Question

4. Suppose that the inverse demand function for movies is p = 120 – Q1 for college students and p = 120 – 2Q2 for other town residents. What is the town’s total demand function (Q = Q1 + Q2 as a function of p)?

Answer 1

The total demand function for the town is the sum of the demand functions for the two groups.

Let suppose that $P=100$. Then, from the first and second demand functions we get:

$$Q_1=120-100=20,$$ $$Q_2=\dfrac{120-100}{2}=10.$$

Therefore, $Q=Q_1+Q_2=20+10=30.$

Then, let's do the same calculations for $P=60$:

$$Q_1=120-60=60,$$ $$Q_2=\dfrac{120-60}{2}=30,$$ $$Q=Q_1+Q_2=60+30=90.$$

As we can see from calculations, we obtain two points on total demand curve: ($x_1$, $y_1$) and ($x_2$, $y_2$) or (30, 100) and (90, 60), respectively. Let's use the point slope formula to find the total demand function:

$$y-y_1=m(x-x_1),$$

here, $y=P$, $y_1=100$, $x=Q$, $x_1=30$, $m$ is the slope of the line.

So, we can rewrite the equation:

$$(P-100)=m(Q-30).$$

The slope of the line can be found as follows:

$$m=\dfrac{\Delta y}{\Delta x}=\dfrac{60-100}{90-30}=-\dfrac{40}{60}.$$

Then, substituting the slope into the equation of the line, we get:

$$(P-100)=-\dfrac{40}{60}(Q-30),$$ $$3P-300=60-2Q,$$ $$P=120-0.67Q.$$

Answer:

$P=120-0.67Q.$