The demand and supply functions for three goods are given as follows:
Dx=100-3Px+Py+3Pz
Dy=80 +Px-2Py-2Pz
Dz=120+3Px-Py-4Pz
Sx=-10+Px
Sy=-20=3Py
Sz=-3+2Pz
Q1: Determine the equilibrium prices and quantities of all three goods.
• The government decides to:
a. Impose a 25% tax on X
b. Impose a 5Rs unit Tax on Y
c. Gives a 10% subsidy on good Z
• Analyze the impact of each of these three policies separately on equilibrium prices and Quantities.
• Also calculate changes in consumer and producer surpluses and the amount of revenue earned by the government.
Q2: Repeat this exercise when polices (a, b),(b,c) & (a, b,c) are jointly implemented. Which policy choice is best? Why?
Q: 3 Provide theoretical justification (using diagrams) of all results obtained.
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To solve this problem, we need to follow the steps below:
1. Determine the equilibrium prices and quantities:
To find the equilibrium prices and quantities, we need to set the demand functions equal to the supply functions for each good and solve the resulting system of equations.
a. For good X:
Demand: Dx = 100 - 3Px + Py + 3Pz
Supply: Sx = -10 + Px
Setting Dx = Sx:
100 - 3Px + Py + 3Pz = -10 + Px
Simplifying the equation:
4Px - Py - 3Pz = 110
b. For good Y:
Demand: Dy = 80 + Px - 2Py - 2Pz
Supply: Sy = -20 + 3Py
Setting Dy = Sy:
80 + Px - 2Py - 2Pz = -20 + 3Py
Simplifying the equation:
Px - 5Py + 2Pz = -100
c. For good Z:
Demand: Dz = 120 + 3Px - Py - 4Pz
Supply: Sz = -3 + 2Pz
Setting Dz = Sz:
120 + 3Px - Py - 4Pz = -3 + 2Pz
Simplifying the equation:
3Px - Py + 6Pz = 117
Now we have a system of three equations with three unknowns (Px, Py, Pz). Solve this system of equations to find the equilibrium prices and quantities for all three goods.
2. Analyze the impact of each policy separately:
a. Imposing a 25% tax on X:
To analyze the impact of this policy, we need to calculate the new equilibrium prices and quantities after applying the tax.
Applying a 25% tax on X means that the new supply function for X is given by: Sx = -10 + 0.75Px (since the tax reduces the price received by producers)
Using the same approach as before, set the new demand and supply functions equal and solve for the new equilibrium quantity and price for good X.
b. Imposing a 5Rs unit tax on Y:
To analyze the impact of this policy, we need to calculate the new equilibrium prices and quantities after applying the tax.
Applying a 5Rs unit tax on Y means that the new supply function for Y is given by: Sy = -20 + 3Py - 5 (since the tax is a fixed amount)
Using the same approach as before, set the new demand and supply functions equal and solve for the new equilibrium quantity and price for good Y.
c. Giving a 10% subsidy on good Z:
To analyze the impact of this policy, we need to calculate the new equilibrium prices and quantities after applying the subsidy.
A 10% subsidy on Z means that the new supply function for Z is given by: Sz = -3 + 0.9Pz (since the subsidy increases the price received by producers)
Using the same approach as before, set the new demand and supply functions equal and solve for the new equilibrium quantity and price for good Z.
Calculate the changes in consumer and producer surpluses and the amount of revenue earned by the government for each policy by comparing the original equilibrium to the new equilibrium for each good.
3. Repeat the exercise when policies (a, b), (b, c), and (a, b, c) are jointly implemented:
To analyze the impact of joint policies, follow the same steps as above but apply all selected policies simultaneously. Determine the new equilibrium prices and quantities for all three goods, calculate changes in consumer and producer surpluses, and the amount of government revenue for each combination of policies.
Remember to compare the results for each combination and analyze which policy choice is best based on the overall impact on equilibrium prices, quantities, consumer and producer surpluses, and government revenue.
4. Provide theoretical justification using diagrams for all results obtained:
To provide theoretical justification for the results obtained, use supply and demand diagrams to illustrate the effects of the different policies on the equilibrium prices and quantities for all three goods. Show the initial equilibrium and the new equilibrium after each policy change, and explain the changes observed based on the shifts in the supply and demand curves caused by the policy interventions.