Given the supply equation: Ps = 50 + 5Qs and demand equation: Pd = 120 – 3Qd
• Calculate
• i) The price at equilibrium
• ii) Consumer surplus /CS
• iii) Producer surplus/PS
• iv) Social surplus
• v) Using a graph illustrate the CS&PS
To find the equilibrium price and quantity, we set the supply and demand equations equal to each other:
50 + 5Qs = 120 - 3Qd
Simplifying this equation, we get:
8Qs = 70 - 3Qd
Now, let's solve for Qd to find the quantity at equilibrium:
8Qs + 3Qd = 70
3Qd = 70 - 8Qs
Qd = (70 - 8Qs) / 3
Next, we substitute this value of Qd in either the supply or demand equation to find the equilibrium price. Let's use the demand equation:
Pd = 120 - 3Qd
Pd = 120 - 3((70 - 8Qs) / 3)
Pd = 120 - (70 - 8Qs)
Pd = 120 - 70 + 8Qs
Pd = 50 + 8Qs
Now we have the price at equilibrium: P = 50 + 8Qs.
i) The price at equilibrium is P = 50 + 8Qs.
ii) To calculate consumer surplus (CS), we need to find the area under the demand curve and above the equilibrium price. The formula for CS is:
CS = 0.5 * (P - equil price) * Q
In this case, the equilibrium price is 50 + 8Qs. So, the consumer surplus is given by:
CS = 0.5 * (120 - (50+8Qs)) * Q
iii) To calculate producer surplus (PS), we need to find the area above the supply curve and below the equilibrium price. The formula for PS is:
PS = 0.5 * (equil price - P) * Q
In this case, the equilibrium price is 50 + 8Qs. So, the producer surplus is given by:
PS = 0.5 * ((50 + 8Qs) - (50)) * Q
iv) Social surplus is the sum of consumer surplus and producer surplus:
Social Surplus = CS + PS
v) To illustrate the consumer surplus and producer surplus on a graph, we can plot the demand and supply curves, and shade the areas representing the CS and PS.
To answer these questions, we'll need to find the equilibrium quantity and price by equating the supply equation (Ps) and demand equation (Pd). Let's go step by step:
i) Finding the equilibrium price (Pe):
Setting Ps equal to Pd and solving for Qs will give us the equilibrium quantity. Substituting this value back into either the supply or demand equation will then give us the equilibrium price (Pe).
50 + 5Qs = 120 - 3Qd ... (equation 1: supply equation)
120 - 3Qd = 50 + 5Qs ... (equation 2: demand equation)
To find Qs, we can solve equation 1 for Qs:
5Qs = 120 - 3Qd - 50
5Qs = 70 - 3Qd
Qs = (70 - 3Qd) / 5
Next, we substitute the derived value of Qs into equation 2 to find Pe:
120 - 3Qd = 50 + 5[(70 - 3Qd) / 5]
120 - 3Qd = 50 + 14 - 3Qd
120 - 14 - 50 = 3Qd - 3Qd
56 = 0
As we can see, there is no solution to this equation, meaning the supply and demand curves do not intersect. As a result, there is no equilibrium price in this scenario.
ii) Consumer Surplus (CS):
Consumer surplus is the difference between what consumers are willing to pay for a good/service and what they actually pay. To calculate CS, we need to find the area between the demand curve and the equilibrium price (which we don't have in this case).
Since there is no equilibrium price, consumer surplus cannot be directly calculated.
iii) Producer Surplus (PS):
Producer surplus is the difference between what producers receive for a good/service and the minimum price they are willing to accept. Again, since there is no equilibrium price, producer surplus cannot be directly calculated.
iv) Social Surplus:
Social surplus is the combined consumer and producer surplus. As we cannot calculate consumer and producer surplus separately, we also cannot calculate the social surplus in this case.
v) Graphical Representation:
Without an equilibrium price, we cannot plot the supply and demand curves to show the consumer and producer surpluses.
In summary, without the presence of an equilibrium price, we cannot calculate consumer surplus, producer surplus, or social surplus, nor can we graphically represent them.
To find the price at equilibrium, we set the supply equation equal to the demand equation and solve for Q:
Ps = Pd
50 + 5Qs = 120 - 3Qd
Simplifying, we can rearrange the equation to solve for Qs:
5Qs + 3Qd = 70
5Qs = -3Qd + 70
Qs = (-3/5)Qd + 14
Now, substitute this value for Qs in the supply equation:
Ps = 50 + 5((-3/5)Qd + 14)
Ps = 50 - 3Qd + 70
Ps = -3Qd + 120
To find the equilibrium, we set Qs and Qd equal to each other:
(-3/5)Qd + 14 = Qd
(-8/5)Qd = -14
Qd = (5/8) * 14
Qd = 8.75
Now, substitute this value for Qd in the demand equation to find the price at equilibrium:
Pd = 120 - 3 * 8.75
Pd = 120 - 26.25
Pd = 93.75
Therefore, the price at equilibrium is $93.75.
To calculate consumer surplus (CS), we need to find the area between the demand curve and the price at equilibrium.
At equilibrium, the quantity demanded (Qd) is 8.75 and the price (Pd) is 93.75. To find the consumer surplus, we need to calculate the difference between the maximum price consumers are willing to pay and the price at equilibrium.
To find the maximum price consumers are willing to pay, substitute Qd = 0 in the demand equation and solve for Pd:
Pd = 120 - 3 * 0
Pd = 120
Consumer surplus (CS) = (1/2)(120 - 93.75)(8.75)
CS = 1/2 * 26.25 * 8.75
CS = 114.84
Therefore, the consumer surplus is $114.84.
To calculate producer surplus (PS), we need to find the area between the supply curve and the price at equilibrium.
At equilibrium, the quantity supplied (Qs) is 8.75 and the price (Ps) is 93.75. To find the producer surplus, we need to calculate the difference between the price at equilibrium and the minimum price producers are willing to accept.
To find the minimum price producers are willing to accept, substitute Qs = 0 in the supply equation and solve for Ps:
Ps = 50 + 5 * 0
Ps = 50
Producer surplus (PS) = (1/2)(93.75 - 50)(8.75)
PS = 1/2 * 43.75 * 8.75
PS = 190.89
Therefore, the producer surplus is $190.89.
To calculate the social surplus, we add the consumer surplus and producer surplus:
Social surplus = CS + PS
Social surplus = 114.84 + 190.89
Social surplus = 305.73
Therefore, the social surplus is $305.73.
To illustrate the consumer surplus (CS) and producer surplus (PS) on a graph, we'll use the demand and supply equations and plot them on a graph.
On the x-axis, we'll represent quantity (Q) and on the y-axis, we'll represent price (P).
The demand curve (Pd) can be plotted as a downward-sloping line with intercept 120 on the y-axis and slope -3.
The supply curve (Ps) can be plotted as an upward-sloping line with intercept 50 on the y-axis and slope 5.
The equilibrium point is where the demand curve intersects the supply curve, at Q = 8.75 and P = 93.75.
To illustrate the consumer surplus, we'll shade the area between the demand curve and the price at equilibrium.
To illustrate the producer surplus, we'll shade the area between the supply curve and the price at equilibrium.
Note: It's not possible for me to draw graphs as a text-based AI, but I hope the explanation helps you in visualizing the CS and PS on a graph.