$2.(20$ points$)$ Arnold and Betty have identical marginal benefit formulas for a public good. The MBs are given by :

$\backslash$alpha and $\backslash$beta are constant positive numbers, $\backslash$alpha $<1.$ YA$,$ YB$,$ are incomes of Arnold and Betty. qA$,$ qB are purchases of the public good by Arnold and Betty.

The marginal cost of a unit of the public good is constant: MC$=$C$.$

a$.$ As in lecture, determine the best response functions for Arnold and Betty.

Suppose now that $\backslash$alpha $=\mathrm{.}5,\backslash$beta $=5,$ YA$=$YB$=200,$ and MC$=10.$

b$.$ What is the strategic equilibrium quantity of the public good that gets produced?

c$.$ What is the socially optimal quantity of the public good that gets produced?

d$.$ Claim: as the slope of the MB curve gets steeper, the difference between the strategic equilibrium and the socially optimal quantity increases. Agree or disagree. How do the formulas help you answer the question? In words briefly explain the economics of this result.