Percentage changes
It is often easier to think in terms of percentage changes than in terms of the original numbers:
rather than noting that your wages went from $ 500 to $ 515 per week and the CPI went from 200 to 210, it
conveys more meaning to say that your nominal or money wages went up by 3 percent and inflation was 5
percent. It is easier to see that your purchasing power (or real wages) declined by 2 percent.
We will use a “percentage change operator” (%Δ) which simplifies arithmetic by changing:
a. Division into subtraction: if real wages = nominal wages divided by the CPI, we know that the
percentage change in real wages is the percentage change in nominal wages minus the percentage change in the
CPI.
%Δ REAL WAGES = %Δ NOMINAL WAGES - %Δ CPI
b. Multiplication into division: revenue is equal to price multiplied by quantity sold, and
the percentage change in revenue is the percentage change in price times the percentage change in quantity.
%ΔREVENUE = %Δ PRICE + %Δ QUANTITY
c.Exponentiation into multiplication. Take the Cobb-Douglas production function,
The percentage change in output, when both capital and labor change will be:
Q = K.25 L.75
%Δ Q = 0.25 * %Δ K + 0.75 %Δ L
Note that these expressions are all approximate rather than exact, and the approximation is worse
when the percentage changes are larger. For example, consider a change in revenue which results from price
increasing from $ 100 to $ 110 (or by 10 percent) and quantity sold increasing from 50 units to 55 units (also by
10 percent) The percentage change formula leads us to expect a percentage increase of 20 percent in revenue.
But if you carry out the exact calculation, you find revenue went from $ 5000 to $ 6050, or 21 %
It will help to derive the percentage change formula from the changes, and to relate those changes to
the graph:
Δ (PQ) = P (ΔQ) + Q (ΔP)
To translate this into percentage changes, we use the trick of multiplying each term by one:
and remember that %Δ X = Δ X / X.
multiply the left hand side by PQ divided by PQ, and we get PQ * Δ (PQ) / PQ = PQ * % Δ (PQ)
multiply P Δ Q by Q / Q and we get PQ * %Δ Q
and multiply Q (ΔP) by P/P and we get QP * %ΔP
Put all this together and we get:
PQ * [%Δ (PQ)] = PQ (%ΔQ) + PQ (%ΔP)
Divide both sides of the equation by PQ, and we get:
%Δ (PQ) = %Δ P + %ΔQ
Refer to the equation in terms of changes:
Δ (PQ) = P (ΔQ) + Q (ΔP)
The first term is shown in light blue on the graph:
the original price of 100 times the change in quantity (55 -50) gets us a change of 500, or 10 percent of the
original revenue.
The second term is the original quantity of 50 times the change in price (110 – 100); again we get a
change in revenue of 500 (another 10 percent of the original revenue), shown in magenta.
What we have overlooked is the area shown in green, which is ΔP * ΔQ = $ 10 * 5 = $ 50, which is
the missing revenue. So more correctly,
%Δ (PQ) = %Δ P + %ΔQ
+ %Δ P * %ΔQ
The final term is usually small enough to ignore for practical purposes; if you add it, be sure to treat all
percentages as the decimals they really are (“per cent” means per hundred, “5 percent” = 5/100 = 0.05).
In this example,
%Δ (PQ) = %Δ P + %ΔQ + %Δ P * %ΔQ
= .10 + .10
+ .10 * .10
=
.20 + .01 = .21 or 21 percent.
Addendum for international economics, Specific factors model.
In chapter 3 of Feenstra and Taylor, we find them examining the effect of a change in price on factor returns.
