In the previous note, we showed that for linear demand and supply curves of the form P = a - bQ_D and P = c + dQ_S, the equilibrium quantity and price, respectively, were: Q_E = and P_E = . What happens to these values if demand increases? If supply increases? Perhaps the easiest way to model these changes is to add a term to the right hand side of the demand and supply relationships. For example, consider the equation P = a’ - bQ_D, where a’ differs from a by some amount D a. That is, a’ = a + D a. If D a is positive, we can interpret this as an increase in demand: the price at which consumers demand any given quantity is greater than previously. Likewise, consider the supply curve P = c’ + dQ_S, where c’ = c - D c. The minus sign on D c may be curious, but consider: if D c is positive, the price at which producers supply any given amount is lower than before. In other words, an increase in supply can be modeled by subtracting a positive amount D c from the supply price.

Inserting these new values of a’ and c’ into the general solution for equilibrium quantity, we find Q_E + D Q = Q_E’ = = . Since we know that Q_E = , we can subtract this from both sides to obtain D Q = . It is now apparent that any factor that increases demand (a positive D a) or increases supply (a positive D c) will increase equilibrium quantity (D Q > 0). Likewise, anything that decreases demand or supply (a negative D a or a negative D c) will reduce equilibrium quantity (D Q < 0). If D a and D c are of opposite sign, equilibrium quantity will move in the direction of whichever change is greater. For example, equilibrium quantity will increase if a demand increase is greater than a supply decrease, so that (D a + D c) > 0.

P_E’ = P_E + D P = = . As before, we note that P_E = , and subtracting from both sides we conclude that D P = . Here we see that an increase in demand (D a positive) or a decrease in supply (D c negative) will raise the price. If both demand and supply increase (both D a and D c positive), the impact on equilibrium price depends on which change is relatively greater, that is, it depends on whether (dD a - bD c) is greater than or less than zero.