputs of rent seekers are assumed to be unobservable, which implies that rent seekers cannot coordinate their saving decisions. Unobservability of a rent seeker’s inputs also implies that a rent seeker cannot expropriate the goods controlled by another rent seeker. Therefore, one may think of x as the return on hoarding which allows rent seekers protection from theft by others.¹⁰ We assume that x < (1- π^Pγ)f’(i) ∀ i and n. Thus, the marginal productivity of the production technology is higher than the return on hoarding.¹¹

The probability π^R that a particular rent seeker is the first to approach a producer is defined by the ratio of the total mass of producers, (1- n), divided by the total number of rent seeking cases in the economy, λn. This probability can be expressed as

The ratio (1-n)/λn also captures the degree of crowding out in the economy. To see this, note that for (1-n) > λn, as the fraction of producers in the economy exceeds the fraction of rent seeking cases, with probability 1 each rent seeker can expropriate from a producer (π^R = 1). When there are fewer rent seeking cases in the economy than there are producers, each rent seeker can capture the full potential rent given by the rent seeking technology from the X producers he approaches. However, when λn > (1-n), crowding out occurs as each rent seeker competes with others for rents and the expected rent to each rent seeker falls below the full potential rent given by the rent seeking technology. One can view this as rent seekers coming up against a capacity constraint – when the capacity of rent seekers to expropriate from producers exceeds the number of producers, the expected rent collected by each rent seeker shrinks.

Let ñ denote the value of n for which the total fraction of rent seeking cases in the economy just equals the fraction of producers. The probability π^P that a producer is approached by a rent seeker can then be expressed as

and the probability π^R that a rent seeker expropriates from a producer as

Note that $\tilde{n}=\frac{1}{1+\lambda }\le 1/2$ for λ ≥ 1. The second period expected return from rent seeking can then be written as π^R λγ f(i) + xs.

B. Equilibrium

In this section we describe individual decision making. First, consider the optimization problem faced by an agent who chooses to become a rent seeker. The budget constraints faced by a rent seeker in each period of his life, are

where s represent the savings of the rent seeker. The first order condition for utility maximization for a rent seeker is given by the Kuhn-Tucker condition

If the agent’s initial endowment w is sufficient to provide the desired first period consumption in light of the intertemporal tradeoff, equation (7) is always satisfied with equality and the agent hoards a positive amount. Suppose this is the case. Define as V^R(w) the maximum utility attained by an agent if he chooses to engage in rent seeking (given an initial endowment, w). When instead the agent chooses to become a producer, he solves

where V^P(w) denotes the maximum utility attained by a producer taking w as given. The first order condition for utility maximization for a producer is

The above equation holds with equality as a producer must invest when young to consume in the second period of his life.

An equilibrium in this environment is given by an allocation of the population between rent seeking and production and values of s and i such that each agent maximizes his lifetime utility taking as given w, x, θ, λ and γ. To study the allocation of agents between rent seeking and production, we examine the relationship between V^R(w) and V^P(w) as a function of n.

We first examine the possibility of corner equilibria. We can show that there does not exist an equilibrium in which all agents engage in rent seeking. To see this, note that, for a given w, with n = 1 rent seekers and no production, π^R = 0 as rent seekers crowd each other out, while π^P = 1. As a result, the rent accruing to each rent seeker is zero. Because the return to hoarding is less than the return to production, for any given θ > 0, the maximum utility attained by a producer, V^P(w), exceeds the maximum utility attained by a rent seeker, V^R(w).

Consider now an equilibrium in which there is no rent seeking. For such an equilibrium, it is necessary that the maximum utility from production, V^P(w), is greater than or equal to the utility from being the only rent seeker, V^R(w). At n = 0, equation (4) suggests that π^R = 1 and the first rent seeker to approach producers can capture λγ f(i) in rents, while π^P = 0. From equations (5), (6) and (8) it can be seen that a necessary condition for such an equilibrium is that the cost of entering into rent seeking, 6, is high relative to the expected return from rent seeking (determined by λγ). The following example illustrates conditions under which an equilibrium with no rent seeking can arise.

