Question

Rusty coaches runners (r) who sometimes get injured and Terrell offers physical therapy (t) at a nearby clinic. More runners means Terrell has to spend less on promoting his services. Rusty charges pr = 120 and faces cost function C(r) = r^2 . Terrell charges pt = 80 and faces cost function C(t, r) = 5t^2 − 2rt. What is the socially optimal number of runners coached and therapy patients treated?

Answer 1

Profit of Rusty==pr*r-C(r)

Profit of Terrel==pt*t-C(t)

Combined profit=

Combined profit=

A social planner will try to choose r and t such that combined profit is maximized

So,

Set

120-2r+2t=0

4r=120+2t

r=60+t

Now,

Set r=60+t

Set

200-8t=0

t=25

Now we have derived that

r=60+t=60+25=85

So,

Socially optimal number of runners coached=r=85

Socially optimal number of therapy patients treated=t=25