# Task 5. 20 points

A beginner economist-cryptographer received a cryptogram from the ruler, which contained another secret decree on the introduction of a commodity tax on a certain market. The cryptogram specified the amount of tax revenue to be collected. It was also emphasized that a larger amount of tax revenue could not be collected on this market. Unfortunately, the economist-cryptographer decrypted the cryptogram with an error - the digits in the tax revenue amount were determined in the wrong order. Based on the incorrect data, a decision was made to introduce a per-unit tax on the producer in the amount of 90 monetary units per unit of the product. It is known that the market demand is given by $Q_d = 688 - 4P$, and the market supply is linear. Additionally, it is known that a change in price by one unit results in a change in the quantity demanded that is 1.5 times smaller than the change in the quantity supplied. After the tax was introduced, the producer's price decreased to 64 monetary units.

1) Restore the market supply function.
2) Determine the amount of tax revenue collected at the chosen rate.
3) Determine the rate of the quantity tax that would allow the ruler's decree to be fulfilled.
4) What are the tax revenues that the ruler indicated to collect?
# Solution:

1) Let the supply function be linear $Q_{s}=c+d P$. It is known that $1.5 \cdot 4=d$. We find that $d=6$. If a per-unit tax $t=90$ is introduced, then $P_{s}=64.688-4\left(P_{s}+90\right)=6 P_{s}+c$; $0.1 c+32.8=P_{s}=64 ; c=-312$. The market supply function is $Q_{s}=6 P-312$. (8 points).
2) It is known that $P_{s}(t=90)=64$. Therefore, $Q_{s}=6 P-312=72, T=72 \cdot 90=6480$. (4 points).
3) Let $P_{s}=P_{d}-t, 688-4 P_{d}=6 P_{d}-6 t-312, P_{d}=100+0.6 t ; \quad Q_{d}=288-2.4 t$. Tax revenues are $T=Q \cdot t=288 t-2.4 t^{2}$. This is a downward-opening parabola, and the maximum of the function is achieved at $t^{*}=60$. (4 points).
4) $T_{\max }=288 \cdot 60-2.4 \cdot 60 \cdot 60=8640$. (4 points).

Penalties: An arithmetic error was made -2 points, lack of justification for the maximum -5 points.