We can now describe the interior equilibrium in this environment shown in Figure 1. Note that ñ can never be a stable equilibrium. While V^R(w) can equal V^P(w) at ñ as is the case when the V^P curve is tangential to the V^R curve (at ñ), such an equilibrium is unstable as a small change in n from its equilibrium value drives the economy to the corner solution with no rent seekers.

Given that V^P(w) > V^R(w) at n = 1 and the assumption that V^R(w) > V^P(w) at n = 0 the difference V^R(w) − V^P(w) is maximized when n = ñ This in addition to the continuity of the utility functions yield Proposition 1.

View Full Size

Unique Equilibrium with Rent Seeking

Citation: IMF Working Papers 2001, 015; 10.5089/9781451843262.001.A001

• Download Figure
• Download figure as PowerPoint slide

Hence, a unique equilibrium occurs when V^R(w) and V^P(w) intersect at n^* > ñ.¹⁵

C. Properties of Equilibrium

In this section we examine the relationship between agents’ wealth and their choice between rent seeking and production. In the presence of the lump sum cost θ in the rent seeking sector and in the absence of credit markets, it can be shown that only relatively wealthy agents will choose to engage in rent seeking. The following lemma states that the second period (earned) income of a rent seeker is higher than the earned income of a producer.

Lemma 2 states for a given w, at the point where agents are indifferent between rent seeking and production, the consumption of a rent seeker when young is lower than the consumption (period 1) of a producer. This result occurs due to the presence of the lump sum cost required to enter into rent seeking.

Let Ω(w) denote the difference between the utility of an agent with endowment w if he chooses to engage in rent seeking and his utility if he chooses to become a producer.

where s^* denotes the value of saving that satisfies with equality the first order condition for a rent seeker (equation (7)) and i^* represents the value of a producer’s investment that satisfies his first order condition (equation (9)). Therefore, Ω(w) represents the difference between the maximum utility obtained from rent seeking, V^R(w), and the maximum utility obtained from production, V^P(w).

The derivative of Ω(w) with respect to w is given by

Where u₁^R and u₁^P represent the marginal utility from engaging in rent seeking and in production, respectively. From Lemma 2 we know that at the point where agents are indifferent between the two occupations, the marginal utility of consumption when young is higher for a rent seeker than for a producer, so that u₁^R> u₁^P and Ω′(w) > 0. Because in equilibrium some agents follow each career path (from Proposition 1), there must exist some borderline endowment $\overline{{w}}$ defined by ${\Omega }(\overline{{w}})=0$, such that agents are indifferent between rent seeking and production. The above analysis indicates that Ω(.) has a positive slope at ${w}=\overline{{w}}$; that is, when rent seeking and production offer nearly the same utility, the relative attraction of rent seeking to production increases with an increase in initial wealth. Therefore, Lemma 2 along with the continuity of the utility functions ensures that V^R(w) and V^P(w) have a single crossing.

We can now characterize the allocation of agents between rent seeking and production. There exists a unique level of endowment such that for ${w}\ge \overline{{w}}$, agents will choose to engage in rent seeking, and for all ${w}<\overline{{w}}$, they will choose production. This result is summarized in Proposition 2.

Proposition 2 states that the rich become rent seekers. Rent seeking is limited to agents with relatively high initial income due to the presence of the lump sum cost required to enter this sector. Note that it is not the higher wealth itself that creates incentives for wealthy agents to enter into rent seeking but the ability to protect their wealth from expropriation by others. Payment of the fixed cost enables rent seekers not only to extract some portion of the proceeds from market production but also ensures protection of their wealth. The rich, by virtue of their higher initial income, therefore, have a greater incentive to engage in rent seeking.

