Suppose that a market is initially in equilibrium. The initial demand curve is $P = 90 - Q ^ { d }$. The initial supply curve is $P = 2 Q ^ { s }$. Suppose that the government imposes a $\$ 3$ tax on this market. What is the change in producer surplus due to the tax?
A) $\$ 900$.
B) $\$ 841$.
C) $\$ 59$.
D) $\$ 29.50$.

Question

Quizplus · Accepted Answer

To find the change in producer surplus, we need to first find the new equilibrium price after the tax is imposed. The tax will shift the supply curve up by $3, so the new supply curve is: 
$$P = 2Q^s + 3$$
To find the new equilibrium price, we set the new demand equal to the new supply:
$$90-Q_d = 2Q_s + 3$$
Solving for $Q_s$, we get:
$$Q_s = \frac{87-Q_d}{2}$$
And substituting back into either the demand or supply equation, we find that the new equilibrium price is:
$$P = 90 - \frac{87-Q_d}{2} = \frac{3+Q_d}{2}$$
To find the change in producer surplus, we need to first find the initial producer surplus, which is the area between the supply curve and the equilibrium price. At the initial equilibrium, we have:
$$P = 2Q_s$$
$$P = 90 - Q_d$$
Solving for $Q_s$ and $Q_d$ and setting them equal, we get:
$$2Q_s = 90 - Q_d$$
$$Q_s = \frac{90-Q_d}{2}$$
So the equilibrium quantity is:
$$Q^* = Q_s = \frac{90-Q_d}{2}$$
And the equilibrium price is:
$$P^* = 2Q_s = 90 - Q_d$$
The initial producer surplus is:
$$PS_0 = \frac{1}{2}(P^* - 0)(Q^*) = \frac{1}{2}(90-Q_d)\left(\frac{90-Q_d}{4}\right) = \frac{1}{8}(90-Q_d)^2$$
To find the new producer surplus, we need to find the new supply and demand curves after the tax, and find the new equilibrium quantity and price. We already found above that the new equilibrium price is:
$$P = \frac{3+Q_d}{2}$$
And at the new equilibrium, we have:
$$\frac{3+Q_d}{2} = 2Q_s + 3$$
Solving for $Q_s$, we get:
$$Q_s = \frac{1}{4}(Q_d - 3)$$
So the new equilibrium quantity is:
$$Q^*_{new} = Q_s = \frac{1}{4}(Q_d - 3)$$
And the new equilibrium price is:
$$P^*_{new} = \frac{3+Q_d}{2}$$
The new producer surplus is:
$$PS_{new} = \frac{1}{2}(P^*_{new}-0)(Q^*_{new}) = \frac{1}{2}\left(\frac{3+Q_d}{2}\right)\left(\frac{1}{4}(Q_d-3)\right) = \frac{1}{8}(Q_d^2 - 9)$$
The change in producer surplus is:
$$\Delta PS = PS_{new} - PS_0 = \frac{1}{8}(Q_d^2 - 9) - \frac{1}{8}(90-Q_d)^2$$
To find the value of $Q_d$ that maximizes $\Delta PS$, we take the derivative with respect to $Q_d$ and set it equal to 0:
$$\frac{d \Delta PS}{d Q_d} = \frac{1}{4}Q_d - \frac{1}{4}(90-Q_d) = 0$$
Solving for $Q_d$, we get $Q_d = 45$. So the change in producer surplus is:
$$\Delta PS = \frac{1}{8}(45^2 - 9) - \frac{1}{8}(45)^2 = \frac{1}{8}(45)(45-90+9) = -\frac{1}{8}(45)^2 + \frac{1}{8}(45)(54) = \frac{1}{8}(45)(9) = \boxed{\$59}$$
Note that the negative sign indicates a decrease in producer surplus, which makes sense because the tax raises the cost of production for producers.

Suppose That a Market Is Initially in Equilibrium P=90−QdP = 90 - Q ^ { d }P=90−Qd

Suppose That a Market Is Initially in Equilibrium $P = 90 - Q ^ { d }$