6. In a perfectly competitive market, the demand function for a certain good is $\mathrm{Q}_{\mathrm{d}}(\mathrm{p})=150-\mathrm{p}$, and the supply function for this good is: $\mathrm{Q}_{\mathrm{s}}(\mathrm{p})=3 \mathrm{p}-10$. As a result of a sharp increase in the number of consumers of this good, the demand for it increased by a factor of $\alpha$ at every possible price level, under all other equal conditions. As a result, the price of the good increased by $25 \%$. Find the value of $\alpha$. (8 points)
# Solution:

The new demand function will be $Q_{d}^{\text {new }}(p)=a(150-p)$.

Find the new equilibrium price from the condition of equality of demand and supply: $3 \mathrm{p}-10=\mathrm{Q}_{s}(\mathrm{p})=\mathrm{Q}_{d}^{\text {new }}(\mathrm{p})=a(150-\mathrm{p}): \quad \mathrm{p}^{\text {new }}=\frac{150 a+10}{3+a}$. Note that the initial price $\mathrm{p}_{0}=40$ (when $\alpha=1$). Then, knowing that the price increased by $25 \%$, we write the equation $\mathrm{p}^{\text {new }}=\frac{150 \alpha+10}{3+\alpha}=40 \cdot(1+0.25)=50$. We get, $a=1.4$.

## Grading Criteria:

Correctly found the old equilibrium price - 2 points.

Correctly formulated the equation to find $\alpha$ - 4 points.

Correctly found the value of $\alpha$ - 2 points.