# Problem 4. 25 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. In the cryptogram, the amount of tax revenue to be collected was specified. 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 amount of tax revenue were determined in the wrong order. Based on the incorrect data, a decision was made to introduce a per-unit commodity tax on the consumer in the amount of 30 monetary units per unit of the good. It is known that the market supply is given by $Q_s = 6P - 312$, and the market demand is linear. In the situation where no taxes are imposed, at the equilibrium point, the price elasticity of market supply is 1.5 times higher than the absolute value of the price elasticity of the market demand function. 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 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 demand function be linear $Q_{d}=a-b P$. It is known that $1.5 \cdot\left|E_{p}^{d}\right|=E_{p}^{s}$. For linear demand functions, using the definition of elasticity, we get: $1.5 \cdot \frac{b P_{e}}{Q_{e}}=\frac{6 P_{e}}{Q_{e}}$. From this, we find that $b=4$.

If a per-unit tax $t=30$ is introduced, then $P_{d}=118 . a-4 P_{d}=6\left(P_{d}-30\right)-312 ; 0.1 a+$ $49.2=P_{d}=118$; from which $a=688$.

The market demand function is $Q_{d}=688-4 P$.

2) It is known that $P_{d}(t=30)=118$. Therefore, $Q_{d}=688-472=216, T=216 \cdot 30=6480$.
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 revenues are $\quad T=Q \cdot t=288 t-2.4 t^{2}$. The graph of the function is a parabola opening downwards, 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 demand curve - 3 points.

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

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

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

Correctly found the demand 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 item even if the rest of the solution is correct.

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

Found the rate that maximizes tax revenues - 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 that will be sold at the rate that maximizes tax revenues - 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 revenues at this tax rate - 2 points. If the tax revenues are found but their value does not align with the condition in the problem about the 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 item 3) gives a maximum, not a minimum of the objective function, is not justified.