Suppose that the market demand and supply functions are given by:
Qd = -500P + 5000
and Qs = 400P-400
Find out the effects of imposition of specific sales tax of 18% on equilibrium price and quantity.
We are given the market demand and supply functions:
Demand function: \( Q_d = -500P + 5000 \)
Supply function: \( Q_s = 400P - 400 \)
At equilibrium, quantity demanded \( Q_d \) equals quantity supplied \( Q_s \):
\( -500P + 5000 = 400P - 400 \)
Solving for \( P \):
\( 5000 + 400 = 400P + 500P \)
\( 5400 = 900P \)
\( P = \frac{5400}{900} = 6 \)
Now, substitute \( P = 6 \) into either the demand or supply equation to find the equilibrium quantity. Using the demand function:
\( Q_d = -500(6) + 5000 = -3000 + 5000 = 2000 \)
Thus, the initial equilibrium price is \( P = 6 \) and the equilibrium quantity is \( Q = 2000 \).
A specific sales tax of 18% means that the price paid by consumers increases by 18% of the price. Let the price received by producers be \( P_s \) and the price paid by consumers be \( P_c \), where:
\( P_c = 1.18P_s \)
Substitute \( P_c = 1.18P_s \) into the demand function and supply function to find the new equilibrium.
Demand function becomes: \( Q_d = -500(1.18P_s) + 5000 \)
Supply function becomes: \( Q_s = 400P_s - 400 \)
At equilibrium, \( Q_d = Q_s \), so:
\( -500(1.18P_s) + 5000 = 400P_s - 400 \)
Simplify and solve for \( P_s \):
\( -590P_s + 5000 = 400P_s - 400 \)
\( 5000 + 400 = 400P_s + 590P_s \)
\( 5400 = 990P_s \)
\( P_s = \frac{5400}{990} = 5.45 \)
Thus, the price received by producers is \( P_s = 5.45 \).
Now, calculate the price paid by consumers \( P_c \):
\( P_c = 1.18 \times 5.45 = 6.43 \)
Substitute \( P_s = 5.45 \) into the supply function:
\( Q_s = 400(5.45) - 400 = 2180 - 400 = 1780 \)
Thus, the new equilibrium quantity is \( Q = 1780 \).
- The equilibrium price paid by consumers rises from \( P = 6 \) to \( P_c = 6.43 \).
- The equilibrium quantity decreases from \( Q = 2000 \) to \( Q = 1780 \).
Thus, the imposition of the specific sales tax leads to an increase in the price paid by consumers and a decrease in the equilibrium quantity.
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