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Consider two consumers, LaTeX: h=1,2h = 1 , 2,with preferences LaTeX: U^h=\ln\left(x^h ight)+\ln\left(G ight)U h = ln ( x h ) + ln ( G ) where LaTeX: x^hx h is consumption of a private good and LaTeX: GG consumption of a public good. Consumer LaTeX: hh has income LaTeX: M^hM h andpays a fraction LaTeX: \tau^h h of the cost of the public good. So, if amount LaTeX: GG of the public good is provided the budget constraint of LaTeX: hh isLaTeX: M^h=x^h+\tau^hGM h = x h + h G.

(i) If LaTeX: M^h=10M h = 10 and LaTeX: \tau^h=0.25 h = 0.25, how many units of public good does consumer 1 demand?

(ii) The fractions LaTeX: \tau^1 1 and LaTeX: \tau^2 2 must sum to 1. If LaTeX: M^1=10M 1 = 10 and LaTeX: M^2=15M 2 = 15, what faction LaTeX: \tau^2 2 ensures that both consumers demand the same quantity of the public good?

(iii) What is the quantity of public good provided for the shares found in (ii)?

(iv) Now let the quantity of public good be chosen to maximize the weighted utilitarian social welfare function LaTeX: W=U^1+\beta U^2.W = U 1 + U 2 . If the cost of the public good is shared equally between the two consumers, what value of LaTeX: \beta makes the quantity found in (iii) optimal?