The total return to labor is w*L, the wage rate times the amount of labor; the change in that total return can be
approximated fairly closely by
%Δ (wL) = %Δ w + %ΔL
The return to land is written as Rt * T, the rate of return to land times the amount of land. You can think of Rt as
the rental of an average acre of land, and you multiply by all the arable land in the country to get the return to
landowners. If the landowners hire labor and sell their crops, their total return is:
Rt * T = P Q - w L
Changes in the total return can be calculated as:
Δ (Rt * T) = Δ (P Q ) - Δ( wL) and we can expand this to:
Rt ΔT + T ΔRt = P (ΔQ) + Q (ΔP) - L(Δ w) - w(ΔL)
Since ΔT = 0, we can rewrite this as:
T ΔRt = P (ΔQ) + Q (ΔP) - L(Δ w) - w(ΔL)
We can use the same trick as before, and multiply each term of the equation by 1 –
for the left hand side of the equation, by Rt /Rt, for the first term on the right, by Q/Q, and so on
Rt * T * (ΔRt/Rt) = PQ (ΔQ/Q) + PQ (ΔP/P) - wL(Δ w/w) - wL(ΔL/L)
and then divide through by Rt * T to get what SHOULD BE the equations which open the section
on the “Change in the Rental on Capital” and the “Change in the Rental on Land” on page 83 and 84.
You will note that in simplifying, the author is effectively assuming that the amount of labor on the land and in
the factories remains the same, so ΔL and therefore ΔQ remain constant.
Although the nature of the changes (if the price of agricultural goods rises, landowners win and
capitalists lose) is correct, the magnitude of the change reported can be very misleading. It is as if we had left out
not just the green area, but the light blue area as well, in calculating changes and percentage changes.
This is not just a matter of algebra – the only reason the wage will change is if workers, whose
marginal value product and hence their wage fell in the contracting industry, show up in the employment lines
for the expanding industry, lowering the wage in that industry as well. We cannot consistently assume that the
wage falls without the allocation of the labor force changing.
The adjustments that have to be made are clear qualitatively:
1. The expanding industry is expanding because it can get more total profit by expanding, so if
manufacture is expanding, %Δ Rk will be greater than indicated by the text formula.
2. The contracting industry is contracting so that it will not lose as much as before, so if agriculture is
contracting, %Δ Rk will be less negative than indicated by the text formula.
3. If we can assume a Cobb-Douglas production function, we can give an EXACT answer to the text
problems (for example, Chapter 3, problem 5). This is because the share of labor in a Cobb-Douglas
model is given by the exponent on labor. A worked example of Chapter 3, Problem 5 follows
Chapter 3, Problem 5.
Initial data:
Manufacturing:
Sales revenue = Pm * Qm = 150
Payments to labor = w * Lm = 100
Payments to capital = Rk * K = 50
Note that labor gets 2/3 of the sales revenue, so the Cobb-Douglas production in manufacturing
would be of the form:
Qm = Am * Lm2/3
and we can find that the MPLm relation is of the form: MPLm = 2/3 * Am / Lm1/3
We can apply the percentage change operators to both the above formulas to get:
(note that constants such as Am drop out since they don't change):
%Δ Qm
= 2/3 %Δ Lm
%Δ MPLm = - 1/3 %Δ Lm
Agriculture:
Sales revenue = Pa * Qa = 150
Payments to labor = w * La = 50
Payments to capital = Rt * T = 100
Note that labor gets 1/3 of the sales revenue, so the Cobb-Douglas production in manufacturing
would be of the form: Qa = Aa * La1/3
Repeating the above steps, we will get:
%Δ Qa
= 1/3 %Δ La
%Δ MPLa = - 2/3 %Δ La
Solution:
We are given that the price of agricultural goods goes up by 10 percent and the wage goes up by 5
percent as a result of the agriculture sector's increased demand for labor.
Since Pa * MPLa = wage, we can again use the percent change operator to conclude that:
%Δ Pa + %Δ MPLa = %Δ w
%Δ Pa - 2/3 %Δ La = %Δ w
10 % - 2/3 %Δ La = 5% and so - 2/3 %Δ La = - 5% or
%Δ La = 7.5
%Δ Qa = 1/3(%Δ La) = 7.5 / 3 = 2.5
For manufacturing, we also start from Pm * MPLm, so
%Δ Pm + %Δ MPLm = %Δ w
0 + %Δ MPLa = %Δ w
0 - 1/3 %Δ Lm = 5 %
%Δ Lm = - 15 %
%Δ Qm = 2/3(%Δ Lm) = 2/3 * -15 % = -10 %
We now have the full information needed to apply the formula:
Manufacturing:
Sales revenue = Pm * Qm = 150
Payments to labor = w * Lm = 100
Payments to capital = Rk * K = 50
%Δ Lm = - 15 %
%Δ Qm = 2/3(%Δ Lm) = 2/3 * -15 % = -10 %
Agriculture:
Sales revenue = Pa * Qa = 150
Payments to labor = w * La = 50
Payments to capital = Rt * T = 100
%Δ La = 7.5 %
%Δ Qa = 1/3(%Δ La) = 7.5 / 3 = 2.5 %
For agriculture:
Rt * T * (ΔRt/Rt)
100 * %Δ Rt
100 * %Δ Rt
100 * %Δ Rt
%Δ Rt
=
=
=
=
PQ (ΔQ/Q) + PQ (ΔP/P) - wL(Δ w/w) - wL(ΔL/L)
150 * 0.025 + 150 * 0.10 - 50 * 0.05 - 50 * 0.075
3.75 + 15.0 - 2.5 - 3.75
15.0 – 2.5 = 12.5
= 12.5 / 100 = .125 or 12.5 percent.
