Download Algebraic Approach to Supply and Demand

Economics 201
Algebraic Approach to Supply and Demand
F2000PS
I would like to introduce supply and demand in simple algebraic form. This approach makes
clear that both price and quantity are simultaneously determined (and that the choice of which
variable to put on the vertical axis is, indeed, truly arbitrary). It also allows you to get a sense of
how a shift in supply or demand works its way through the system. Finally, for our purposes, it
makes some of the discussion in later chapters regarding the incidence—“who actually pays”—a
tax more straightforward. In doing the following, be sure to “work it out” to your satisfaction!
Let’s represent demand as a linear (because they are easier to work with) equation:
D: Qd =  – Pd +  Inc,
where the notation should be fairly straightforward: Q denotes quantity, P denotes price, the
superscript, d, denotes “demand,” and Inc represents Income. The Greek letters are “parameters”
and represent the specific magnitudes that define the relationship; in what follows we will use
specific values for these parameters.
Let’s represent supply similarly:
S: Qs =  +  Ps +  Tech,
here, Tech denotes a shift variable in the supply relationship; perhaps it can be thought of as an
index of technological progress.
For the present—until we hit the section dealing with sales taxes—we can treat the two prices as
being the same, i.e., Pd = Ps = P. Furthermore, in equilibrium, Qd = Qs, i.e., the quantity
demanded at the going price equals the quantity supplied at that price.
Therefore, in equilibrium:
Qd =  - P +  Inc =  +  P + Tech = Qs
Solving for P:
 -  Inc - Tech = ( P + P)
P = [1/( + )] [ -  Inc -  Tech].
This is the Equilibrium Price.
Then “plug” the value for price obtained back into the demand and supply relationships. The Qvalues you get for each of the Supply and Demand relationships should be the same; this is your
Equilibrium Quantity.
OVER
Now let’s try some cases.
D: Qd =   P +  Inc
S: Qs =  +  P +  Tech
We need specific values for Inc and for T. Let Inc = 10 (measured, say, in $1000s), and let the
index of technology, Tech = 2.0.
Demand, then, is
Qd =  - P +  (10) = 1300 – 200 P, [graph this in a rough sort of way]
and Supply is
Qs =  +  P +  (2) =  60 + 50 P [graph this also].
In equilibrium,
250 P = 1360, or P = 5.44.
Plugging P = 5.44 into the demand equation yields Qd = 1300 – 200 (5.44) = 212.
Plugging P = 5.44 into the supply equation yields Qs =  60 + 50 (5.44) = 212.
What if: Income increases from 10 to 20? Demand then shifts to the right (“increases”) by 100—
at any Price the quantity demanded is now 100 more than previously. Do you understand how
this is so?
The new equilibrium price will be
P = (1/250) 1460, or P = 5.84, and the equilibrium quantity will be 232.
Notice, as expected, both price and quantity are higher in the new equilibrium.
Try the following:
D: Qd =   P +  Inc, Inc = 30
S: Qs =  +  P +  Tech, Tech = 3
Find the equilibrium Price and Quantity. What happens if Inc increases to 50?—if Tech increases
to 7?—if both Inc and Tech increase as above?
Graph what is going on in the above examples. The graphs (if carefully drawn) and the equations
will give the same results; they do represent exactly the same market behaviors. If the graphs are
“rough and ready,” they will still be informative.
In a later handout, when we discuss taxes, the possibility of the Price faced by sellers being
different from the Price faced by buyers (Pd ≠ Ps) will be explicitly considered.