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1muflon1 ♦
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In addition the reason why we can observe increase in money supply over long periods of time is that central banks try to maintain price stability, which many of them define as a yearly inflation of around $2\%$. For that to happen money supply needs to constantly increase as well. Consequently, even though it is correct to say that private banks create most of the money in our economic system, the central bank is ultimately ultimately in control in how much money is being created. They do this either through bank regulation, interest rate setting or by control of monetary base - this is also explained in the McLeay Radia, & Thomas (2014) you provide link to in your Q. If a central bank would be hell bent on it and would purposefully want to keep $M$ constant, they could do it (by unreasonably hiking interest rate, regulating borrowing in a such way as to make it nearly impossible etc). Conversely, if central banks think there is not enough money supply in the economy they can increase it regardless of bank lending (e.g. direct purchase and monetization of government debt). Note here that central bank can actually do this even without using private banks. For example, central bank can buy government debt which creates new money. Thus central bank can always move money supply wherever it want's to in principle (even if private banks would not want to lend).

In addition the reason why we can observe increase in money supply over long periods of time is that central banks try to maintain price stability, which many of them define as a yearly inflation of around $2\%$. For that to happen money supply needs to constantly increase as well. Consequently, even though it is correct to say that private banks create most of the money in our economic system, the central bank is ultimately ultimately in control in how much money is being created. They do this either through bank regulation, interest rate setting or by control of monetary base - this is also explained in the McLeay Radia, & Thomas (2014) you provide link to in your Q. If a central bank would be hell bent on it and would purposefully want to keep $M$ constant, they could do it (by unreasonably hiking interest rate, regulating borrowing in a such way as to make it nearly impossible etc). Conversely, if central banks think there is not enough money supply in the economy they can increase it regardless of bank lending (e.g. direct purchase and monetization of government debt).

In addition the reason why we can observe increase in money supply over long periods of time is that central banks try to maintain price stability, which many of them define as a yearly inflation of around $2\%$. For that to happen money supply needs to constantly increase as well. Consequently, even though it is correct to say that private banks create most of the money in our economic system, the central bank is ultimately ultimately in control in how much money is being created. They do this either through bank regulation, interest rate setting or by control of monetary base - this is also explained in the McLeay Radia, & Thomas (2014) you provide link to in your Q. If a central bank would be hell bent on it and would purposefully want to keep $M$ constant, they could do it (by unreasonably hiking interest rate, regulating borrowing in a such way as to make it nearly impossible etc). Conversely, if central banks think there is not enough money supply in the economy they can increase it regardless of bank lending (e.g. direct purchase and monetization of government debt). Note here that central bank can actually do this even without using private banks. For example, central bank can buy government debt which creates new money. Thus central bank can always move money supply wherever it want's to in principle (even if private banks would not want to lend).

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1muflon1 ♦
1muflon1 ♦
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In addition the reason why we can observe increase in money supply over long periods of time is that central banks try to maintain price stability, which many of them define as a yearly inflation of around $2\%$. For that to happen money supply needs to constantly increase as well. Consequently, even though it is correct to say that private banks create most of the money in our economic system, the central bank is ultimately ultimately in control in how much money is being created. They do this either through bank regulation, interest rate setting or by control of monetary base - this is also explained in the McLeay Radia, & Thomas (2014) you provide link to in your Q. If a central bank would be hell bent on it and would purposefully want to keep $M$ constant, they could do it (by unreasonably hiking interest rate, regulating borrowing in a such way as to make it nearly impossible etc). Conversely, if central banks think there is not enough money supply in the economy they can increase it regardless of bank lending (e.g. direct purchase and monetization of government debt).

In addition the reason why we can observe increase in money supply over long periods of time is that central banks try to maintain price stability, which many of them define as a yearly inflation of around $2\%$. For that to happen money supply needs to constantly increase as well. Consequently, even though it is correct to say that private banks create most of the money in our economic system, the central bank is ultimately ultimately in control in how much money is being created. They do this either through bank regulation, interest rate setting or by control of monetary base - this is also explained in the McLeay Radia, & Thomas (2014) you provide link to in your Q. If a central bank would be hell bent on it and would purposefully want to keep $M$ constant, they could do it (by unreasonably hiking interest rate, regulating borrowing in a such way as to make it nearly impossible etc). Conversely, if central banks think there is not enough money supply in the economy they can increase it regardless of bank lending (e.g. direct purchase and monetization of government debt).

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1muflon1 ♦
1muflon1 ♦
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Question : Is what I wrote above accurate?

There is a bit more nuance to it but 1-3 points are 'corect-ish'.

I guess what I'm getting at is: Is there some connection to the bank making loans, and those loans being used to fund additional productive activities* than otherwise possible (say, given the agent's initial budget constraints), and long run growth in the money supply?

