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Demand and Supply
Overview
In chapter 2, we deal with demand and supply analysis in perfectly competitive markets.
Perfectly competitive markets consist of a large number of buyers and sellers.
The transactions of any individual buyer or seller are so small, in comparison to the overall volume of the good or the service traded in the market, that the buyer or seller has no choice but to takes the price set by the market.
Demand Curves
Market Demand Curve: A curve that shows us the quantity of good that consumers are willing to buy at different prices.
Law of Demand : The negative relationship between the price of a good and the quantity demanded, when all other factors that influence demand are held fixed.
As illustrated in the following graph…
The direct demand curve will generally take the linear form…
Q = a – bP where ‘a’ = vertical intercept, and ‘b’ = slope
Rearranging and solving for P, we get…
This is the Inverse Demand Curve
Vertical intercept slope ba^^ b^1 *^ Q
Example
Direct Demand Curve: Q = 100 ‐2P
Inverse Demand hence becomes:
The “Choke Price” is the price at which Q=0, or simply put, at what price consumers demand 0 units of the good. Setting Q=0, the “Choke Price” = 50
And so, the graph looks like a straight line...
Q -100 = -2p 2P =100 - Q
0 =
Or more generally... Demand: Q = a – bP, then Inverse Demand :
P
Q
Choke Price (a/b) P (^) ba Qb
0 ab Qa Q a
a
P ab
Example
Demand Curve: Qd^ = 500 – 4P
Supply Curve: QS^ = ‐ 100 + 2P
Before finding the equilibrium output and price level…
Let us sketch these curves on the same graph with quantity on the horizontal axis and price on the vertical axis.
Solving for P: 4p = 500 - Qd p = 500/4 – Qd^ / Solving for P: 2p = Qs^ + 100 p = Qs^ /2 + 50
Inverse Demand Curve →
Inverse Supply Curve →
When P=0, Q d^ =500‐4.0=
P = 100
Q = 100
At what price and quantity do you reach equilibrium? Q S^ = Q d 500 – 4P = ‐ 100 + 2P 600 = 6P 100 = P
And then take this p=100 and plug it into either the demand or supply curve to find the equilibrium quantity… Q S^ = 500 – 4(100) = 100
And so, equilibrium occurs at P=100 and Q=
P=
Comparative Statics: An Increase in
Demand, for any given price
An increase in demand as the one depicted above can originate from an increase in income, or in the consumer’s preference for the good. For any given price, the quantity that consumers demand has now gone up.
You can visually see that by extending a long horizontal dotted line which maintains your focus on a given (fixed price). The point where the dotted line crosses each demand curve represents the quantity demanded.
A decrease in supply, for any given price
A decrease in supply might originate from an increase in production costs, which lead producers of the good to supply lower amounts of the good at any given price. [Follow similar graphical representation as above]
Extend a horizontal dotted line at the price p=$10. The quantity supplied is lower after the increase in production costs (S 2 ) than before the increase (S 1 ).
Example: Market for Aluminum Demand: Qd^ = 500 – 50P + 10I where P=price and I=income Supply: QS^ = ‐ 400 + 50P Let’s analyze the equilibrium when income is I=10 and how it is affected when income decreases to I= First, Equilibrium when I = 10…
Plug in I=10 into Q d^ to get Q d^ = 500 – 50P + 10(10) = 600 – 50P Equating Qd^ =QS^ we obtain… 600 ‐5p=‐ 400 ‐50p 1000=100p Or p=$10→Q S^ = -400 + 50($10) = 100
A change in both the demand and
supply curve
Interpretation…
A decrease in demand and an increase in supply:
Unambiguously produce a reduction in the equilibrium price; but…
the effect on the equilibrium quantity is more ambiguous. In the figure, the outward shift in the supply curve dominates the inward shift in the demand curve, producing an overall increase in the equilibrium quantity. Otherwise, the equilibrium quantity would decrease.
Can you draw a figure in which Demand increases and Supply decreases, but equilibrium output decreases?
Price Elasticity of Demand
Now that we know how to construct a demand curve, to what extent does a change in one variable affect quantity demanded?
Price Elasticity of Demand: A measure of the percentage change of quantity demanded as a result of a 1% increase in price, holding all other determinants of demand constant.
We can express the elasticity of demand as…
Notice that the first term is simply the derivative of quantity demanded with respect to P.
Let’s look at an example…
P
Q
0
1 0 Q
Q Q Q where Q^^^ and 0
1 0 P
P P P
P (^)
Example
Suppose that we have a price change that results in the following changes in quantity demanded…
When P=10, quantity is Q= When P=12, quantity falls to Q=45 (So ∆Q = ‐5)
What is the elasticity of demand?
∆1% in P ∆0.5% in Q (less-than-proportional decrease)
Why is εQ,P always negative?
