Supply Equation: Ps = 40 + 4Qs
and Demand Equation: Pd = 100 –
2Qd
Find :
a)Qe and pe
2)CS,PS and social welfare
3)Illustrate (2) on a graph
To find the equilibrium quantity and price, set the supply equation equal to the demand equation and solve for Q:
40 + 4Qs = 100 - 2Qd
Simplify by adding 2Qd to both sides:
2Qd + 4Qs = 100 - 40
2Qd + 4Qs = 60
Divide both sides by 2:
Qd + 2Qs = 30
Rearrange the equation to solve for Qd:
Qd = 30 - 2Qs
Now substitute this expression for Qd into the demand equation:
Pd = 100 - 2(30 - 2Qs)
Simplify:
Pd = 100 - 60 + 4Qs
Pd = 40 + 4Qs
Since the demand price (Pd) and supply price (Ps) are equal at equilibrium, we can write:
Pe = Pd = Ps
Pe = 40 + 4Qs
Substitute Pe back into one of the equations to find Qe:
Pe = 40 + 4Qe
40 + 4Qe = 100 - 2Qe
Combine like terms:
6Qe = 60
Qe = 10
Substitute Qe back into the demand or supply equation to find Pe:
Pd = 40 + 4(10)
Pd = 80
a) The equilibrium quantity (Qe) is 10 and the equilibrium price (Pe) is 80.
Now, to find consumer surplus (CS), producer surplus (PS), and social welfare, we need to calculate the areas under the demand and supply curves.
CS is the area above the price line and below the demand curve. It is given by:
CS = 0.5 * (Pe - Pd) * Qe
CS = 0.5 * (80 - 40) * 10
CS = 0.5 * 40 * 10
CS = $200
PS is the area above the supply curve and below the price line. It is given by:
PS = 0.5 * (Pd - Ps) * Qe
PS = 0.5 * (80 - 40) * 10
PS = 0.5 * 40 * 10
PS = $200
Social welfare is the sum of CS and PS:
Social welfare = CS + PS
Social welfare = $200 + $200
Social welfare = $400
b) On a graph, plot the supply equation (Ps = 40 + 4Qs) and the demand equation (Pd = 100 – 2Qd). Label the axes as quantity (Q) and price (P). Label the equilibrium point as (Qe, Pe) which is (10, 80). Shade the areas representing CS and PS as calculated above.
To find the equilibrium quantity (Qe) and price (Pe), we need to set the supply and demand equations equal to each other and solve for Qe, and then substitute the value of Qe back into either the supply or demand equation to find Pe. Let's do the calculations step by step:
Step 1: Set the supply and demand equations equal to each other:
Ps = Pd
40 + 4Qs = 100 – 2Qd
Step 2: Solve for Qe:
40 + 4Qe = 100 – 2Qe
6Qe = 60
Qe = 10
Step 3: Substitute Qe into either the supply or demand equation to find Pe:
Pd = 100 – 2Qd
Pd = 100 – 2(10)
Pd = 100 – 20
Pd = 80
Pe = Pd = 80
Therefore, the equilibrium quantity (Qe) is 10 and the equilibrium price (Pe) is 80.
Now, let's calculate consumer surplus (CS), producer surplus (PS), and social welfare.
Consumer Surplus (CS):
CS represents the difference between the value consumers are willing to pay for a good and the price they actually pay. In this case, to find consumer surplus, we need to find the area between the demand curve and the equilibrium price.
Step 1: Calculate the area of the triangle formed between the demand curve (Pd = 100 – 2Qd) and the equilibrium quantity (Qe):
CS = (1/2) * (Qe) * (Pd)
CS = (1/2) * 10 * 80
CS = 400
Producer Surplus (PS):
PS represents the difference between the cost of production for producers and the price they receive for the goods. In this case, to find producer surplus, we need to find the area between the supply curve and the equilibrium price.
Step 1: Calculate the area of the triangle formed between the supply curve (Ps = 40 + 4Qs) and the equilibrium quantity (Qe):
PS = (1/2) * (Qe) * (Pe - cost of production)
PS = (1/2) * 10 * (80 - 40)
PS = 200
Social Welfare:
Social welfare indicates the overall welfare or benefit derived by society from the production and consumption of a good. Social welfare is the sum of consumer surplus and producer surplus.
Social welfare = CS + PS
Social welfare = 400 + 200
Social welfare = 600
Now, let's illustrate this on a graph.
To find Qe and pe, we first need to set the supply (Ps) equal to the demand (Pd):
40 + 4Qs = 100 - 2Qd
To solve for Qe, we can rearrange the equation as follows:
4Qs + 2Qd = 100 - 40
4Qs + 2Qd = 60
Since Qs = Qd at equilibrium, we can simplify the equation to:
6Qe = 60
Qe = 10
Substituting Qe back into one of the original equations (either supply or demand), we can solve for pe:
Ps = 40 + 4Qs
Ps = 40 + 4(10)
Ps = 40 + 40
Ps = 80
Therefore, Qe = 10 and pe = 80.
Now, let's calculate consumer surplus (CS), producer surplus (PS), and social welfare.
Consumer Surplus (CS):
To calculate CS, we need to find the area between the demand curve and the equilibrium price. We can use the formula:
CS = 0.5 * (pe - Pd) * Qe
CS = 0.5 * (80 - (100 - 2Qe)) * Qe
CS = 0.5 * (80 - (100 - 20)) * 10
CS = 0.5 * (80 - 80) * 10
CS = 0.5 * 0 * 10
CS = 0
Producer Surplus (PS):
To calculate PS, we need to find the area between the supply curve and the equilibrium price. We can use the formula:
PS = 0.5 * (Ps - pe) * Qe
PS = 0.5 * (40 + 4Qe - 80) * Qe
PS = 0.5 * (40 + 4(10) - 80) * 10
PS = 0.5 * (40 + 40 - 80) * 10
PS = 0.5 * 0 * 10
PS = 0
Social Welfare:
Social Welfare is the combination of CS and PS:
Social Welfare = CS + PS
Social Welfare = 0 + 0
Social Welfare = 0
Now, let's illustrate this on a graph:
On the horizontal axis, plot the quantity (Q) and label it accordingly.
On the vertical axis, plot the price (P) and label it accordingly.
Draw the demand curve starting at intercept 100 on the P-axis and -2 on the Q-axis.
Draw the supply curve starting at intercept 40 on the P-axis and 4 on the Q-axis.
Find the points where the supply and demand curves intersect. This is the equilibrium point (Qe, pe).
Mark Qe on the Q-axis, and pe on the P-axis.
To illustrate consumer surplus, shade the area between the demand curve and the equilibrium price (pe) up to the quantity (Qe).
To illustrate producer surplus, shade the area between the supply curve and the equilibrium price (pe) from Qe to the right end of the supply curve.
Since CS and PS are 0, there will be no shaded areas on the graph.