Not in the long-run. There is a relationship between growth of money supply and economic activity but not in the long-run only in the short run. This is because in the long run prices are flexible and they will quickly adjust to increase in money supply (see further discussion of that in any standard macro text like Blanchard et al Macroeconomics ch 8-10).

If someone sees fit to answer, ideally, I'd like some math. Or if there is some model that can show the players (consumers, firms, banks etc.) and their interactions, that would be great.

It is possible to describe this with a model that is built upon individual interactions of consumers, firms and so on, but that would take too much space as even very simple micro founded macro models are quite large. For example, a basic microfounded (i.e. explicitely including interaction between households, firms etc) macro model introduced Woodford, M. (2011). Interest and prices, takes over 30 pages and that is a graduate text that does not explain basics that one would probably have to cover to explain it to someone with no economic training. Consequently, let me offer more concise model (which will not have qualitatively different results from more nuanced microfounded model) where I will skip description of individual interaction of individuals and just directly impose assumptions on behavior of aggregates that could otherwise be derived from individual interactions of individuals and firms (if you want more nuanced models look at above mentioned Woodford ch 3 or at Romer Advanced Macroeconomics Ch 6, 7 and 11).

Thus I will introduce simplified model based of above mentioned Blanchard et al. Let us start by description of money market. The money market can be described using:

$$M/P = L(Y,i) \tag{1}$$

Where $M$ is money supply, $P$ price level (so $M/P$ is real money supply), $L$ is money demand, $Y$ is real output and $i$ interest rates.

The money demand will be given as follows:

$$L= f_1 Y - f_2 i \tag{2}$$

Because money demand will increase with economic activity, and decrease with interest rates. This is what would normally be microfounded, but I will just assume it as a shortcut without proving it. However, intuitively it is not unreasonable assumption, when aggregate output (which is economically also equal to income) increases there is more economic activity so people will demand higher real money balances as there will be more transactions in the economy. Next if the interest rate is high people will demand less loans and thus also money, and will save more and thus hold less money balances.

Next, goods market will be given by:

$$Y = C + I +G \tag{3}$$

where $C$ is consumption, assumed to follow $C=c_0 + c_1 (Y-T)$ with $0<c_1<1$. Where $Y-T$ is income after taxes. Again aggregate consumption would typically be derived based on individual interactions, but it is reasonable to assume people consume more if their disposable income increases.

$I$ is the investment assumed to follow $I = \bar{I} +d_1 Y- d_2 i$. Once again this would normally be microfounded, but it is reasonable to say that investment increases when output increases, and investment decreases when interest rate $i$ increases as it is harder for firms to get loans and so on.

Finally $G$ is government spending, there will be no additional assumptions on its behavior for your question it is not relevant.

So given above assumptions we will have:

$$Y = c_0 + c_1 (Y-T) + \bar{I} +d_1 Y- d_2 i +G \tag{4}$$

Solving for $Y$ so we get expression for goods market equilibrium yields:

$$Y = \frac{1}{1-c_1 -d_1}\left( c_o + \bar{I} + G - c_1T \right) - \frac{d_2}{1-c_1-d_1} i \tag{5} $$

Now finally, in an economy from a macro perspective both the goods market equilibrium and money market equilibrium above must be also in equilibrium together.

Recall that money market equilibrium was given by equation (1) substituting (2) into (1) we get:

$$ M/P=f_1Y−f_2i \tag{6}$$

Solving 6 for $i$, substituting into $5$ and once again solving for $Y$ so we isolate output we get:

$$ Y = \frac{1}{(1-c_1 - d_1) \frac{f_2}{ d_2} + f_1} \frac{M}{P} + \frac{1}{1-c_1-d_1}+d_2 \frac{f_1}{f_2} \left( c_o + \bar{I} + G - c_1T \right) \tag{7} $$

Examining how $Y$ varies with respect to real money balances we get that:

$$\frac{dY}{d(M/P)} = \frac{1}{(1-c_1 - d_1) \frac{f_2}{ d_2} + f_1} \tag{8}$$

So indeed increase in amount of money will increase output, in fact there even be multiplier effect because the higher the output is the more demand for money there will be etc.

However, note the above does not hold in a long run. This is not captured by the brief and simplistic model above but while in short run we can assume $P$ is fixed, in a long run $P$ will most certainly not be fixed, and in fact it will increase in response to increase in $M$ (see discussion of that in Blanchard et al ch 9). Hence, in a long run increase in money supply cannot stimulate output (which would via multiplier stimulate more demand for money and again more output). Consequently, and answer to your question is no (although I am not sure what you mean by long run - in economics long run is defined as time period where $P$ if flexible and can adjust).