[Note for bibliographic reference: Melberg, Hans O. (1998), Rent seeking, social waste and economic models, www.oocities.org/hmelberg/papers/981208.htm]  

Rent seeking, social waste and economic models

by Hans O. Melberg


Introduction
This is another paper in the series in which I examine the value of formal models in economics. As in the previous papers I want to examine whether the formal model generates surprising and counterintuitive implications (see "Pay according to need or effort" and "Cooperatives: A short model with surprising implications"). If the results are true, surprising and unavailable (or difficult to derive) using verbal reasoning, then formal modeling is justified. Since I want to avoide generalizations at this stage, I focus on one model, more specifically: a short model of rent-seeking behaviour.

The model
The model below is based on lecture notes from a class given by Professor K.O. Moene (University of Oslo) in 1998. Assume that two firms are competing about a fixed sum (R) and that the amount one firm receives depends on:
- how much it spends on lobbying activities (x)
- how effective the spending is (b) (i.e. lobbying by one form may be more effective than another and this is reflected in different values of b)
- how much the other firms spends on lobbying and how effective it is

We also assume that this is not a "winner-takes-all" situation. That is, the more you spend on lobbying the more you get, but the other firm does not get zero even if it spends less on lobbying than you. Formally we could express the expected net value of lobbying (v) as follows:

(1) v₁ = [(b₁ x₁) / (b₁ x₁ + b₂ x₂)] R - x₁

In short, you receive a share of the total cake that is proportional to your share of lobbying weighted by the effectiveness of each firms' lobbying (the first term) minus the money you spend on lobbying (x₁). Similarly, the expected value for the second firm is:

(2) v₂ = [(b₂ x₂) / (b₁ x₁ + b₂ x₂)] R - x₂

Before we go on, ask yourself the following question: What is the point of writing equations (1) and (2)? Couldn't the interpretation stand alone without the equations? There is nothing in the equations that could not be expressed in words. This is true, but we the equations may be useful when we want to answer questions as: "Exactly how much would the firm spend on lobbying?" "How does the difference in b (the effectiveness of lobbying) affect the amount spent on lobbying?" "How large is the social waste?" and "What happens if there are more than two firms?"

Try to answer these questions intuitively first. My intuition, in any case, was that almost all the rent would be wasted in the lobbying game, that the firm with the highest lobbying effectiveness would exploit this by spending more than the other firm. It turns out that if we do the math the answers are no so intuitive.

Consider first the question of how much the firms will spend on lobbying activities. The firm wants to maximize its expected net gain. Mathematically we find the expression for the maximum by setting the partial derivative of (1) and (2) with respect to the choice variable (x₁ and x₂ respectively) equal to zero. This gives:

(3a) dv₁/dx₁ = [b₁ (b₁ x₁ + b₂ x₂) - b₁ (b₁ x₁)] / [(b₁ x₁ + b₂ x₂)²] R - 1 = 0

(4a) dv₂/dx₂ = [b₂ (b₁ x₁ + b₂ x₂) - b₂ (b₂ x₁)] / [(b₁ x₁ + b₂ x₂)²] R - 1 = 0

Which can be simplified to:

(3b) dv₁/dx₁ = [(b₁ b₂ x₂) / (b₁ x₁ + b₂ x₂)²] R - 1 = 0

(4b) dv₂/dx₂ = [(b₁ b₂ x₁) / (b₁ x₁ + b₂ x₂)²] R - 1 = 0

From (3b) and (4b) we see that in equilibrium x₁ = x₂ (Why? Both equations should be zero. So setting (3b) = (4b) and eliminating the common terms leaves us with x₁ = x₂). This is somewhat surprising. In our model the firms spend exactly the same amount on lobbying activities. This result holds even if one firm is more effective at lobbying than the other: They still spend the same amount on lobbying! This is surprising and not obvious without doing the math.

Setting x = x₁ = x₂ and using equation (3b) we find that the amount used on lobbying is given by the following equation:

(5) x = [(b₁ b₂) / (b₁ + b₂)²] R

Social waste (L) can be defined as the total amount used on "socially unproductive" lobbying activities. Since each firm in our equilibrium spends x, the total waste is simply two times x, or:

(6) L = 2x = [(b₁ b₂) / (b₁ + b₂)²] 2 R

Now, we would like to examine the waste in three different situations; when the two firms are equally effective at lobbying (b₁=b₂), when one is more effective than the other (b₁>b₂), and when one firm is not effective at all (b=0).

If b₁=b₂ (=b), then (6) becomes

(6a) L = [(b b) / (2b)²] 2 R

or, simplified:

(6b) L = 1/2 R

In short, the firms waste half of the cake (R) on lobbying activities.

 

If b₁>b₂, then the waste goes down. For instance, if b₁=b₂/2 the total waste becomes 4/9 R and the larger the difference is, the less the waste is. This is interesting. It implies that a situation with one large and powerful firm generates less waste in terms of influence activities than two equally powerful firms. However, I also have some serious questions about the sensitivity of this conclusion to changes in the model (see below).

Finally, if b₁=0 there is no social waste. One firm receives the rent without any significant lobbying since the other firm will not spend money on ineffective lobbying.

Generalizations and criticism
The case of two firms can easily be extended to many firms. The conclusion is that once we allow more firms, the return to lobbying decreases and so does social waste (your share of total lobbying if all firms did the same amount of lobbying becomes small when there are many firms).

A more serious problem occurs if the model is extended to include the wasteful effects of lobbying, not only the waste in terms of the money spent on lobbying. For instance, a firm that lobbies in favor of a more permissive law on pollution than is socially optimal, not only engages in unproductive activities (the money it spends trying to change the law). The consequences (not only the process) the lobbying may also be negative. Moreover, firms may have opposing interests; some firms want to be able to pollute a river, another firm may want to use clean water. If this is the case we must re-consider the result that one powerful firm is "better" than two equally powerful firms are. True, the waste in terms of money spent on lobbying may be higher with two equal firms, but the end result may be a law that is better in terms of balancing interests (since the lobbying activities cancel out when the firms have opposing interests). This is not to argue that the results of the model are wrong (mathematically speaking the results follow from the setup), only that the model itself needs to be realistic to generate policy implications. It must be extended to include the bad effects of lobbying as well as the waste of the process of lobbying.

Finally, the whole setup may need some change if the nature of the game is "winner-takes-all" instead of the "benefit proportional to lobbying" result implied by equation (1) and (2). One might capture some of this with the use of b and the terms "expected" net gain, but I suspect that a different setup is needed if the game is of the "winner-takes-all" type.

Conclusion
I believe we have an example of a model that is both easy, rich on interesting implications and which generates results that cannot be found using verbal reasoning alone. I am, however, suspicious about generating policy implications from mathematical models that are sensitive to changes in the setup (for instance when we include firms with opposing interests and the bad effects of lobbying). It is not enough to say that the model generates surprising results, these results must also be true in the real world to make the model really interesting.

[The same comment was made about the model "Cooperatives" in which the model predicted fewer workers when the price of a product increased. This is counterintuitive, but it is also counter-factual. The use of the model in this case may simply be to force us to be more precise about the mechanisms and generate further research to see what we have forgotten in the original setup.]

See Economic Journal 1998 (vol. 108), p. 1826 -- for several articles on formalism and models in economics.

[Note for bibliographic reference: Melberg, Hans O. (1998), Rent seeking, social waste and economic models, www.oocities.org/hmelberg/papers/981208.htm]