HOW TO DERIVE EQUILIBRIUM QUANTITY AND PRICE

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Equilibrium, as you probably know, is the point where the downward-sloping demand curve intersects the upward-sloping supply curve. 

In clear terms, equilibrium occurs when the price and quantity offered by the seller are the same as the price and quantity of the buyer.

In my previous post, I explained equilibrium with the aid of demand schedule, demand curve, supply schedule, and curve. 

But, we didn't learn how to determine the equilibrium quantity and price from the demand and supply function. So today, we shall be learning just that. 

Example 1
Determine the equilibrium price and quantity given the following demand and supply functions:
$Q_d=210-12p$
$Q_s=40+8p$

Solution:
Equibrium is where $Q_d=Q_s$, hence
$210-12p=40+8p$
$210-40=8p+12p$
$170=20p$

Divide both side by $20$
$\frac{170}{20}=\frac{20p}{20}$
$8.5=p$
$8.5=p$
$p=8.5$

Equilibrium quantity will be
$Q_s=40+8(8.5)$
$Q_s=108$

So, the equilibrium price is $8.5$, equilibrium quantity is $108$.

Example 2
Given that $Q_d=60-\frac{2p}{3}$ and $Q_s=2p+28$, find the equilibrium price and quantity.

Solution:
You know that equilibrium is the point where $Q_d=Q_s$. Hence, 
$60-\frac{2p}{3}=2p+28$

Taking the L.C.M of the left-hand side of the equation.
$\frac{180-2p}{3}=2p+28$

Cross multiplying.
$180-2p=3(2p+28)$
$180-2p=6p+84$
$180-84=6p+2p$
$96=8p$

Divide both side by 8
$\frac{96}{8}=\frac{8p}{8}$
$12=p$
$p=12$

Substituting the value of p in the demand function, 
$Q_s=2p+28$
$Q_s=2(12)+28$
$Q_s=24+28$
$Q_s=52$

Therefore, the equilibrium price and quantity are $12$ and $52$ respectively.

Example 3
Find the equilibrium quantity and price given the following functions.
$Q_d=220-\frac{2p}{3}$
$Q_s=80+\frac{4p}{3}$

Solution:
In equilibrium, $Q_d=Q_s$
$220-\frac{2p}{3}=80+\frac{4p}{3}$

Taking the L.C.M
$\frac{660-2p}{3}=\frac{240+4p}{3}$

For simplicity, let multiply 3 to both sides
$\frac{660-2p}{\require{cancel}\bcancel{3}}\times\require{cancel}\bcancel{3}=\frac{240+4p}{\require{cancel}\bcancel{3}}\times\require{cancel}\bcancel{3}$
$660-2p=240+4p$
$660-240=4p+2p$
$420=6p$

Divide both side by 6
$\frac{420}{6}=\frac{6p}{6}$
$70=p$
$p=70$

To solve for equilibrium quantity,
You would substitute $p$ in the demand function.
$Q_d=220-\frac{2(70)}{3}$
$Q_d=220-\frac{140}{3}$

Taking the L.C.M
$Q_d=\frac{660-140}{3}$
 $Q_d=\frac{520}{3}$
 $Q_d=173.3$

Therefore, the equilibrium price and quantity is $70$ and $173.3$

Example 4
Given that $Q_d=20+24p$, $Q_s=100+8p$, find the equilibrium price and quantity.

Solution
$20+24p=100+8p$

Collecting like terms
$24p-8p=100-20$
$16p=80$

Divide booth side by $16$
$\frac{16p}{16}=\frac{80}{16}$
$p=5$

To obtain equilibrium quantity, we would substitute $p$ in the demand function
$Q_d=20+24(5)$
$Q_d=20+120$
$Q_d=140$

Thus, the equilibrium price is $5$, the equilibrium quantity is $140$

Example  5
Given the demand function $Q_d=120-3p$ and supply function as $Q_s=50+4p$. Find the equilibrium quantity and price

Solution
As usual, we equate both function
$120-3p=50+4p$
$120-50=4p+3p$
$70=7p$
$\frac{7p}{7}=\frac{70}{7}$
$p=10$.

To obtain the demand function, we will insert the value of p in any of the two functions. Let use the supply function

$Q_s=50+4p$
$Q_s=50+4(10)$
$Q_s=90$

Accordingly, the equilibrium price and quantity are 10 and 90 respectively.

Example 6
Given the market demand $Q_d=100-2p$, and market supply $P=\frac{Q_s}{2}+10$, Find:
1. the equilibrium price and quantity.
2. Determine whether there is a shortage or surplus where p=25
3. Determine whether there is a shortage or surplus where p=35

Solution
First of all, we evaluate the market supply $p=\frac{Q_s}{2}+10$,
$p=\frac{Q_s+20}{2}$
$2p=Q_s+20$
$2p-20=Q_s$
$Q_s=2p-20$

At equilibrium, quantity demanded equals quantity supplied. Therefore, we equate both the demand and supply function.
$100-2p=2p-20$
$100+20=2p+2p$
$120=4p$
$\frac{120}{4}=\frac{4p}{4}$
$p=30$

To get the equilibrium quantity, we substitute the value of p in the demand function 
$Q_d=100-2(30)$
$Q_d=100-60$
$Q_d=40$
Therefore, the equilibrium price and quantity are 30 and 40 respectively.

2. Moving on, $Q_s=2p-20$, $Q_d=100-2p$, 
Since $p=25$,
$Q_s=2(25)-20$
$Q_s=50-20$
$Q_s=30$

To quantity demanded now,
$Q_d=100-2(25)$
$Q_d=100-50$
$Q_d=50$
Since the quantity demanded  (50) is greater than the quantity supplied (30), we say there is a shortage of 20. 

3. When $p=35$, 
$Q_d=100-2(35)$
$Q_d=100-70$
$Q_d=30$
$Q_s=2(35)-20$
$Q_s=70-20$$
$Q_s=50$
Here, the quantity demanded (30) is lesser than the quantity supplied  (50), we have a surplus of 20.

Example 7
The demand and supply function for his goods are given by:

$P_d=100-0.5Q_d$ 

$P_s=10+0.5Q_s$

What would be the effect of introducing a price ceiling of N40 to the market.

Solution 

A price ceiling is the price above which no legal trade can occur. 

If a price ceiling of 40 is introduced, it simply means that buyer and seller can not trade above a price of 40. 

Hence, we substitute 40 for p in both the demand and supply function.

Since p=40 

Inserting it in the demand function 

$40=100-0.5Q_d$ 

Taking like terms 

$0.5Q_d=100-40$ 

$Q_d=\frac{60}{0.5}$.

$Q_d=120$ 

Inserting price in the supply function 

$40=10+0.5Q_s$ 

$30=0.5Q_s$ 

$Q_s=60$ 

The quantity demanded (120) is greater than the quantity supplied (60). So, the effect on the market is that they will be a shortage of 60

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