# Problem 5. Maximum 20 points

The commander of a tank battalion, in celebration of being awarded a new military rank, decided to invite soldiers to a tank festival, where the main delicacy is buckwheat porridge. The commander discovered that if the soldiers are lined up by height, there is a certain pattern in the change of their demand functions. The demand for buckwheat porridge of the shortest soldier is given by $Q d=510-5.1 P$, the next tallest soldier's demand is $Q d=520-5.2 P$, the next one's is $Q d=530-5.3 P$, and so on. The commander's individual demand for buckwheat porridge is $Q d=500-5 P$. The battalion commander always tries to feed as many soldiers as possible. Apart from the commander and the invited soldiers, no one else demands buckwheat porridge. At the festival, 25 perfectly competitive firms offer buckwheat porridge, with each firm's supply function being $Q s=302 P$. The battalion commander, after consulting with his family, decided to make a surprise - initially, the guests and the commander will make decisions about consuming buckwheat porridge, assuming that everyone will pay for it themselves, but at the end of the celebration, the commander's family will pay the total bill from their savings, which amount to 2,525,000 monetary units. If this amount is insufficient, the commander will ask all guests to equally share the remaining bill. It is known that the equilibrium price is set at 20 monetary units. Assume that if guests, like the commander, have to make a certain fixed payment, this payment does not affect their demand for buckwheat porridge.

(a) How is the individual demand of the 60th soldier invited to the celebration described? How will the aggregate demand for buckwheat porridge be described if the commander plans to invite 40 soldiers and does not refuse the porridge himself? Explain the construction of the demand function in this case.

(b) How many soldiers did the commander invite to the celebration? Did the guests have to pay part of the bill? If so, how much did each of them pay?
# Solution

(a) The commander tries to feed as many soldiers as possible, which means he will invite relatively short soldiers first - all other things being equal, their consumption of porridge 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 $P \in [0 ; 100]$, so to find the market demand, it is sufficient to sum the individual demand functions.

The individual demand of the 60th soldier is $Q_{d}=1100-11 P$. To find the demand of 40 soldiers, we can use the formula for the arithmetic progression:

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

$Q_{d}=\frac{1410-14.1 P}{2} \cdot 40$

$Q_{d}=28200-282 P$

Adding the commander's demand, we get the market demand: $Q_{d}=28700-287 P$.

Answer: $Q_{d}=1100-11 P, Q_{d}=28700-287 P$.

(b) 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 $Q_{s}(20)=Q_{d}(20)$, so $Q_{d}=7550 \cdot 20=151000$.

From the previous part, we know that

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

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

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

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

Obviously, $n>0$. 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.

## Grading Criteria

(a) Individual demand is formulated - (3p), market demand is formulated - (5p).

(b) Market supply function is formulated - (2p), equilibrium quantity is determined - (2p), market demand function in terms of the number of guests is formulated - (2p), maximum number of guests is determined - (3p), behavior of guests and their payment of the bill is determined - (3p).

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