Demand–supply analysis (calculation)

Managerial Economics study material

On a competitive market the equation of the demand curve is 𝑄 = 1600 − 2 ∙ 𝑃, while supply
is given by 𝑄 = 𝑃 − 200. (𝑃 is the price of the product in dollars.)
a) What is the equilibrium price and quantity?
b) What is the value of consumer surplus and producer surplus in the equilibrium?
c) How do consumer price, producer price, quantity, consumer surplus and producer sur-
plus change if the government introduces a tax of 60 dollars per unit? How much tax
revenue can the government collect, and how much deadweight loss appears as a result?
d) How do price, consumer surplus and producer surplus change if the government max-
imizes the quantity at 300 units? How much deadweight loss appears as a result?

Equilibrium price can be found at the intersection of the demand and supply curves, that is, the
equation 1600 − 2 ∙ 𝑃 ∗ = 𝑃∗ − 200 has to hold. By adding 2 ∙ 𝑃∗ and then 200 for both sides
of the equation, we get that 3 ∙ 𝑃 ∗ = 1800. After dividing both sides by 3, we get the equilib-
rium price: 𝑃∗ = 600 dollars. The equilibrium quantity can be calculated by substituting back
this price to the demand (or the supply) curve: 𝑄 ∗ = 1600 − 2 ∙ 600 = 400 units.
For the calculation of consumer and producer surplus we need a graph of the demand
and supply curves. To do that, first we have to express the inverse demand and inverse supply
curves (rearrange the demand and supply curves in such a way that we express 𝑃 as a function
of 𝑄 in both cases), since price is measured on the vertical, and quantity is measured on the
horizontal axis of demand–supply graphs. Demand is 𝑄 = 1600 − 2 ∙ 𝑃, which also means that
2 ∙ 𝑃 = 1600 − 𝑄. After dividing both sides by 2, we get the equation of the inverse demand
curve: 𝑃 = 800 − 0.5 ∙ 𝑄. Similarly, supply is 𝑄 = 𝑃 − 200, which also has to be rearranged.
In this example this can be done simply by adding 200 to both sides of the equation, therefore,
inverse supply is 𝑃 = 200 + 𝑄.
Consumer surplus can be calculated as the area of the orthogonal triangle shaded in blue
on the graph (below the demand curve and above the horizontal straight line at the equilibrium
(800−600)∙400
price): 𝐶𝑆 ∗ = = 40,000 dollars. Similarly, producer surplus can be calculated as
2
the area of the orthogonal triangle shaded in red on the graph (above the supply curve and below
(600−200)∙400
the horizontal straight line at the equilibrium price): 𝑃𝑆 ∗ = = 80,000 dollars.
2

If the government levies a tax of 60 dollars per unit on the producers, then it means that
out of the price the consumers pay (henceforth denoted as 𝑃𝐶 ) the producers have to pay 60
dollars to the government and they can keep only what remains after that. That is, the price the
producers can keep is 𝑃𝑃 = 𝑃𝐶 − 60.1 The demand curve shows the relationship between the
quantity demanded and the price that the consumers pay, which remains unchanged: 𝑄 =
= 1600 − 2 ∙ 𝑃𝐶 . However, supply curve shows the relationship between the quantity supplied
and the price that the producers can keep and use to cover their costs: 𝑄 = 𝑃𝑃 − 200. If we use
the relationship between consumer and producer prices (that is, we express quantity supplied
as a function of the price the consumers pay), we get that 𝑄 = (𝑃𝐶 − 60) − 200 = 𝑃𝐶 − 260.
Graphically, this means that the supply curve shifts upwards, exactly with the amount of tax
per unit (60 dollars).

