6. The supply of a certain good on a competitive market can be described by the function $\boldsymbol{Q}^{S}(\boldsymbol{p})=2+8 \boldsymbol{p}$ (where $\boldsymbol{Q}$ is the volume of sales, and $\boldsymbol{p}$ is the price per unit of the product). It is known that at a price of 2 monetary units, consumers were willing to purchase no more than 8 units of the product, and at a price of 3 monetary units, no more than 6 units of the product, with the demand curve being linear.

a) What will be the price of the product, and what volume will be sold at equilibrium?

b) To increase consumption of this product, the government introduced a per-unit subsidy for its producers in the amount of $s=1$ for each unit of the product produced. By how many units will the sales volume increase? (14 points).
# Solution:

Let's find the demand function. Let $Q^{D}=\boldsymbol{k} \boldsymbol{p}+\boldsymbol{b}$. We have $8=2 \boldsymbol{k}+\boldsymbol{b}, 6=3 \boldsymbol{k}+\boldsymbol{b}$. We get $k=-2, b=12$, i.e., $Q^{D}(p)=-2 p+12$.

a) Market equilibrium: $Q^{D}(\boldsymbol{p})=Q^{S}(p),-2 p+12=2+8 p, p^{*=1,} Q^{*}=10$.

b) $\boldsymbol{s}=\mathbf{1}$, i.e., $\mathrm{Q}^{\mathrm{S}}(\mathrm{p})=2+8(\mathrm{p}+1)$. We have $\boldsymbol{Q}^{D}(\boldsymbol{p})=\boldsymbol{Q}^{S}(\boldsymbol{p})$, i.e., $\boldsymbol{p}^{* *=2 / 10=0.2, Q^{* *}=\mathbf{11.6}}$.

Then the sales volume increased by $11.6-10=1.6$ units.

Answer: a) $p^{*}=1, Q^{*}=10$; b) by $1.6$ units.

## Grading Criteria:

a)

1) Correctly found the demand function: 4 points
2) Correctly found the equilibrium (price and quantity of goods): 3 points. If an arithmetic error was made in finding the equilibrium, 2 points are given
b)
3) Correctly accounted for the market's reaction to the introduction of the subsidy: 4 points
4) Correctly found the new equilibrium quantity of goods: 2 points. If an arithmetic error was made in finding the equilibrium sales volume, 1 point is given.
5) Correctly calculated the change in sales volume: 1 point