Thus, despite the fact that agents are ex ante identical in terms of preferences and abilities, two classes of agents emerge in equilibrium. It is the initial income (w) of an agent that determines his (or her) decisions whether to engage in rent seeking or production, and how much to consume and save. Hence, the initial distribution of endowments determines aggregate output in the economy and the measure of agents engaged in rent seeking.

and in production

Given that hoarding has a lower marginal product than production, the greater the proportion of rent seekers in the economy, the smaller is aggregate output. Hence, the total level of output in the economy is negatively related to the size of the rent seeking sector.

A. Multiple Equilibria

In the previous sections we abstracted from multiple equilibria in order to focus on the relationship between agents’ initial wealth and their choice between rent seeking and production. To illustrate the possibility of multiple equilibria in rent seeking and production, we modify the assumption that V^R(w) > V^P(w) at n = 0. Such a situation may arise if, for example, the amount expropriated from producers γ is low or if the cost to enter into rent seeking, θ, is sufficiently high (see Example in Section II.B). If V^P(w) > V^R(w) at n = 0, such that production is relatively more attractive than being the only predator, and V^R(w) > V^P(w) for some 0 < n < 1, V^P(w) and V^R(w) can intersect twice (Figure 2). If V^P(w) > V^R(w) ∀ 0 ≤ n ≤ 1, the two curves will not intersect at all (Figure 3).

View Full Size

Multiple Equilibria

Citation: IMF Working Papers 2001, 015; 10.5089/9781451843262.001.A001

• Download Figure
• Download figure as PowerPoint slide

Consider first the case where they intersect twice as in Figure 2. As discussed in Section II.A, the return to production is decreasing in the number of rent seekers in the economy for n < ñ. However, an increase in production may also increase the expected return to rent seeking since rents increase, and for n > ñ the crowding out among rent seekers becomes less severe. Hence, a decline in rent seekers stimulates production and increases returns to both activities, giving rise to multiple equilibria in rent seeking and production.²¹ One corresponds to an equilibrium with positive levels of rent seeking as discussed in Section II.B. A second is an equilibrium with no rent seeking. There is a third equilibrium with a positive level of rent seeking and production where V^P(w) = V^R(w). Such an equilibrium is unstable as a small change in n from its equilibrium value drives the economy to one of the two stable equilibria of positive or no rent seeking.

View Full Size

Unique Equilibrium without Rent Seeking

Citation: IMF Working Papers 2001, 015; 10.5089/9781451843262.001.A001

• Download Figure
• Download figure as PowerPoint slide

The multiplicity of equilibria suggests that two economies with identical technologies of production and rent seeking may arrive at entirely different outcomes. If an economy starts out with few rent seekers (n < n in Figure 2), as the relative return to production exceeds that to rent seeking, all agents choose to become producers. If n > n, rent seeking is more attractive than production, and the economy ends up in an equilibrium with a positive level of rent seeking and smaller aggregate output.

Note that it is possible that V^P(w) > V^R(w) ∀ 0 ≤ n≤ 1 as in Figure 3. In this case the only stable equilibrium is the one with no rent seeking. Such a situation may arise if, for example, the fixed cost to entering into rent seeking, θ, is sufficiently high for any given w or the fraction γ expropriated from producers is sufficiently low.

B. Endogenous θ

The equilibrium allocations between production and rent seeking in our analysis so far relied on the assumption of a fixed cost θ of entering into rent seeking. However, the model could easily be extended to endogenize θ. In all cases considered below, it is assumed that there is a maximum potential number of rent seeking cases (R^*) which is ultimately a function of the existing set of rules and regulations and the degree of discretionary power of public officials. R^* is assumed to be at least as large as the maximum number of rent seeking cases in the economy given the rent seeking technology and the fraction of rent seekers (i.e., R^*≥n^* λ).

