# Task 4. 25 points

A novice economist-cryptographer received a cryptogram from the ruler, which contained another secret decree on the introduction of a commodity tax on a certain market. In the cryptogram, the amount of tax revenue to be collected was specified. 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 producer in the amount of 90 monetary units per unit of the product. It is known that the market demand is given by $Q d=688-4 P$, and the market supply is linear. In the situation where there are no taxes, the price elasticity of market supply at the equilibrium point is 1.5 times higher than the absolute value of the price elasticity of the market demand function. 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 $1.5 \cdot\left|E_{p}^{d}\right|=E_{p}^{s}$. Using the definition of price elasticity for linear demand functions, $1.5 \cdot$ $\frac{4 P_{e}}{Q_{e}}=\frac{d P_{e}}{Q_{e}}$. We find that $d=6$. If a per-unit tax $t=90$ is introduced, then $P_{s}=64$. Therefore, $688-4\left(P_{s}+90\right)=6 P_{s}+c ; 0.1 c+32.8=P_{s}=64 ; c=-312$.

The market supply function is $Q_{s}=6 P-312$.

2) It is known that $P_{s}(t=90)=64$. Therefore, $Q_{s}=6 P-312=72$. Then the government revenue is $T=72 \cdot 90=6480$.
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 revenue will be $T=Q \cdot t=288 t-2.4 t^{2}$. The graph of the function is a parabola opening downwards, and the maximum of the function is achieved at $t^{*}=60$.
4) $T_{\max }=288 \cdot 60-2.4 \cdot 60 \cdot 60=8640$.

## Grading Criteria:

1) Correctly determined the slope of the supply curve - 3 points.

Correctly determined the purchase price with the incorrect tax - 1 point.

Correctly determined the sales volume with the incorrect tax - 2 points.

Correctly determined the intercept of the supply curve - 3 points.

Correctly found the supply function - 1 point.

If an error/typo is made in the solution that leads to results contradicting economic logic, only 3 points are awarded for this section even if the rest of the solution is correct.

2) Correctly found the tax revenue - 3 points.
3) Explained how to find the rate that maximizes tax revenue - 4 points.

Found the rate that maximizes tax revenue - 4 points. If the rate found contradicts logic, even if the value of $t$ is correct, these 4 points are not awarded.

4) Found the quantity of goods sold at the rate that maximizes tax revenue - 2 points. If the quantity found contradicts logic, even if the value of $t$ is correct, these 2 points are not awarded. Found the tax revenue at this tax rate - 2 points. If the tax revenue is found but does not align with the condition in the problem about mixed-up digits, these 2 points are not awarded.

## Penalties:

An arithmetic error was made in calculations that did not lead to results contradicting economic logic -1 point.

The fact that the tax rate found in section 3) gives a maximum, not a minimum, of the objective function is not justified.