# Supply and Demand: Elasticities and Comparative Statics

Before trying to do this exercise, be sure to read my post "The Logarithmic Harmony of Percent Changes and Growth Rates" and to study the section on prices elasticities of supply and demand in the "Growth Rates and Percent Changes" Powerpoint file (beginning on slide 21). The first key equation resulting from the algebra in that Powerpoint file is

(1)                       %ΔP = [%ΔD - %ΔS] / [η + ε]

The key notation here is

%ΔP = Platonic percent change in price = Δ ln(P)

%ΔQ = Platonic percent change in price = Δ ln(Q)

%ΔD = Platonic percent horizontal (quantity-metric) expansion of the demand curve.

%ΔS = Platonic percent horizontal (quantity-metric) expansion of the supply curve.

η = eta = price elasticity of supply (the slope of the supply curve when the horizontal axis is ln(Q) and the vertical axis is ln(P)).

ε= epsilon = price elasticity of demand (minus the slope of the demand curve when the horizontal axis is ln(Q) and the vertical axis is ln(P)). Note that for a normal downward sloping demand curve, this price elasticity of demand is expressed as a positive number.

Now, let's solve for %ΔQ using

%ΔQ = %ΔD - ε (%ΔP).

Substituting in %ΔP = [%ΔD - %ΔS] / [η + ε] yields

%ΔQ = %ΔD - (ε [%ΔD - %ΔS] / [η + ε]) = (η/[η + ε]) %ΔD + ( ε/[η + ε]) %ΔS

Try doing the same substitution into

%ΔQ = %ΔS + η (%ΔP)

You should get the very same answer, since the percent change in the equilibrium quantity must equal both the percent change in quantity demanded and the percent change in quantity supplied.  That is,

(2)               %ΔQ = (η/[η + ε]) %ΔD + ( ε/[η + ε]) %ΔS

The demand and supply elasticities epsilon and eta are called "parameters." %ΔD and %ΔS are the exogenous changes. %ΔP and%ΔQ are the endogenous changes. Your task is to find the exogenous changes corresponding to the parameters and exogenous changes in each row of the table just below. Please look at the answers way below only after you have calculated all of your proposed answers.

Here are the answers. Let me know if you think I have made an error in constructing the answer table.