# Solving 2 equations, 2 unknowns (Money Growth = Inflation + Real growth rate)

#### CHRISTMAS

##### New member
Money Growth = Inflation + Real growth rate
Inflation = Expected inflation + 1 x (real growth rate - potential growth rate)

I know:
Money growth = 7%
Expected inflation = 10%
Potential growth rate = 10%

Hence:
7 = Inflation + real growth rate
Inflation = 10 + 1 x (real growth rate - 10)

What are inflation and real growth?
Could you show me the steps, please

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#### JeffM

##### Elite Member
Your questions are so lacking in detail and context that they are impossible to answer.

#### tkhunny

##### Moderator
Staff member
Have you considered substituting the expression for "Inflation" from the second equation into "Inflation" from the first equation?

#### CHRISTMAS

##### New member
An answer key tell me that inflation = 7%; real growth = 0%

I have tried several things, but am still not seeing something!

#### JeffM

##### Elite Member
An answer key tell me that inflation = 7%; real growth = 0%

I have tried several things, but am still not seeing something!
Why don't you give us the complete problem in its exact words.

#### CHRISTMAS

##### New member
The original problem as requested

Parts a + b are easy to follow. Having trouble seeing how the solved for c, but it looks like they found that the growth rate = 0%, plugged this into the formula for Money Growth --> inflation then = 7%

I wrote to an economist, who said that in c. "The SRAS curve must cross the original point at which Inflation = 10% and Growth=3%. This means that in the SRAS formula E[pi] =10%, not the lung run 4%. This makes sense ofcourse, the short run expectation is what inflation was (i.e. 10%)."

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#### CHRISTMAS

##### New member
Here is an online solution that I found, still not seeing it

Here is an online solution that I found, still not seeing it

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#### JeffM

##### Elite Member
Parts a + b are easy to follow. Having trouble seeing how the solved for c, but it looks like they found that the growth rate = 0%, plugged this into the formula for Money Growth --> inflation then = 7%

I wrote to an economist, who said that in c. "The SRAS curve must cross the original point at which Inflation = 10% and Growth=3%. This means that in the SRAS formula E[pi] =10%, not the lung run 4%. This makes sense ofcourse, the short run expectation is what inflation was (i.e. 10%)."
The whole problem is a mess, which may be part of the reason that you are having trouble. It seems quite clear that there is additional information provided outside of what you have given us, which may in fact eliminate much of the mess in what you have shown. (It won't eliminate all of it: the author is sloppy about the difference between change and rate of change.) As nearly as I can see, there are six variables in the model, but we seem to be provided provided with only two equations, neither of which is given any justification.

Let's start by naming the variables in algebraic notation.

$$\displaystyle \text {rate of change in aggregate demand for the current year is } d \%.$$

$$\displaystyle \text {actual rate of change in aggregate supply for the current year is } a \%.$$

$$\displaystyle \text {rate of change in monetary aggregate for the current year is } m \%.$$

$$\displaystyle \text {actual rate of inflation for the current year is } i \%.$$

$$\displaystyle \text {expected rate of inflation for the current year is } e \%.$$

$$\displaystyle \text {sustainable rate of potential annual change in aggregate supply is } p \%.$$

Good so far? This is just naming the variables in standard algebraic notation. (I have no idea why economists do not use the simple notation of normal algebra.)

One equation given in the text is:

$$\displaystyle \text {EQUATION 1: } m = d + i \implies d = m - i.$$

Although no explicit theoretical justification for this equation is given in the statement of the problem itself, it can easily be derived from a very rigid quantity theory of money. (I am aware of no empirical support for a theory so rigid.)

No justification is given for the other equation explicitly identified as an equation in the statement of the problem itself, and I have some difficulty in guessing what the detailed justification may be. (It may have been specified outside the statement of the problem.) In any case, it is quite clear that the equation is:

$$\displaystyle \text {EQUATION 2: } i = e + \{ 1 * (a - p)\} \implies a = i - e + p.$$

The first part of the problem asks what m will equal if i = 10 and d = 3. (Remember I have defined the variables as percents.) Well, m = 3 + 10 = 13. Quite simple because we are given two of the variables in equation 1. The second part of the problem asks what m will be given i = 4 and d = p. What is going on here is that p is not really being treated as a variable but as a constant. Unfortunately, the problem statement nowhere discloses what the numeric value of that constant is. Based on the answer given, we can deduce that p = 3. (This value may be disclosed somewhere in the text that surrounds the problem statement.) Based on that deduction, we realize that there are actually only 5 variables and that equation 2 can be re-written as

$$\displaystyle \text {EQUATION 2a: } a = i - e + 3.$$

However, there is a third equation that the problem statement hints at, but neither explicitly discloses nor justifies, namely

$$\displaystyle \text {EQUATION 3: } s = d.$$

This presumably is justified by a very rigid application of Say's Law to the very short term (an application that Say may never have intended and that almost certainly cannot be supported empirically no matter what is almost certainly the approximate validity of Say's Law in the very long term.)

That lets us re-state equation 2 again to

$$\displaystyle \text {EQUATION 2b: } d = i - e + 3.$$

Finally, we have the author's statement that we already know the value of e. Actually, we do not unless it is given somewhere OUTSIDE the problem statement. It appears that he is assuming some sort of lagged expectation function, in which the market's expectation for the current year is identical to actuality for a previous year. (That is, he is assuming that no one paid any attention to the actions of the central bank. And the economist to whom you talked has interpreted the problem the same way.) Thus, it appears then that there is another undisclosed equation, namely

$$\displaystyle \text {EQUATION 4: } e = \text { previous year's actual inflation } = 10.$$

So we can restate equation 2 again as

$$\displaystyle \text {EQUATION 2c: } d = i - 10 + 3 = i - 7.$$

Now we really do have two equations in two unknowns because we are assuming that m = 7, meaning that

$$\displaystyle \text {EQUATION 1a: } d = 7 - i.$$

Finally, after a bunch of work that is nowhere explicitly stated in the problem itself, we have a solvable system of two linear equations with two variables that can easily be solved by elimination:

$$\displaystyle d = i - 7 \text { and}$$

$$\displaystyle d = 7 - i \implies$$

$$\displaystyle 2d = 0 \implies d = 0 \implies 0 = i - 7 \implies i = 7.$$

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