Question

You are an economic advisor to the Treasurer of the United States. Congress is considering increasing the sales tax on gasoline by $.05 per gallon. Last year motorists purchased 15 million gallons of gas per month. The demand curve is such that every $.01 increase in price decreases sales by 100,000 gallons per month. You also know that for every $.01 increase in price, producers are willing to provide 50,000 more gallons of gasoline to the market. The legislature has stated that the $.05 tax will increase government revenues by $750,000 per month and raise the price of gasoline by $.05 per gallon. Please explain why this is correct.

Given the information, the supply and demand curves can be described as follows:

Qs = 15,000,000 + 5,000,000(ps - p0)

Qd = 15,000,000 - 10,000,000(pd - p0)

Where do these formulas come from? Specifically, I don't understand where the 5 and -10 million come from.

Answer 1

The supply and demand equations are given as and .

Let us start deducing the given equations, and match the information given. For P0 be the previous (equilibrium) price, we may see that for P=P0, we would have and , ie the demand and supply are equal at 15,000,000 units. This is the same as last year purchase. This means that for the same price as before, the demand and supply would also be the same as before. This matches with the information.

The rate of change of supply with respect to price is , meaning that for increase in price by $1, the increase in quantity would be 5,000,000 units. This also means that, for increase in price by $1/100 or $0.01, the increase in quantity would be 5,000,000/100 units or 50,000 units. This also matches.

The rate of change of demand with respect to price is , meaning that for the increase in price by $1, the decrease in quantity would be 10,000,000 units, meaning that for the increase in price by $1/100 or $0.01, the decrease in quantity would be 10,000,000/100 units or 100,000 units. It matches again.

Imposing a tax on the sellers would change the supply curve as or . The new equilibrium would be where or or or or or . For T=$0.05, we have the price as . The quantity would be or or . The tax revenue would be dollars. Hence, the revenue would increase by $741,666.67, while the price would increase by $0.05/3, both not as stated.

The formulas are derived as below.

The decrease in demand by 100,000 for an increase in price by $0.01 would mean that for an increase in price by $0.01*100 or $1, the demand would increase by 10,000,000. Similarly, the increase in supply by 50,000 for an increase in price by $0.01 would mean that for an increase in price by $0.01*100 or $1, the supply would increase by 5,000,000. In this case, for a unit increase in price, the change in demand or supply is the slope, and hence, the slope of demand would be 10,000,000 and slope of supply would be 5,000,000.

The equation of a line is , where m is slope and (x1,y1) are any point on the line. The slope of the demand is -100,000*100 or -10,000,000, which is the decrease in Q for an increase in price by $0.01*100 or $1. Hence, for initial price and quantity be (P0,Q0) we have or or or , which is the stated equation of demand. Again, the slope of supply is 50,000*100 or 5,000,000, which is the increase in Q for an increase in price by $0.01*100 or $1. Hence, for the initial price and quantity, we have or or , which is the stated equation of supply.