# Problem 5. Maximum 20 points

The commander of a tank battalion, in celebration of being awarded a new military rank, decided to organize a mass celebration, inviting subordinate soldiers. Only the soldiers whom the commander personally invites can attend. The main delicacy at the celebration is buckwheat porridge. However, after consulting with his family, the tank battalion commander decided to make a surprise - initially, guests, as well as the commander himself, will make decisions about consuming buckwheat porridge, assuming that they will pay for it themselves, but at the end of the celebration, the commander will pay the total bill from his family's savings. He discovered that if the soldiers are lined up by height, there is a certain pattern in the changes in their demand functions. The demand for buckwheat porridge of the shortest soldier is given by \( Q_d = 510 - 5.1P \), the next tallest \( Q_d = 520 - 5.2P \), the next \( Q_d = 530 - 5.3P \), and so on. The commander will first invite the shortest tankers. The individual demand of the commander for buckwheat porridge is given by \( Q_d = 500 - 5P \).

(a) How is the individual demand of the 45th soldier invited to the celebration defined? How will the aggregate demand for buckwheat porridge be defined if the commander invites 45 soldiers and does not refuse the porridge himself?

(b) The commander can invite the soldiers to a local cafe, the only one in the region. The marginal costs of producing buckwheat porridge are 0. The commander's family is willing to spend no more than 2,525,000 monetary units on the celebration. What is the maximum number of guests whose demand for buckwheat porridge will be satisfied at the celebration if the commander eats the porridge with everyone?

(c) After estimating how many soldiers he can feed in the cafe, the battalion commander decided to invite the military personnel to a tank festival. In addition to the commander and the soldiers he invites, no one else demands porridge. Buckwheat porridge at the festival is offered by 25 perfectly competitive firms, each with a supply function \( Q_s = 302P \). The commander's family is willing to spend the entire amount allocated for the celebration, but if it is not enough, the commander will ask all guests to equally share the remaining bill. It is known that the equilibrium price was set at 20 monetary units. How many soldiers did the commander invite to the festival? Did the guests have to pay part of the bill? If so, how much did each of them pay? Assume that if the guests, like the commander, have to make a certain fixed payment, the payment does not affect their demand for buckwheat porridge.

(d) Which option (b or c) would the soldiers prefer if each of them made a decision about the amount of buckwheat porridge to consume according to their demand function as described in the problem? Justify your answer.
# Solution

(a) The commander tries to feed as many soldiers as possible, which means he will primarily invite relatively short soldiers - all other things being equal, their porridge consumption is less.

Note that the individual demand of soldiers is determined by the formula $Q_{d}=500+10 n-(5+0.1 n) P$, where $n$ is the ordinal number of the soldier. Also note that all consumers are willing to purchase the product at $P \in [0; 100]$, which means that to find the market demand, it is sufficient to sum the individual demand functions.

The individual demand of the 45th soldier is $Q_{d}=950-9.5 P$. To find the demand of 45 soldiers, we can use the formula for the arithmetic progression:

$Q_{d}=\frac{(510-5.1 P + 500 + 10 n - (5 + 0.1 n) P)}{2} \cdot n$

If $n=45$, then:

$Q_{d}=32850-328.5 P$

Adding the commander's demand, we get the market demand: $Q_{d}=33350-333.5 P$.

Answer: $Q_{d}=950-9.5 P, Q_{d}=33350-333.5 P$.

(b) For the canteen, the profit maximization problem coincides with the revenue maximization problem, since marginal costs are zero. If the commander invites $n$ soldiers, then the market demand will be:

$Q_{d}=500-5 P + \frac{1010 n + 10 n^2 - (10.1 n + 0.1 n^2) P}{2}$

$Q_{d}=5 n^2 + 505 n + 500 - (0.05 n^2 + 5.05 n + 5) P$

$T R=(5 n^2 + 505 n + 500) P - (0.05 n^2 + 5.05 n + 5) P^2$

The graph of $T R$ is a parabola opening downwards, since for $n \geq 0$ it holds that $0.05 n^2 + 5.05 n + 5 > 0$. The maximum value is at the vertex of the parabola:

$$
\begin{aligned}
& P^{*}=\frac{-5 n^2 - 505 n - 500}{-2(0.05 n^2 + 5.05 n + 5)}=50 \\
& T R_{\max }=T R(50)=125 n^2 + 12625 n + 12500 \\
& T R_{\max } \leq 2525000 \\
& 125 n^2 + 12625 n + 12500 - 2525000 \leq 0 \\
& n^2 + 101 n - 20100 \leq 0 \\
& (n + 201)(n - 100) \leq 0 \\
& n_{1}=-201, n_{2}=100
\end{aligned}
$$

Obviously, the number of guests cannot be negative or a non-integer, so the maximum number of guests the commander can invite is 100.

Answer: 100.

(c) Knowing the individual supply of one firm, we can find the market supply: $Q_{s}=302 P \cdot 25=7550 P$

It is known that $Q_{s}(20)=Q_{d}(20)$, which means $Q_{d}=7550 \cdot 20=151000$.

From the previous part, we know that

$Q_{d}=5 n^2 + 505 n + 500 - (0.05 n^2 + 5.05 n + 5) P$

$Q_{d}(20)=4 n^2 + 404 n + 400=151000$

$(4 n + 1004)(n - 150)=0$

$n_{1}=-251, n_{2}=150$

The commander invited 150 guests.

$T R=151000 \cdot 20=3020000>2525000$ - the expenses for buckwheat porridge exceed the family budget, which means the soldiers will help pay part of the bill. Each soldier will contribute an amount of $\frac{495000}{150}=3300$ monetary units.

Answer: the commander invited 150 guests, each of whom paid 3300 monetary units.

(d) Let's find the consumer surplus (CS) for the soldier with number $i$ in each case. For a linear demand function, $C S=\frac{(P_{\max }-P_{e}) Q}{2}$. The demand function of each soldier is linear, and $P_{\max }=100$.

In part (b), consumers do not pay for the buckwheat porridge, i.e., $P_{e}=0$.

$Q_{i}^{1}=500+10 i-(5+0.1 i) \cdot 50=250+5 i$

$C S_{i}^{1}=(P_{\max }-P_{e}) \cdot 0.5 \cdot Q_{1}^{i}=(100-0) \cdot 0.5(250+5 i)=12500+250 i$.

In part (c), consumers do not pay for the buckwheat porridge directly, but spend 3300 monetary units.

$Q_{i}^{2}=500+10 i-(5+0.1 i) \cdot 20=400+8 i$

$C S_{i}^{2}=(100-0) \cdot 0.5(400+8 i)-3300=16700+400 i$

It is clear that even with the expenses for partial payment of the bill, for any $i \geq 0$ it holds that $C S_{i}^{2}>C S_{i}^{1}$.

Answer: all soldiers prefer the option (c).

## Grading Criteria

(a) Individual demand is formulated - (1p), market demand is formulated - (2p).

(b) The firm's problem is formulated - (1p), the optimal price is determined - (2p), the maximum number of guests is determined - (3p).

(c) The market supply function is formulated - (1p), the equilibrium quantity is determined - (1p), the market demand function in terms of the number of guests is formulated - (2p), the maximum number of guests is determined - (2p), the behavior of guests and their payment of the bill is determined - (2p).

(d) A justified comparison of benefits is formulated, from which a conclusion is drawn about the preference of a certain option - (3p).

A penalty of 1 point for arithmetic errors that did not lead to significant distortion of the results. Alternative solutions may be evaluated for the full number of points if they contain a correct and justified sequence of actions.