Question 3. In many situations an economist may not have access to an actual demand function but does have a rough idea of the price elasticity of demand. This can be useful for making a back of the envelope estimate of the impact of a price change.

For example, suppose the current price of widgets is $4 with 1000 units sold per month. Further suppose that a recent empirical study has calculated the price elasticity of demand for widgets as -1.2 (often reported without the minus sign as 1.2). Many empirical studies assume a constant elasticity functional form for the demand function. This takes the form:

Qd = aP^e,

where e is the price elasticity of demand. With the widgets example, e equals -1.2.

Suppose a manager wants to know the impact of raising the price to $5. This can be solved by first solving for a. You have Qd and P, so you have the equation:

1000 = a*4^-1.2

You can then solve for a:

a = 1000/(4^-1.2) = 5278

This gives the demand function:

Qd = 5278*P^-1.2

Plug in P=$5, to get:

Qd = 765.

Your problem: A city is experiencing a drought and wants to reduce water usage by raising the price per 1000 gallons. Current monthly usage averages 5000 gallons per household, at a price of $3 per 1000 gallons. The price elasticity of demand for water is estimated to be -0.20. Calculate the average monthly household usage if the city raised its price to $4 per 1000 gallons. Show your work.