Rearranging Equations in Economics

Introduction

Economists write equations to describe economic models. However, often the variable we want to study is not by itself on the left-hand side of the equation. In a few simple steps, that variable can be moved.

An example

A business knows the equation for quantity demanded based on price:

`Q_d = 100 - 2P`

But the business wants to find out how what price will be needed to create a certain level of demand.

That requires rearranging the equation so that P is by itself on the left-hand side:

`P = 50 - Q/2`

How do I rearrange equations?

Rules for rearranging equations

Rule #1: You can perform an operation to one side of an equation as long as you perform the same operation to the other side of the equation.

RULE #2: Isolate your intended subject by moving or canceling out parts of the equation that get in the way.

RULE #3: Undo an equation in reverse order of the order of operations.

An example

Solve the following equation for X. In other words, isolate X.

`Y = (5X - 3)/2`

First, ask how this equation was initially created. What operations were done to X to formulate this equation, and in what order?

  • First, we multiplied X by 5.
  • Next, we subtracted 3.
  • Lastly, we divided it all by 2.

To undo what has been done to X, we have to work in reverse order.

  • First, undo the division by multiplying by 2.
  • Next, undo the subtraction by adding 3.
  • Finally, undo the multiplication by dividing by 5.

`X = (2Y + 3)/5`


Steps for rearranging equations

  1. Read the equation and identify its variables.
    Consider the following demand function:

    $Q = 100 - 2P$

    The unknown variables are Q (quantity) and P (price).

  2. Determine which variable you want to be the dependent variable, or "subject" of the equation. That is, which variable are we going to isolate?
    As it stands, Q (quantity) is the subject of our demand function. However, suppose we want to know what price will be for a given quantity.

    Then we want to isolate P in the equation.

  3. Rearrange this equation so that the variable you selected is by itself, and on the left-hand side of the equals sign, using Rules #1, #2 and #3.
    Let's solve the following equation for P, one step at a time.

    $Q = 100 - 2P$

    1. Add 2P to both sides.

      $Q \color{red}{+ 2P} = 100 - 2P \color{red}{ + 2P}$
    2. Cancel out the 2P's on the right hand side.

      $\require{cancel}Q + 2P = 100 \color{red}{\cancel{- 2P + 2P}}$

      $Q + 2P = 100$
    3. Subtract Q from both sides.

      $Q + 2P \color{red}{ - Q} = 100 \color{red}{ - Q}$
    4. Cancel out the Q's on the left hand side.

      $\require{cancel}\color{red}{\cancel{Q}} + 2P \color{red}{\cancel{- Q}} = 100 - Q$

      $2P = 100 - Q$
    5. Divide both sides by 2.

      $2P\color{red}{/2} = (100 - Q)\color{red}{/2}$
    6. Cancel out the 2's on the left hand side and simplify.

      $\require{cancel}\color{red}{\cancel{2}}P\color{red}{\cancel{/2}} = (100 - Q)/2$

      $P = (100 - Q)/2$

      $P = 100/2 - Q/2$

      $P = 50 - Q/2$
    We're done! Now we have a new equation for P in terms of Q.

Where are rearranged equations used in Macroeconomics

Below are a few worked-out examples of rearranging equations in a macroeconomics context.

GDP and Consumer Spending

GDP can be written as the sum of consumer spending (C), investment (I), government purchases (G), and net exports (NX):

$GDP = C + I + G + NX$

An economist may want to look at how C, consumer spending, is affected by the other variables. We can rearrange this equation to solve for consumer spending.


Nominal vs. Real Pay

Real pay, that is pay corrected for inflation, is calculated using the following formula, where Nominal Pay is the dollar amount received, not corrected for inflation. CPI is the Consumer Price Index:

$\text{Real Pay} = (\text{Nominal Pay}/CPI) \times 100$

An economist may want to know nominal pay, given information on real pay. We can rearrange this equation to produce a new equation for nominal pay.



The Equation of Exchange and the Velocity of Money

Economic theory puts forward the equation of exchange as:

$M \times V = P \times Y$

where M is the nominal stock of money, V is the velocity of money, P is the nominal price level, and Y is the flow of real transactions.

An economist may want to graph the velocity of money --- i.e., how money moves through the economy in a year.


Twin Deficits

GDP can be written as the sum of consumer spending (C), investment (I), government purchases (G), and net exports (NX):

$GDP = C + I + G + NX$

GDP can also be written as the sum of consumer spending (C), saving (S) and taxes (T):

$GDP = C + S + T$

Since each equation is equal to GDP, we can set the two equations equal to each other:

$C + S + T = C + I + G + NX$

Now let's say we want to look at net exports.

Where are rearranged equations used in Microeconomics?

Below are a few worked-out examples of rearranging equations in a microeconomics context.

The formula for elasticity

We can use the formula for elasticity to predict the percentage change in quantity demanded for a given percentage change in price.

Elasticity of demand is calculated using the following formula:

$Elasticity = \% \text{change in Q} / \% \text{change in P}$

"Change" can also be represented by the delta symbol, "Δ". So let's rewrite the equation:

$Elasticity = \%\Delta Q / \%\Delta P$

We can rearrange this equation to solve for the percent change in Q, or %ΔQ.


Finding Total Cost when you know Average Total Cost

Often an economic model will calculate average total cost using the formula:

$ATC = TC / Q$

where ATC = average total cost; TC = total cost; Q = quantity

But what if we know ATC, and we want to compute TC?


Budget Constraints

A budget constraint represents all the combinations of two goods, X and Y, that a consumer can afford with his income, I.

Suppose the price of good X is PX and the price of good Y is PY. The consumer's budget constraint is commonly written in the following format:

$P_X X + P_Y Y = I$

Written in this way, the budget constraint says that the sum of a consumer's expenditure on X and his expenditure on Y, must equal his income.

If, for example, we want to graph the budget constraint with good Y on the vertical axis and good X on the horizontal axis, it would be useful to solve for good Y.

Next Steps...


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