Equilibrium in a single market model
A single market model has three variables: the quantity demanded of the commodity (Q_{d}), the quantity supplied of the commodity (Q_{s}) and the price of the commodity (P). equilibrium is assumed to hold in the market when the quantity demanded (Q_{d}) = Quantity Supplied (Q_{s}) . It is assumed that both Q_{d }and Q_{s} are functions. A function such as y = f (x) expresses a relationship between two variables x and y such that for each value of x there exists one and only one value of y. Q_{d} is assumed to be a decreasing linear function of P which implies that as P increases, Q_{d} decreases and Vice Versa. Q_{s} on the other hand is assumed to be an increasing linear function of P which implies that as P increases, so does Q_{s}.
Mathematically, this can be expressed as follows:
Q_{d} = Qs
Q_{d} = a - bP where a,b > 0. ............................(i)
Q_{s }= -c + dp where c,d >0. ...........................(ii)
Both the Q_{d} and Q_{s} functions in this case are linear and can be expressed graphically as follows:
Once the model has been constructed it can be solved.
At equilibrium,
Qd = Qs
\a - bP = -c + dP
= a + c
b + d
To find the equilibrium quantity , we can substitute into either function (i) or (ii).
Substituting into equation (i) we obtain:
= a - b (a+c) = a (b+d) - b (a+c) = ad -bc
b + d b + d b + d
Taking a numerical example, assume the following demand and supply functions:
= 100 - 2P
Q_{s} = 40 + 4P
At equilibrium, Q_{d} = Q_{s}
100 - 2 = 40 + 4
6 = 60
= 10
Substituting P = 10, in either equation.
Q_{d} = 100 - 2 (10) = 100 - 20 = 80 = Q_{s}
A single market model may contain a quadratic function instead of a linear function. A quadratic function is one which involves the square of a variable as the highest power. The key difference between a quadratic function and a linear one is that the quadratic function will yield two solution values.
In general, a quadratic equation takes the following form:
ax^{2 }+ bx + c = 0 where a ¹ 0.
Its two roots can be obtained from the following quadratic formula:
X_{1}, X_{2} = -b + √( b^{2} - 4ac)
2a
Given the following market model:
Q_{d} = 3 - P^{2}
^{2} = 6P - 4
At equilibrium:
3 - P^{2} = 6P - 4
P^{2 }+ 6P - 7 = 0
Substituting in the quadratic formula:
a =1, b = 6, c = -7
= - 6 +Ö 6^{2} - 4 (1 x - 7)
2 x 1
=
P = 1 or -7 (ignoring -7 since price cannot be negative)
= 1
Substituting = 1 into either equation:
Q_{d} = 3 - (1)^{2} = 2 = Q_{s}
= 2