# Assignment 5.

## Maximum 15 points

In a small state called Konyr, the beloved good "ravot" is sold in a perfectly competitive market. The Ministry of Economic Development of this state is conducting a study on the state of the economy and is collecting data on various markets, including the market for "ravot." Analysts from the ministry have gathered the following information about the "ravot" market:

- Demand and supply are always represented by linear functions and change only seasonally. Demand varies in each season, while supply changes only twice a year. In spring and autumn, supply is the same as in summer.
- When the demand for "ravot" is halved in spring compared to the demand in summer, the volume of its sales on the market decreases by one and a half times.
- In spring, the total revenue of "ravot" producers is 90 monetary units of Konyr.
- In winter, when supply is halved compared to the other seasons and demand falls by half compared to the spring demand, the quantity of "ravot" sold on the market is only 5 units of this good.
- At any time of the year, producers of "ravot" would be willing to sell it only if its price was more than 4 monetary units. At a price of 4 or less, producers do not produce "ravot."

You need to help the Ministry of Economic Development of Konyr answer the following questions from its government:

1) If the price of "ravot" is "frozen" at the winter price level for the entire year, what will be the shortage of this good in the summer?
2) If there is no intervention in the pricing of the "ravot" market, what will be the total expenditures of Konyr residents on the purchase of this good in the summer?
3) At what minimum price of "ravot" will Konyr residents refuse to buy it? Does this price depend on the time of year?

#
# Solution and Evaluation Criteria:

Let the demand function in spring be: $Q_{\text {spring }}^{D}(p)=a-b p$, then the demand functions in winter and summer are, respectively: $Q_{\text {winter }}^{D}(p)=(a-b p) / 2$ and $Q_{\text {summer }}^{D}(p)=2(a-b p)$.

Let the supply function in spring, summer, and autumn be: $Q_{\mathrm{B}, \mathrm{N}, 0}^{s}(p)=c+d p$, then the supply function in winter is $Q_{\text {winter }}^{s}(p)=(c+d p) / 2$.

Note that the equilibrium price of the good in spring and winter was the same, as the same value of the variable - price - is the solution to both equations:

$$
\begin{gathered}
Q_{\text {spring }}^{D}(p)=a-b p_{\text {spring }}=Q_{\text {B, }, 0}^{s}(p)=c+d p_{\text {spring }} \\
Q_{\text {winter }}^{D}(p)=\frac{a-b p_{\text {winter }}}{2}=Q_{\text {winter }}^{s}(p)=\frac{c+d p_{\text {winter }}}{2}
\end{gathered}
$$

Since demand and supply in spring were twice as high as in winter, the equilibrium quantity sold in spring was twice that of winter. Given that 5 units of the good were sold in winter, 10 units were sold in spring. Therefore, the sales volume of the good in summer is $1.5 * 10 = 15$.

It is known that the total revenue of producers in spring was 90 monetary units, so the price of the good in spring was $90 / 10 = 9$ monetary units. The price in winter, as we have already determined, was also 9.

Now, using the equilibrium data for winter and spring, we will find the supply functions for each period, considering that producers will not sell the good at a price of 4 or lower:

$Q_{\mathrm{B}, \mathrm{n}, \mathrm{o}}^{s}(9)=c+9 d=10$,

$Q_{\mathrm{B}, \pi, 0}^{s}(4)=c+4 d=0$.

From this, we find: $c=-8, d=2$ and $\quad Q_{\mathrm{B},, 0}^{s}(p)=2 p-8, \quad Q_{\text {winter }}^{s}(p)=p-4$

Considering that 15 units of the good will be sold in summer, we find the equilibrium summer price from the supply function:

$Q_{\mathrm{B}, \mathrm{N}, \mathrm{O}}^{S}(p)=2 p_{\text {summer }}-8=15$ and $p_{\text {summer }}=11.5$

Now we will find the demand functions, given the known equilibrium quantities and prices in spring and summer:

$Q_{\text {spring }}^{D}(9)=a-9 b=10$,

$Q_{\text {summer }}^{D}(11.5)=2(a-11.5 b)=15$.

From this, we find that $a=19, b=1$. Therefore, $Q_{\text {spring }}^{D}(p)=19-p$ and $Q_{\text {summer }}^{D}(p)=2(19-p)$.

Now we can answer the questions in the assignment:

1) If the price of the good is "frozen" at the winter price, the quantity demanded in summer will be 2(19-9)=20 units. The supply (as previously found) is only 10 units. Therefore, the shortage of the good will be $20-10=10$ units. (Maximum 2 points)
2) To determine the residents' expenses on purchasing the good in summer, we need to multiply the price of the purchased good by its quantity. In summer, 15 units of the good were sold at a price of 11.5. Therefore, the expenses on purchasing the good in summer were $11.5 * 15 = 172.5$ monetary units. (Maximum 2 points)
3) Residents will refuse to buy the good, as can be seen from the found demand functions, if the price of the good is 19 monetary units or more. Therefore, the required price level is 19. This price does not depend on the time of year. (Maximum 2 points)

For a logically justified solution and correct determination of the supply functions, a maximum of 5 points.

For a logically justified solution and correct determination of the demand functions, a maximum of 4 points.

Each arithmetic error in the solution is penalized by a reduction of 1 point.