Where represents the slope of the demand curve, which is negative by the Law of Demand.
Where since both p>0 and Q>0.
Hence, the product (‐)٠(+) must be negative, making the elasticity εQ,p a negative number.
Example
Let’s look at another example of elasticity… Demand curve Q = 100 – 2(P)
Let’s find the elasticities when...
A) P=40, so Q= 20
B) P=25, so Q= 50
C) P=10, so Q= 80
Remember that our demand curve is in the form Q = a – bP , and that the first term of our elasticity equation is simply the derivative with respect to P (i.e., ‐b) so our elasticity equation is…
So now, we simply plug in the particular values of P and Q to find the elasticity for each price.
Q
P P Ed Q *
P^ Q ^2
Remember, elasticity of demand does not equal the slope. That is, a linear demand curve (constant slope) can have different elasticities, as we just showed.
Why not use slope of the demand function,
rather than EQ, p?
The reason we use price elasticity of demand (and not simply the slope of the demand curve) is because by using the former we can produce a unit‐free measure of how sensitive is the demand curve to changes in prices.
Indeed, note that units cancel out when you use the formula of price‐elasticity of demand, but they wouldn’t if you were simply using the slope of the demand curve.
We need a unit‐free measure of Qd^ and Price to be able to compare the changes in one with the changes in the other.
Qd in tons Qin tons P inUS $ P inUS $
Qd Q P P
A number
In this way we can also compare the sensitivity of demand to price changes across different goods:
Measured in different units (uranium versus steel)
Consumed in different countries (prices expressed in different currencies)
Etc…
Examples of price‐elasticities of demand:
Intuition : a 10% increase in the price of cigarettes produces a 1.07% reduction in the quantity demanded. Hence, we can claim that demand for cigarettes is relatively insensitive to price changes.
Cigarettes ‐0. Pet food ‐0. Air travel, leisure ‐1. Air travel, business ‐0.
Constant Elasticity Demand Curve
Constant Elasticity Demand Curve: A demand curve of the form Q = aP‐b^ where ‘a’ and ‘b’ are positive constants. The term –b is the price elasticity of demand along all points in this curve.
Let us see why… First, recall the formula of price elasticity;
d
dQ
dP
P
Q
Constant Elasticity Demand Curve
Therefore,
Constant in P
Q a * p ^ b
We will just find , and then plug into the formula;
dQ dP^ d
Query
Consider the demand Curve Qd^ = 5P‐^1 The elasticity of demand along this demand curve: A. Is inelastic B. Is elastic C. Is unitary elastic D. Falls as the price falls
What about changes in the price of other goods? For instance, how the change in the price of Pepsi change the quantity of Coke demanded?
Cross‐Price Elasticity of Demand: The percentage change of the quantity of one good demanded (good i ) that results from a 1% increase in the price of another good (good j ).
Example of Cross‐Price elasticity of demand (Application 2.4 in textbook) ∆1% in the price of Nissan Sentra
produces a change in the demand of…
Ford Escort (^) BMW 735
Example of Cross‐Price elasticity of
demand (Application 2.4 in textbook)
∆1% in the price of Nissan Sentra produces a…
∆0.454% in the quantity demanded of Ford Escort ∆0% in the quantity demanded of Lexus or BMW 735 Generally…
If εQi,Pj >0, goods i and j are regared as substitutes. (i.e. two brands of mineral water, Nissan Sentra and Ford Escort)
If εQi,Pj <0, goods i and j are regarded as complements. (i.e. left and right shoe, cars and gasoline, etc)
Application: Coke vs. Pepsi
Application: Coke vs. Pepsi
Price Elasticity of Demand єQ,P
Cross‐Price Elasticity of Demand
Income Elasticity of Demand
Intuition…
Coke Pepsi
Therefore, a ∆1% in the Pcoke produces…
1.47% in the quantity demanded of Coke, but… ∆0.64% in the quantity demanded of Pepsi (So people, on average, regard Coke and Pepsi as substitutes!)
On the other hand, a ∆1% in the income level produces… ∆0.58% in the quantity demanded of coke
(You can repeat a similar analysis starting with ∆1% in the price of Pepsi)
As you might have guessed, we can even extend this analysis to how changes in prices affect the quantity supplied. This is called the Price Elasticity of Supply.
It shows us how a 1% change in price affects the amount of the good supplied. Or, how price sensitive is supply?
Back of Envelope calculations about linear demand curves: Now that we know how a demand curve is constructed and how to use its various parts to analyze the relationship between quantity demanded, quantity supplied, and price, we can work backwards to actually construct a demand curve. Consider that we know the prevailing Q, the P, and εQ,P. We want to find the value of a and b in Q=a-bp , where… ‘a’ (vertical intercept or choke price)
‘b’ (slope of demand which we can derive from elasticity)
Q d^ =a-bP
