Brad buys apples and bananas. His utility function is given by

U $($ a$,$ b $)=2$ a$\mathrm{.}375$ b$\mathrm{.}625$

The price of a lb$.$ of apples $=$ $$5/$lb$.$

The price of a lb$.$ of bananas $=$ $$5/$lb$.$

Brad has $$80$ to spend on apples and bananas.

a$.$ What is the equation of Brad$$s budget line?

b$.$ Solve for Brad$$s optimal bundle

c$.$ If a tax raises the price of apples to $$10/$lb$.,$ what will Brad$$s new optimal bundle

be$?$

d$.$ What share of Brad$$s budget was spent on apples before the tax? After the tax?

e$.$ How are Brad$$s optimal bundles related to his utility function?

$2.$ The American cigarette industry is an example of an industry where the most important

effect of advertising is to convince consumers who already smoke to switch brands. Two

cigarettes must decide whether it is worth it to bear the cost of advertising.

If they don$$t advertise, they earn a profit of $500.$

If both advertise, no brand switching occurs. Advertising costs $250$ so each earn a profit

of $250.$

If one advertises and the other doesn$$t$,$ the one that advertises earns $750,$ while the one

that doesn$$t$,$ earns $0.$

Call the $2$ firms Phillip Morris and Brown and Williamson.

a$.$ Write out the payoff matrix for the game.

b$.$ Do the two firms have dominant strategies to play? Explain.

c$.$ What is the Nash equilibrium of this game? Explain.

d$.$ Congress passed a law forbidding the advertising of cigarettes on TV effective

January $1,1971$ in order to protect people from $$dangerous messages$$ over the

airways. In what way did this law benefit cigarette manufacturers? Explain.