2.e. The Algebra of Supply & Demand
Using the equation for a straight line, y = mx + b, we can determine the equations for the supply and demand curve to be the following:
· Demand: P = 15 – Q
· Supply: P = 3 + Q
To solve for the equilibrium price and equilibrium quantity, set the demand equation equal to the supply equation.
15 – Q = 3 + Q
Q* = 6
Plug Q back into either the demand or supply equation to solve for P
P* = 15 – 6 = 9
To calculate the amount of shortage resulting from a price ceiling at $6, set the supply and demand curve both equal to $6
Demand: 6 = 15 – Q
Q = 9
Supply: 6 = 3 + Q
Q = 3
THE AMOUNT OF SHORTAGE IS 9 – 3 = 6 UNITS
To calculate the new equilibrium after imposing a $5 tax, set the new demand curve equal to the supply curve. (Remember to solve for the equation of the new demand curve first.)
Equation of new demand curve: P = 10 – Q
10 – Q = 3 + Q
Q = 3.5
Plug Q into either the demand or supply curve equation to solve for Ps (price sellers will pay)
P = 10 – 3.5 = 6.5
You can now calculate the price that buyers will pay (Pb).
To calculate Pb, set the new Q value equal to the old demand curve
Pb = 15 – Q = 15 – 3.5 = 11.5
Q. What if a price floor of $12 was imposed? What would be the amount of surplus?
A. Set both the demand curve and supply curve equal to the price floor.
Demand: 12 = 15 – Q
Q = 3
Supply: 12 = 3 + Q
Q = 9
THE AMOUNT OF SURPLUS IS 9 – 3 = 6 UNITS
Q. What if there was a subsidy of $2/ unit? What would be the new quantity supplied (Qs)? What price would suppliers pay (Ps)? What price would buyers pay (Pb)?
A. To solve for Qs, set the equation of the new demand curve, equal to the supply curve.
Equation of new demand curve: P = 17 – Q
17 – Q = 3 + Q
Qs = 7
Plug Qs into the old demand curve to solve for Pb
Pb = 15 – Q = 15 – 7 = 8
Plug Qs into the supply curve to solve for Ps
Ps = 3 + Q = 3 + 7 = 10
You can use these values to even calculate DWL
Area = ½ BH = ½ (2)(1) = 1