caine consumption to seizures. The magnitude of the derivative ∂Q/∂X depends on the shape of the demand function as controlled by the demand elasticity parameter b, on the shape of the cost function as controlled by the power-function parameter d, and on the composite parameter g, as well.
We wish to focus here on the role played by parameter d, determining the shape of the cost function. For this purpose, it simplifies matters to choose certain values for the other parameters. In particular, suppose that b = -0.5 and that a = c = 1. The value b = -0.5 is the RAND baseline value for the demand elasticity. Setting a = c = 1 implies that g = 1. With these choices for (a, b, c), equations (8) and (10) reduce to
Observe that setting X = 0 in equation (11) yields Q = 1 as the equilibrium consumption of cocaine for all values of the parameter d. Evaluating the derivative (12) at the point Q = 1 reveals how consumption falls as seizures rise above zero. The result is
Thus, the response of cocaine consumption to seizures increases with the value of d. Under the RAND assumption that d = 1, consumption of cocaine responds least to seizures. The derivative value (∂Q/∂X)Q=1 = -1/2 that holds in this case implies that, as seizures rise from 0 equilibrium cocaine consumption falls by half the amount seized. In contrast, if the value of d is much larger than 1, then the derivative (∂Q/∂X)Q=1 is close to -1, implying that equilibrium cocaine consumption falls by almost all the amount seized.
In the market simulation in Figure 2 (in Section 2 of report), the parameter d is set to 1 to produce the two downward sloping average cost curves and to 4 to produce the two upward sloping average cost curves. These values for d are selected to illustrate the sensitivity of predictions to the assumed shape of the average cost curve. The actual shape of the curve is unknown.