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.

Slide1.png

 

 

 

 

 

 

 

 

 

 

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

Slide2.png

The Effects of Shifts in the CF(r) Curve on the Long-Run International Finance Diagram Are Not Ambiguous After All

  • Assume that the Supply of Loanable Funds curve is upward-sloping
  • Assume that the I(r) curve is downward sloping. 
  • Both of these assumptions are crucial to what follows. 
  • Now, Consider a case when the CF(r) curve shifts out. (After you read the rest of this post, make sure you work through the case when the CF(r) curve shifts back for yourself. Actually, in general you should always work through the opposite directions of things as part of your studying. The simplest way to make an exam question is to do the opposite direction from what was covered in class.)
  • Since the Demand for Loanable Funds is the horizontal sum of the I(r) curve and the CF(r) curve, the outward shift in the CF(r) curve also shifts the Demand-for-Loanable-Funds out. 
  • This raises the interest rate r.  
  • The increase in r moves things up and left along the CF(r) curve. 
  • It might seem that the outward shift in the CF(r) curve combined with a shift up and to the left along the CF(r) curve might make the direction the quantity of CF goes ambiguous. But not so! 
  • Since the I(r) does not shift, the increase in r lowers the quantity of investment I. 
  • Since the Supply-of-Loanable-Funds curve does not shift, the increase in r raises the quantity of saving S. 
  • S = I + CF, since physical investment and sending funds abroad with CF are the two possible uses of overall national saving. (Remember that national saving is household saving + corporate saving + government saving.)  
  • Since quantity of S increases, while the quantity of I decreases, the only way that the equation S = I + CF can be satisfied is for the quantity of CF to increase. 
  • Thus, the quantity of CF moves in the same direction that the CF(r) curve shifts. 
  • The quantity of CF determines the location of the vertical Supply of Dollars. With the quantity of CF increasing, the Supply of Dollars curve, which is vertical at the quantity of CF, shifts to the right. 
  • A very similar analysis can be used to analyze shifts in the I(r) curve. Try it!