# Task 5.

## Maximum 10 points

In the Kingdom of Consumerland, chips are sold in a perfectly competitive market. King Consumerland not only wants to reduce the consumption of chips in his country but also increase the treasury's revenue. To avoid making a mistake in choosing the tax for chip producers, the king ordered his ministers to gather data on the market. The information they provided to the king is as follows:

- Demand and supply can be represented by linear functions and are stable over time.
- With a per-unit tax for producers at a rate of 4.5 monetary units per weight unit of chips, the treasury is replenished by tax revenues of 22.5 monetary units of Consumerland.
- If the tax were doubled, the treasury's revenue from taxation would increase by 60%.
- In the times when chip production was not taxed, the consumption of chips was 1.2 times higher than at a tax rate of 4.5 monetary units.
- The residents of Consumerland have a positive demand for chips only at a price lower than 20 monetary units.

Can the king, based on this information, set a tax that will maximize the treasury's revenue from taxing chip producers? If yes, how much will be collected by the treasury?

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# Solution and Evaluation Criteria:

Let the inverse demand function for chips be: $P^{D}(Q)=a-b Q$, and the inverse supply function for chips: $P^{S}(Q)=c+d Q$.

Since with a per-unit tax of 4.5 monetary units, the tax revenue amounted to 22.5 monetary units, in equilibrium, 22.5 / 4.5 = 5 weight units of chips were consumed in this case. (maximum 1 point)

Similarly, we can find how many chips were consumed with a tax of $2 * 4.5 = 9$ monetary units:

## $22.5 * 1.6 / 9 = 4$ weight units. (maximum 1 point)

Before the introduction of taxes, according to the condition, 1.2 * 5 = 6 weight units were consumed in equilibrium. (maximum 1 point)

Considering the demand and supply functions, let's find how equilibrium on the market depends on the parameters of these functions before and after the introduction of the tax.

Before the introduction of the tax: $P^{D}(Q)=a-b Q^{*}=P^{S}(Q)=c+d Q^{*}$, from which $Q^{*}=(a-c) /(b+d)$ (maximum 1 point)

After the introduction of the per-unit tax, in equilibrium, the difference between the price for buyers and the price sellers receive for their goods is the tax amount $t$, so: $P^{D}(Q)=a-b Q^{t}-t=P^{S}(Q)=c+d Q^{t}$, from which $Q^{t}=(a-c-t) /(b+d)$ (maximum 1 point)

Let's find the tax amount that will maximize tax revenues in the budget of Consumerland. For this, we need to find the maximum of the function: $T=t Q^{t}=t(a-c-t) /(b+d)$, which represents a downward-opening parabola in (Q,T) coordinates. The maximum of such a function will be achieved at $Q^{t}=(a-c) / 2(b+d)$ with a tax of $t=(a-c) / 2$. (maximum 4 points)

It can be noted that with the introduction of a tax that maximizes its revenue, the equilibrium quantity of the good will be halved compared to what it was before the tax and will become equal to 6 / 2 = 3 weight units. (maximum 1 point)

Considering the linearity of demand and supply (graphically, one can use the properties of similar triangles), the introduction of a tax proportionally reduces the equilibrium quantity of the good. That is, if with a tax of 9 monetary units, the equilibrium quantity decreased by 1.5 times compared to the equilibrium quantity without the tax, then to achieve an equilibrium quantity of 3 weight units, a tax of 9 * 1.5 = 13.5 monetary units must be introduced. (maximum 4 points)

Then the revenue from such a tax, maximizing the budget revenues, will amount to 13.5 * 3 = 40.5 monetary units. (maximum 1 point)

Each arithmetic error in the solution is penalized by a reduction of 1 point in the score.