The textbook simplification is to:
Rt * T * (ΔRt/Rt) = PQ (ΔP/P) - wL(Δ w/w)
100 * (ΔRt/Rt) = 150 (0.10) - 50 (0.05)
100 * (ΔRt/Rt) = 15 - 2.5 = 12.5
%Δ Rt = 12.5 / 100 = .125 or 12.5 percent.
For manufacturing:
Rk * K * (ΔRk/Rk) = PQ (ΔQ/Q) + PQ (ΔP/P) - wL(Δ w/w) - wL(ΔL/L)
50 * %Δ Rk = 150 * (- 0.10) + 150 * 0
- 100 * 0.05 - 100 * -0.15
50 * %Δ Rk = -15 +
0
5
+ 15
50 * %Δ Rk = - 5
%Δ Rk = - 5 / 50 = - .1 = - 10 %
The textbook simplification is to:
Rk * K * (ΔRk/Rk) = PQ (ΔP/P) - wL(Δ w/w)
50 * %ΔRt
= 150 (0) - 100 (0.05)
%ΔRt
= - 5 / 50 = - .10 = - 10 %
We get exactly the same results with the correct formula and the textbook simplification.
Why? The answer lies in the properties of the Cobb-Douglas production function that the
authors are clearly assuming.
Note that the results on quantity change and on labor change exactly cancel out.
The Cobb-Douglas production function and the text simplification of the percentage change calculations.
If we have a Cobb-Douglas production function, the text simplification is legitimate.
Why? The basic secret is that the exponent on labor in a Cobb-Douglas production function will show the
share of labor, wL / PQ
Consider the functions for manufacturing and agriculture:
Qm = Am * Lm2/3
which we will simplify for concreteness to
Qm = 300 * Lm2/3
and we can find that the MPLm relation is of the form: MPLm = 2/3 * 300 / Lm1/3
Since the wage of labor is Pm * MPLm, we can see that:
wLm = Pm * MPLm * Lm or
wLm = Pm * 2/3 * (300 / Lm1/3) * Lm
wLm = Pm * 2/3 * (300 * Lm2/3)
(Note the change in the exponent)
wLm = Pm * 2/3 *Qm
Hence, wL / PQ = 2/3
Notice the key terms that cancel in the manufacturing data:
Rk * K * (ΔRk/Rk) = PQ (ΔQ/Q) + PQ (ΔP/P) - wL(Δ w/w) - wL(ΔL/L)
50 * %Δ Rk = 150 * (- 0.10) + 150 * 0
- 100 * 0.05 - 100 * -0.15
50 * %Δ Rk = -15 +
0
5
+ 15
PQ (ΔQ/Q) - wL(ΔL/L) = -15 + 15
Substitute the relationship wL = 2/3 PQ to get:
PQ (ΔQ/Q) - 2/3 * PQ * (ΔL/L)
and then substitute the relation of the percent change in labor to the percent change in output:
%Δ Qm = 2/3(%Δ Lm)
PQ * 2/3(%Δ Lm) - 2/3 * PQ * (%ΔLm)
With a Cobb-Douglas production function, these two terms will ALWAYS cancel,
justifying the simplification of the percentage change calculations in the textbook –
at least IF the production function is in fact of the Cobb-Douglas form.
Check the similar logic with the agricultural sector as an exercise.