' Question: Calculate the equilibrium price and quantity before…

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Fathima Shifana SK · Accepted Answer

1. Calculate the equilibrium price and quantity before the tax is imposed. The equilibrium occurs where quantity demanded  Q d Q_d Q d ​ equals quantity supplied  Q s Q_s Q s ​ . Given the equations: Q d = 500 − 2 P Q_d = 500 - 2P Q d ​ = 500 − 2 P Q s = 3 P − 200 Q_s = 3P - 200 Q s ​ = 3 P − 200 Set  Q d = Q s Q_d = Q_s Q d ​ = Q s ​ : 500 − 2 P = 3 P − 200 500 - 2P = 3P - 200 500 − 2 P = 3 P − 200 Solve for  P P P : 500 + 200 = 3 P + 2 P 500 + 200 = 3P + 2P 500 + 200 = 3 P + 2 P 700 = 5 P 700 = 5P 700 = 5 P P = 140 P = 140 P = 140 Now, substitute  P = 140 P = 140 P = 140 back into either the demand or supply equation to find  Q Q Q : Using the demand equation: Q d = 500 − 2 ( 140 ) = 500 − 280 = 220 Q_d = 500 - 2(140) = 500 - 280 = 220 Q d ​ = 500 − 2 ( 140 ) = 500 − 280 = 220 So, the equilibrium price before the tax is $140 , and the equilibrium quantity is 220 units . 2. Determine the new equilibrium price and quantity after the tax is imposed. With a per-unit tax of $10 on the good, the supply equation changes. The new supply function  Q s ′ Q_s' Q s ′ ​ incorporates the tax: Q s ′ = 3 ( P − 10 ) − 200 Q_s' = 3(P - 10) - 200 Q s ′ ​ = 3 ( P − 10 ) − 200 Simplify  Q s ′ Q_s' Q s ′ ​ : Q s ′ = 3 P − 30 − 200 = 3 P − 230 Q_s' = 3P - 30 - 200 = 3P - 230 Q s ′ ​ = 3 P − 30 − 200 = 3 P − 230 Set  Q d = Q s ′ Q_d = Q_s' Q d ​ = Q s ′ ​ for the new equilibrium: 500 − 2 P = 3 P − 230 500 - 2P = 3P - 230 500 − 2 P = 3 P − 230 Solve for  P P P : 500 + 230 = 3 P + 2 P 500 + 230 = 3P + 2P 500 + 230 = 3 P + 2 P 730 = 5 P 730 = 5P 730 = 5 P P = 146 P = 146 P = 146 Now, find the new equilibrium quantity: Q d = 500 − 2 ( 146 ) = 500 − 292 = 208 Q_d = 500 - 2(146) = 500 - 292 = 208 Q d ​ = 500 − 2 ( 146 ) = 500 − 292 = 208 The new equilibrium price after the tax is $146 , and the new equilibrium quantity is 208 units . 3. Calculate the tax incidence on consumers and producers. The price consumers pay after the tax is  P c = 146 P_c = 146 P c ​ = 146 . The price producers receive after the tax is  P p = P c − 10 = 146 − 10 = 136 P_p = P_c - 10 = 146 - 10 = 136 P p ​ = P c ​ − 10 = 146 − 10 = 136 . Tax incidence on consumers (change in price paid by consumers): Δ P c = P c − P initial = 146 − 140 = 6 \Delta P_c = P_c - P_{	ext{initial}} = 146 - 140 = 6 Δ P c ​ = P c ​ − P initial ​ = 146 − 140 = 6 Tax incidence on producers (change in price received by producers): Δ P p = P initial − P p = 140 − 136 = 4 \Delta P_p = P_{	ext{initial}} - P_p = 140 - 136 = 4 Δ P p ​ = P initial ​ − P p ​ = 140 − 136 = 4 4. Determine the deadweight loss resulting from the tax. Deadweight loss (DWL) is the loss in total surplus due to the tax. It is calculated using the difference in quantity before and after the tax and the tax per unit. The formula for DWL is: DWL = 1 2 × ( Quantity before tax − Quantity after tax ) × Tax 	ext{DWL} = \frac{1}{2} 	imes (	ext{Quantity before tax} - 	ext{Quantity after tax}) 	imes 	ext{Tax} DWL = 2 1 ​ × ( Quantity before tax − Quantity after tax ) × Tax DWL = 1 2 × ( 220 − 208 ) × 10 	ext{DWL} = \frac{1}{2} 	imes (220 - 208) 	imes 10 DWL = 2 1 ​ × ( 220 − 208 ) × 10 DWL = 1 2 × 12 × 10 = 60 	ext{DWL} = \frac{1}{2} 	imes 12 	imes 10 = 60 DWL = 2 1 ​ × 12 × 10 = 60 So, the deadweight loss resulting from the tax is $60 . Summary Equilibrium price and quantity before tax : $140 and 220 units. New equilibrium price and quantity after tax : $146 and 208 units. Tax incidence : $6 on consumers and $4 on producers. Deadweight loss : $60

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