Annual sales of new cars and light trucks in a certain country from 2015–2019 can be approximated by the following when the price (in constant 2021 dollars) was $p thousand.
q = −0.15p + 73 million vehicles sold
(a)
If car dealers were prepared to supply
q = 3.85p − 161
million vehicles at a sale price of $p, what would have been the equilibrium price (in dollars)?
$
(b)
The actual average sale price in constant 2021 dollars was $58,600 in 2019. Estimate the projected shortage or surplus (in million of vehicles) at that price.
There is an estimated
---Select---
of
million vehicles.

We are given two relationships between price ($p$) and the quantity of vehicles sold ($q$) in millions:

1. **Demand function**: $ q = -0.15p + 73 $
2. **Supply function**: $ q = 3.85p - 161 $

### Part (a): Find the equilibrium price

The equilibrium price occurs when the quantity demanded equals the quantity supplied. This means we set the two equations for $q$ equal to each other:

\[
-0.15p + 73 = 3.85p - 161
\]

Now, solve for $p$.

1. Combine like terms:
\[
73 + 161 = 3.85p + 0.15p
\]
\[
234 = 4p
\]

2. Divide both sides by 4:
\[
p = \frac{234}{4} = 58.5
\]

So, the equilibrium price is **$58,500**.

### Part (b): Estimate the projected shortage or surplus at the actual average sale price of $58,600.

To find whether there is a shortage or surplus, we first calculate the quantity demanded and supplied at $p = 58.6$.

- **Demand at $p = 58.6$**:
  \[
  q_d = -0.15(58.6) + 73 = -8.79 + 73 = 64.21 \text{ million vehicles}
  \]

- **Supply at $p = 58.6$**:
  \[
  q_s = 3.85(58.6) - 161 = 225.61 - 161 = 64.61 \text{ million vehicles}
  \]

Now, compare the supply and demand:

\[
q_s - q_d = 64.61 - 64.21 = 0.4 \text{ million vehicles}
\]

Since the supply is greater than the demand, this indicates a **surplus** of 0.4 million vehicles.

### Final Answer:
- (a) The equilibrium price is **$58,500**.
- (b) There is an estimated **surplus** of **0.4 million vehicles** at a sale price of $58,600.

Would you like more details on the solution process or clarification on any steps?

Here are 5 related questions to think about:
1. How would the equilibrium change if the demand curve shifted due to a major economic event?
2. What happens to the equilibrium price if the supply curve becomes less steep?
3. How would a price ceiling affect the market if set below the equilibrium price?
4. How would inflation adjust the constants in these equations if calculated in future dollars?
5. What other factors could affect car sales that aren't captured in this simple model?

**Tip**: Always check if both demand and supply curves intersect, as equilibrium can only occur at that point.

Annual sales of new cars and light trucks in a certain country from 2015–2019 can be approximated by the following when the price (in constant 2021 dollars) was $p thousand.
q = −0.15p + 73 million vehicles sold
(a)
If car dealers were prepared to supply
q = 3.85p − 161
million vehicles at a sale price of $p, what would have been the equilibrium price (in dollars)?
$
(b)
The actual average sale price in constant 2021 dollars was $58,600 in 2019. Estimate the projected shortage or surplus (in million of vehicles) at that price.

This problem involves calculating the equilibrium price and estimating the vehicle surplus or shortage based on linear demand and supply functions. The solution uses algebraic methods and is suitable for high school students in Grades 10-12.

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