1
In the absence of taxes 𝑃𝑃 = 𝑃𝐶 , therefore it would be pointless to use this distinction in the notation of prices.
 The new equilibrium consumer price is found at the intersection of the demand curve
and the new supply curve, that is, quantity demanded and quantity supplied have to be equal to
each other at this price: 1600 − 2 ∙ 𝑃𝐶 = 𝑃𝐶 − 260. By adding 2 ∙ 𝑃𝐶 and then 260 to both sides
of the equation, we get that 3 ∙ 𝑃𝐶 = 1860. After dividing both sides by 3, the new equilibrium
consumer price is found: 𝑃𝐶 = 620. We know that out of this amount, the producers have to
pay 60 dollars to the government, therefore the new equilibrium producer price (what the pro-
ducers can use to cover their costs) is 𝑃𝑃 = 620 − 60 = 560 dollars. The new equilibrium
quantity is the amount that the consumers are willing to buy at 620 dollars (or what the pro-
ducers are willing to sell if they can keep 560 dollars after each unit): 𝑄𝑡 = 1600 − 2 ∙ 620 =
= 360 units.
Consumer surplus can be calculated as the area of the orthogonal triangle shaded in blue
on the following graph (below the demand curve and above the horizontal straight line at the
(800−620)∙360
new equilibrium consumer price): 𝐶𝑆𝑡 = = 32,400 dollars. Producer surplus can
2

be calculated as the area of the orthogonal triangle shaded in red on the graph (above the original
supply curve and below the horizontal straight line at the new equilibrium producer price) 2:
(560−200)∙360
𝑃𝑆𝑡 = = 64,800 dollars. As 360 units are exchanged on the market if a tax of 60
2

dollars per unit is levied on the product, the government collects a tax revenue of 𝑇 =
= (620 − 560) ∙ 360 = 60 ∙ 360 = 21,600 dollars (the area of the rectangle shaded in green
on the graph). The deadweight loss (that is, the loss of social surplus caused by the reduction

2
Or equivalently, as the area of the orthogonal triangle above the new (upward-shifted) supply curve and the
horizontal straight line at the new equilibrium consumer price. Since the introduction of a per-unit tax shifts the
supply curve parallelly, these areas are exactly the same.
of the number of transactions due to the introduction of the tax) can be calculated by the area
(620−560)∙(400−360)
of the triangle shaded in grey on the graph: 𝐷𝑊𝐿 = = 1200 dollars.
2

We can easily check our results, because (as it also can be seen from the graphs) the
following equation has to hold: 𝐶𝑆 ∗ + 𝑃𝑆 ∗ = 𝐶𝑆𝑡 + 𝑃𝑆𝑡 + 𝑇 + 𝐷𝑊𝐿. In the case of this exam-
ple, this means that 40,000 + 80,000 = 32,400 + 64,800 + 21,600 + 1200, which equation
indeed holds.
Now let us assume that the government uses a different intervention in the market in-
stead of the tax: it maximizes the quantity that can be sold and bought on the market at 300
units. The introduction of such a quota means that the consumers have to compete for the quan-
tity available on the market. This process drives up the price to the maximum level at which the
consumers are willing to buy 300 units.3 This price can be found by substituting the 𝑄𝑞 = 300
units to the inverse demand curve: 𝑃𝑞 = 800 − 0.5 ∙ 300 = 650 dollars.

3
At any lower price quantity demanded is higher than 300 units, which means that there will be unsatisfied con-
sumers who would be willing to offer more in order to being able to buy the product. Their willingness to pay
more drives up the price.
 Consumer surplus can be calculated as the area of the orthogonal triangle shaded in blue
on the following graph (below the demand curve and above the horizontal straight line at the
(800−650)∙300
new equilibrium price): 𝐶𝑆𝑞 = = 22,500 dollars. Producer surplus can be calcu-
2

lated as the area of a trapezoid (above the supply curve and below the horizontal straight line at
the new equilibrium price). To be able to calculate that area, we need the lowest price at which
producers would be willing to sell the 𝑄𝑞 = 300 units, which can be found by substituting this
quantity to the inverse supply curve: 200 + 300 = 500 dollars. Therefore, the value of pro-
(650−200)+(650−500)
ducer surplus is 𝑃𝑆𝑞 = ∙ 300 = 90.000 dollars. The deadweight loss (that
2

is, the loss of social surplus caused by the reduction of the number of transactions due to the
introduction of the quota) can be calculated by the area of the triangle shaded in grey on the
(650−500)∙(400−300)
graph: 𝐷𝑊𝐿 = = 7,500 dollars.
2
 Again, we can easily check our results, because (as it also can be seen from the graphs)
the following equation has to hold: 𝐶𝑆 ∗ + 𝑃𝑆 ∗ = 𝐶𝑆𝑞 + 𝑃𝑆𝑞 + 𝐷𝑊𝐿. In the case of this exam-
ple, this means that 40,000 + 80,000 = 22,500 + 90,000 + 7,500, which equation indeed
holds.