Suppose market demand and market supply are given by the equations:
QD = 40 − P
QS = P − 4
a. How much is total consumer surplus at the equilibrium price in this market?
b. How much is total producer surplus at the equilibrium price in this market?
Please explain what producer surplus is. I have never done a problem with the y-intercept like this and I am confused. For CS, I got 162. No fake answers please (I will report). Thanks much & have a wonderful day!
Answers
Consumer surplus is a measure of benefit to consumers resulting from paying less than the maximum price they are willing to pay.
Producer surplus is a measure of benefit to producers resulting from selling at a price higher than the minimum price at which they are willing to sell.
Referring to the supply-demand curve, these economic surpluses are represented by the area between either curve and the horizontal line fixed at the equilibrium price.
In this case, demand is zero when P = 40; this means there is a consumer surplus whenever P < 40. Similarly, supply is zero when P = 4; then there is a producer surplus whenever P > 4. In particular, any price below this point corresponds to a supply shortage. Realistically, producers would provide the good only if P > 4.
Set supply equal to demand and solve for P to find the equilibrium price:
40 - P = P - 4 ⇒ P = 22
At the minimum price P = 4, consumers demand 40 - 4 = 36 units and producers supply 4 - 4 = 0 units. At equilibrium price, consumers demand 40 - 22 = 18 units and producers supply 22 - 4 = 18 units. Then
CS = 1/2 (36 units - 18 units) ($22/unit - $4/unit) = $162
and
PS = 1/2 (18 units - 0 units) ($22/unit - $4/unit) = $162
As per the details given, both consumer surplus and producer surplus at the equilibrium price are 162.
To find the equilibrium price and quantity in the market, we need to set the quantity demanded equal to the quantity supplied and solve for the equilibrium price ([tex]\(P_{\text{eq}}\)[/tex]):
[tex]\[QD = QS\][/tex]
[tex]\[40 - P_{\text{eq}} = P_{\text{eq}} - 4\][/tex]
Solving for [tex]\(P_{\text{eq}}\)[/tex]:
[tex]\[2P_{\text{eq}} = 44\][/tex]
[tex]\[P_{\text{eq}} = 22\][/tex]
Now that we have the equilibrium price, we can calculate the equilibrium quantity ([tex]\(Q_{\text{eq}}\)[/tex]) by plugging [tex]\(P_{\text{eq}}\)[/tex] into either the demand or supply equation:
[tex]\[Q_{\text{eq}} = 40 - P_{\text{eq}}\][/tex]
[tex]\[Q_{\text{eq}} = 40 - 22 = 18\][/tex]
a. Consumer Surplus:
Consumer surplus represents the difference between what consumers are willing to pay for a good (as indicated by the demand curve) and what they actually pay at the equilibrium price.
Consumer Surplus ([tex]\((CS\)) = \(\frac{1}{2} \times (\text{Base}) \times (\text{Height})[/tex])
where [tex]\(\text{Base}\) = \(Q_{\text{eq}}\) and \(\text{Height}\)[/tex]
= Difference between demand curve and equilibrium price
[tex]\[CS = \frac{1}{2} \times Q_{\text{eq}} \times (\text{Difference between demand curve and equilibrium price})\][/tex]
[tex]\[CS = \frac{1}{2} \times 18 \times (40 - 22)\][/tex]
[tex]\[CS = 162\][/tex]
b. Producer Surplus:
Producer surplus represents the difference between the price at which producers are willing to sell a good (as indicated by the supply curve) and the price they actually receive at the equilibrium price.
[tex]Producer Surplus (\(PS\)) = \(\frac{1}{2} \times (\text{Base}) \times (\text{Height})\)[/tex]
[tex]\[PS = \frac{1}{2} \times Q_{\text{eq}} \times (\text{Difference between equilibrium price and supply curve})\][/tex]
[tex]\[PS = \frac{1}{2} \times 18 \times (22 - 4)\][/tex]
[tex]\[PS = 162\][/tex]
Thus, both consumer surplus and producer surplus at the equilibrium price are 162.
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