Consider a two stage game in which a self-interested higher authority in the first stage decides on the size of the rent seeking sector, n. In the second stage there is free competitive bidding for the rent seeking positions, such that all successful bidders pay the same price and agents choose their profession. Assume that there is no taxation in the economy. The central authority then acts as a revenue maximizing Stackelberg leader, choosing the optimum n to maximize its revenues, nθ(n)²². Given that the central authority takes into account the responsiveness of 0 to n, the equilibrium size of the rent seeking sector will be equal to or greater than ñ.

Suppose instead, in the absence of a strong centralized authority a (relatively large) number of decentralized government units effectively have power to charge agents an entrance fee into rent seeking, each one in its own area, and the number of potential entrants into rent seeking R^* is the same as in the previous case. Each unit can grant one agent the right to engage in rent seeking and it acts independently of the other units. Hence, it does not take into account the effect on the entrance fee θ of increasing the number of rent seekers. Intuitively, the number of rent seekers will be larger, and the equilibrium entrance fee and the threshold initial wealth level $(\overline{{w}})$ lower than in the previous case. In other words, the revenue maximizing central authority will choose a smaller, more exclusive and wealthier club of rent seekers.

This feature of a small group of wealthy elite engaging in rent seeking is observed in a number of developing countries as well as in medieval Europe. Examples of uncoordinated solicitation of bribes and other forms of rent seeking by a large number of government units are seen in many African countries.²³ The large number of units with this power reflects their own low entrance costs into rent seeking. As noted by Shleifer and Vishny (1993), the resulting effect can be an even heavier burden on private producers of rent seeking, and this is even more detrimental to private investment and economic growth.

In the above discussion, it is implicitly assumed that the free competitive bidding in the second stage raises the entrance fee θ to the level where the marginal bidder’s excess utility from rent seeking relative to production is reduced to zero. Hence, if all agents had the same initial wealth (say the same $\overline{{w}}$) there would be no rent seekers. This would be the case regardless of the higher authority’s decision in stage 1 about the size of the rent seeking sector. Only individuals with initial wealth higher than $\overline{{w}}$ would have positive excess utility from rent seeking relative to production at that level of θ. In other words, an important implication of our model is that there must be initial inequality in wealth for rent seeking to take place.

C. Differences in Ability

The model can easily be extended to assume differences in abilities across individuals leading to differing returns in each sector. The link between rent seeking and misallocation of resources in the economy is then reinforced by the finding that the most able persons become rent seekers. In addition, it can be shown that incentives for gambles remain as long as differences in ability cause inequality in the marginal utility of first period consumption across the two activities. In this case, the winners of the lottery (i.e., the most able ones that score highest on the civil service entrance exam) become wealthy rent seekers in the civil service while the losers become producers.²⁴

Differences in ability can be incorporated in a number of ways in our model. Suppose that more able individuals have lower costs of entering into rent seeking (lower θ). Assume that an individual’s ability a is drawn from a continuous density function q(a), where agents with high a have a comparative advantage for rent seeking relative to production. In this case, it can be shown that the return to rent seeking becomes more attractive for individuals above some average ability a^* and less attractive for those below average ability.²⁵

Suppose there exists productivity differences among individuals in both rent seeking and production such that persons with higher ability have higher productivity in both rent seeking and in production than less able individuals. We can formalize this by considering an amended production function, af(i), and rent seeking function, aλγf(i) as in Murphy, Shleifer and Vishny (1991). Both rent seeking and production show increasing net returns to ability. This reflects the existence of a cost element in both sectors that is fixed with regard to variations in the level of ability (the entrance fee θ and the investment I, respectively) whereas revenue in rent seeking and output in production both vary with ability. If the elasticity of the net return to ability is higher in rent seeking than in production, rent seeking will be relatively more attractive to the highly able individuals.

We see that the most able individuals will be attracted to rent seeking, resulting in a misallocation of talent. Aggregate economic activity is decreased because the number of individuals and the amount of capital invested in production is lower; and, for a given number of persons engaged in production, the average ability—and hence productivity—is lower since the most able are engaged in rent seeking.