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Let $M^d (Y,r)=a+bY-cr$ where $M^d = M/P$ is the money demand in the economy. $a,b,c>0$. Derive $LM$.

My try

$M/P=a + b Y - c r$

$b Y = -a + \frac{M}{P} + c r$

$Y = -\frac{a}{b} + \frac{M}{b P} + \frac{c r}{b}$

Is this all there is to it? Equal the equations and solve for $Y$?

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The calculation is correct but the explanation is not simply 'equal the equations and solve for Y'.

Following Blanchard et al Macroeconomics: A European Perspective, 2nd ed. ch 5. by definition

LM curve represents all the combinations of output and interest rate for which the money market is in equilibrium, i.e. the demand for money equals the supply of money.

As given by your question the demand for money is given by $M/P$. Now in your question you do not explicitly label $a+bY−cr$ as a supply for money - but it looks like a supply for money. The reason why you equate the equations is because you want to equate supply and demand for money. If this would be given by some more complex relationship, for example you are given some extra equation for how $M$ or $P$ behaves, you would have to use that too and equating the two equations above would not be enough.

In addition, it is not necessarily to solve it for $Y$, you could as well solve it for $r$. In fact by convention in economics prices go on $y$-axis, so $r$ goes on $y$-axis and output $Y$ would go on $x$-axis, so if you would want to plot LM curve you would actually solve for $r$ not $Y$. This being said if you want to know what is effect of parameters and variables on output (which often is the case) it is not a mistake to solve it for $Y$ instead of $r$.

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    $\begingroup$ @bymathformath yes $\endgroup$
    – 1muflon1 ♦
    Dec 28, 2020 at 15:34
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    $\begingroup$ @bymathformath well regarding that previous assignment it might be that if you would combine the solution with some other identities/theories you could show it’s always lower. But just from what you showed me there it could be either lower or higher. But depending on different theories you use you might be able to prove that M_2>M_1 etc. - if you restrict parameter ranges in that new answer you could show that one derivative is higher than other. Alternatively you can calculate a range of parameter values for which one derivative is higher than another - I don’t the details of your assignment $\endgroup$
    – 1muflon1 ♦
    Dec 28, 2020 at 15:43
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    $\begingroup$ @bymathformath but it can also be just a mistake you should consult that with your teacher $\endgroup$
    – 1muflon1 ♦
    Dec 28, 2020 at 15:46
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    $\begingroup$ @bymathformath right but sometimes assignments on purpose don’t feed you every single relationship and expect you to know that for example by definition I=S or other relationships. Or there could be seemingly innocuous remark by professor that you should always in class assume government runs balanced budget where G=T and so on. Regardless definitely for these sort of questions contact your teacher it’s not like we can know everything they said in class or how they like to create their assignments $\endgroup$
    – 1muflon1 ♦
    Dec 28, 2020 at 16:01
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    $\begingroup$ @bymathformath no Z=C(Y-T)+G+I(r) does not give you IS curve - I mean you can just look at any macro textbook. Then either it’s a mistake in assignment, it’s a trick question and answer is you can’t guarantee it will be bigger or you are supposed to use other well known macro identities/knowledge about behavior of terms to further constrain the coefficient parameters $\endgroup$
    – 1muflon1 ♦
    Jan 4, 2021 at 11:45

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