The aggregate demand for gasoline in the city of Cambridge is given by D=200−20P and the aggregate gasoline supply is S=60P, where P is the price of gasoline and quantities are measured in gallons. To help tackle the issues of climate change, the city sets a bold goal to reduce aggregate gasoline consumption to 80% of its current level.

As a student taking 14.100x, you are asked to help the city of Cambridge come up with a good policy plan in order to achieve this goal. At the same time, you are also concerned about how you will be affected by any new policy since gasoline is in your consumption bundle. For simplicity, let's call all the remaining goods in your consumption "all other goods" (A). Your utility from gasoline (G) and all other goods (A) is U(G,A)=G−−√A−−√. The price of all other goods is $10/unit and your income is $100.

Solve for the current equilibrium price and quantity in the gas market.

P∗=  unanswered 

 

Q∗=  unanswered 

 

What is your optimal consumption bundle (A∗,G∗) at the equilibrium gas price you found?

A∗=  unanswered 

 

G∗=  unanswered 

 

What is your utility from this consumption bundle?

U(A∗,G∗)=  unanswered 

 

A tax policy can achieve the goal of reducing gas consumption to 80% of the equilibrium level found in the first question. Suppose the City makes the suppliers pay a tax of T dollars for each gallon of gas they sell. What size should T be in order to meet the City's target?

T=  unanswered 

 

What is the new equilibrium price?

P′∗=  unanswered 

 

What is your optimal consumption bundle (Atax,Gtax) under this tax policy?

Atax=  unanswered 

 

Gtax=  unanswered 

 

What is your utility from this consumption bundle?

  unanswered 

 

A price ceiling can also be used to keep the quantity supplied low. If Cambridge wants to achieve its goal of reducing gasoline consumption to 80% of the level found in the first question, what should the price ceiling be?

  unanswered 

 

An effective price ceiling will lead to excess demand as consumers demand more gas than firms are willing to supply at the market price. In order to solve this excess demand problem, the city of Cambridge implements rationing: it says that you cannot purchase more than Gtax in Question 4. What will your new optimal consumption bundle (Aceiling,Gceiling) be?

Aceiling=  unanswered 

 

Gceiling=  unanswered 

 

What is your utility from this consumption bundle?

  

unanswered 

 

Consider the tax policy you solved earlier (without the price ceiling or rationing). The government is concerned that this tax hurt consumers too much. In an attempt to mitigate gasoline consumers' losses, they rebate all the tax revenue to consumers. Your rebate is calculated as GtaxT, where Gtax is the amount of gasoline you purchased under the tax policy and T is the tax per gallon that the government collected from suppliers. (Both of these values you calculated earlier.)

What is the value of the rebate?

  unanswered 

 

Your new income, with the rebate you solved above, is 100+rebate, and so your new budget constraint is 4G+10A=100+rebate. Given the new budget constraint, what is your optimal consumption bundle (Arebate,Grebate) under the tax policy? (Assume the rebate was a one-time policy, and you won't get another rebate.)

Arebate=  unanswered 

 

Grebate=  unanswered 

 

What is your utility from this consumption bundle?

  unanswered