# 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 it was impossible to collect a larger amount of tax revenue on this market. Unfortunately, the economist-cryptographer decrypted the cryptogram with an error - the digits in the amount of tax revenue were determined in the wrong order. Based on the incorrect data, a decision was made to introduce a per-unit tax on the consumer in the amount of 30 monetary units per unit of the good. The market supply is given by $Q_s = 6P - 312$, and the market demand 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 consumer price increased to 118 monetary units.

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

1) Let the demand function be linear $Q_{d}=a-b P$. It is known that $1.5 b=6$. We find that $b=$ 4. If a per-unit tax $t=30$ is introduced, then $P_{d}=118 . a-4 P_{d}=6\left(P_{d}-30\right)-312 ; 0.1 a+$ $49.2=P_{d}=118 ; a=688$. The market demand function is $Q_{d}=688-4 P$. (8 points).
2) It is known that $P_{d}(t=30)=118$. Therefore, $Q_{d}=688-472=216, T=216 \cdot 30=6480$. (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, the maximum of the function is reached 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.