7. In his country of Milnlandia, Winnie-the-Pooh decided to open a company that produces honey. Winnie-the-Pooh sells honey only in pots, and it costs him 10 milnovs (the monetary units of Milnlandia) to produce any pot of honey. The inverse demand function for honey is given by $\boldsymbol{P}=310-3 \boldsymbol{Q}$ (where $\boldsymbol{Q}$ is the number of pots, and $\boldsymbol{P}$ is the price of one pot). There are no other honey suppliers.

a) How many pots of honey will Winnie-the-Pooh produce if his main goal is to maximize profit?

b) The government of Milnlandia decided to introduce a tax $\boldsymbol{t}$ milnovs on each pot of honey sold by Winnie-the-Pooh's company. What should be the amount of the tax to maximize the tax revenue for the budget of Milnlandia? (17 points).
# Solution:

a) Profit $=P(Q) \cdot Q - TC(Q) = (310 - 3Q) \cdot Q - 10 \cdot Q = 310Q - 3Q^2 - 10Q = 300Q - 3Q^2$.

Since the graph of the function $300Q - 3Q^2$ is a parabola opening downwards, its maximum is achieved at the vertex: $Q = -b / 2a = -300 / (-6) = 50$.

b) Winnie-the-Pooh maximizes the quantity $P(Q) \cdot Q - TC(Q) = (310 - 3Q) \cdot Q - 10 \cdot Q - tQ = (300 - t)Q - 3Q^2$, $Q = (300 - t) / 6$. The state receives a monetary amount in the form of tax revenue $Q \cdot t = (300 - t) / 6 \cdot t = (50 - 1/6 \cdot t) \cdot t = -1/6 \cdot t^2 + 50t$. The graph of this function is a parabola opening downwards, its maximum is achieved at $t = -50 / (-2/6) = 50 \cdot 6 / 2 = 50 \cdot 3 = 150$.

Answer: a) 50; b) 150.

## Grading Criteria:

a)

1) The problem of profit maximization for a single producer is correctly formulated: 3 points
2) The solution to the profit maximization problem is correctly found and justified: 4 points. The absence of justification for the maximum value at the extremum point results in a 1-point deduction.

b)

1) The consequences of introducing a profit tax for a single producer are correctly accounted for: 3 points
2) The tax amount is correctly found: 7 points. If there is an arithmetic error in calculating the tax, 6 points are given.