One model of the quantity of health services performed (q) as a function of the proportion of health care services that an individual pays (p) is called the uniform induction hypothesis. Let 0 ≤ p ≤ 1 and scale q so that q(0) = 1. The cost of health care to the consumer, in appropriate units, is given by pq. Using this model, an increase in cost to the consumer causes a proportional decrease in the quantity of health services performed, so that dq = -k d(pq) for some constant k.

(a) Use the product rule in the equation above, and then divide by dp, to show that the equation can be rewritten as

(b) Solve the equation in part (a) with the initial condition q(0) = 1, writing the solution in terms of k and p.
(c) Some economists estimate that q(1) = 1/2. Use this to eliminate k in the solution found in part (b), writing the solution in terms of p.
(d) Use the answer to part (c) to estimate the portion of q(0) that consumers will use if they pay 20% of the cost of their health care.

Transcribed Image Text:

dq dp -kq 1 + kp