Competitive markets for illegal goods and services
We first consider a perfectly competitive market for the good. Suppose that firms are risk neutral. Let the competitive equilibrium price be P *. In a competitive equilibrium, producers supply the good up to the point where the expected market price that they receive (which in a perfectly competitive market is equal to their expected marginal revenue) equals their expected marginal cost.
A producer's expected revenue from supplying one unit of the good (that is, their expected marginal revenue) is equal to (1-p)P*, whilst expected unit costs are equal to c + pf. Therefore, equating the expressions for expected marginal revenue and marginal cost above allows us to solve for the market-clearing price:
The equilibrium market price is increasing in the fine and increasing in p, the probability of detection. Following Becker et al. (2006), it is useful to work with the odds ratio, which is the ratio of the probability of being detected to the probability of not being detected:
Note that 1 + в = —. We can then write the competitive equilibrium
market price as:
The amount 6(c + f ) = P* - c is the expected marginal economic value of the goods that are confiscated, and is equal to the excess of price over production cost. Thus the competitive equilibrium market price is equal to the marginal cost of production, plus a term that reflects the expected value of goods that are confiscated.
Even though price exceeds production cost, firms earn zero expected profits in this market - the excess of price over cost compensates them for the additional loss to them that is associated with the possibility or risk that their goods will be confiscated and destroyed, resulting in a loss of revenue.
Specifying a demand curve allows us to pin down the equilibrium quantity. For example, suppose that the demand curve is:
Then the equilibrium quantity is:
The market quantity is decreasing in p and f For a given probability of detection and fine, the extent of the reduction in quantity depends on the elasticity of demand.