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.  

%ΔQ = elasticity %ΔP; %Δ (PQ) = %ΔP+%ΔQ

The price elasticity of supply is defined as %ΔQ/%ΔP when moving along a supply curve. It is typically a positive number. (That is, supply curves slope up.)

The price elasticity of demand is defined as %ΔQ/%ΔP when moving along a demand curve. It is typically a negative number. Economists often talk about price elasticities of demand in terms of their absolute value, so that you have to supply the negative sign yourself from your general knowledge that increases in price reduce the quantity demanded. (That is, demand curves slope down.)

In the exercise below, the last four problems are more difficult, but for the rest, here is the procedure:

1. Figure out whether the elasticity should have a negative sign because it is a demand elasticity or a positive sign because it is a supply elasticity. Together with the absolute value of the elasticity in the second column, that will give you the elasticity with a + or a - in the third column. 
2. Multiply the (Platonic) percent change in the price by the elasticity to find the percent change in the quantity. 
3. Add the percent change in the price and the percent change in the quantity to get the percent change in revenue PQ.  

Of the harder questions, some are missing %ΔP but have %ΔQ. All you need to do there is to divide %ΔQ by the signed elasticity to get %ΔP. Then you can add %ΔP to %ΔQ to get the percent change in PQ.  The last two questions are the hardest. There you need to do a bit of algebra:

%Δ (PQ) = (1 + elasticity) %ΔP

You can solve for %ΔP, and then multiply by the elasticity to get %ΔQ.

The answers are below as a filled-in table. 

MV=PY, or equivalently %ΔM+%ΔV=%ΔP+%ΔY

This equation is called either “the Quantity Equation” or “the Equation of Exchange.” It is a very important equation for understanding the effects of money on the economy, but in this exercise, you can just treat the percent change version of the equation as a mathematical fact. Fill in the blank in each row and then check your answers. 

Note that if you divide each thing in the percent change version of the equation by time elapsed, you will get an equation in growth rates. The growth rate of the aggregate price index P is called inflation. Note also that if Divisia indices are used for money M (which is an aggregate of several different components), prices and quantities, then the percent change version of the equation is primary (most basic) and the MV=PY version would have to be deduced from the percent change version.