Answer to Question #156445 in Microeconomics for Tlotlo Pinaemang

Question

Suppose there is demand for a product is Qd=70-2P and supply is Qs= 10+P. The government imposes a tax of $ 9 per unit for producer. Determine the initial equilibrium, government revenue from the tax, DWL and how the consumer's surplus and producer's surplus have changed.

Answer 1

a) Let's find the initial equilibrium (equilibrium price and quantity). In the equilibrium, $Q_d=Q_s$ and we can find the eqquilibrium price:

$$70-2P_E=10+P_E,$$ $$3P_E=60,$$ $$P_E=\$20.$$

Then, we can find the equilibrium quantity by substituting $P_E$ into the equation for $Q_d$:

$$Q_d=Q_s=Q_E=70-2\cdot20=30.$$

b) Now, the government imposes a tax of $9 per unit for producers. Let's modify our supply function end epress the price $P$ without taxation through the new price $P_1$, when the government tax is taken into account: $P=P_1-9$. Then, substituting $P$ into the supply function $Q_s$, we get:

$$Q_s=10+P=10+P_1-9=P_1+1.$$

Finally, we can calculate the new equilibrium price and equilibrium quantity:

$$70-2P_1=P_1+1,$$ $$3P_1=69,$$ $$P_{E1}=\$23.$$

Then, the new equilibrium quantity (or quantity transacted in the market) can be calculated as follows:

$$Q_{E1}=P_1+1=24.$$

Government revenue can be calculated as follows:

$$GR=Tax\cdot Q_{E1}=\$9\cdot24=\$216.$$

c) Deadweight loss can be calculated as follows:

$$DWL=\dfrac{1}{2}\cdot Tax\cdot(Q_E-Q_{E1}),$$ $$DWL=\dfrac{1}{2}\cdot \$9\cdot(30-24)=\$27.$$

d) Let's plot demand and supply curves in Excel:

Here, orange line is the supply function, the blue curve is the demand function and the gray curve is the new supply function after the taxation.

Let's find the consumer's surplus and the producser's surplus before the taxation:

$$CS=\dfrac{1}{2}\cdot Base\cdot Height,$$ $$CS=\dfrac{1}{2}\cdot (35-20)\cdot 30=\$225,$$ $$PS=\dfrac{1}{2}\cdot Base\cdot Height,$$ $$PS=\dfrac{1}{2}\cdot (20-0)\cdot 30=\$300.$$

Let's find the consumer's surplus and the producser's surplus after the taxation:

$$CS=\dfrac{1}{2}\cdot (35-23)\cdot 24=\$144,$$ $$PS=\dfrac{1}{2}\cdot (23-0)\cdot 24=\$276.$$

Answer:

a) $P_E=\$20, Q_E=30.$

b) $GR=\$216.$

c) $DWL=\$27.$

d) CS and PS before taxation: $CS=\$225, PS=\$300.$

CS and PS after taxation: $CS=\$144, PS=\$276.$