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Wednesday, January 15, 2020

The Monetary Monopoly Model

What I refer to as the Monetary Monopoly Model is the simplest possible mathematical model that captures basic concepts from Modern Monetary Theory (MMT). Despite its simplicity, it gives a good feeling of how a sovereign could pin down the value of a brand new currency (relative to existing currencies, or the value of real goods or services). However, the model makes almost no assumptions about private sector behaviour, and such assumptions would be needed to simulate an existing industrial capitalist society. The reason to start with this model is that the discussion of those behavioural assumptions will drown out the MMT-specific parts of the model.

The model is based on the model presented by Pavlina R. Tcherneva, in "Monopoly Money: The State as a Price Setter."* Tcherneva's discussion follows earlier texts on the imposition of money in European colonies, by other authors; I chose this article solely because it had a convenient mathematical exposition within the article. Readers that are interested in the academic precedents for these concepts are invited to consult the citations in Tcherneva's article. The model presented here uses my preferred notation, but cannot be considered to be an original model.

Model Structure

For readers who are allergic to equations, feel free to jump ahead to the sub-section "Interpretation in a Fictional Country." An intuitive example of how this model works is given therein.

The model determines relationships between discrete-time time series, that is, time series that are defined for time points 0, 1, 2, N. (We can allow an infinite time horizon, but that makes the discussion of what set the series lies in more awkward. For a time horizon $N$, a time series is a vector in ${\mathbb R}^N$.) We denote a time series as $x$, while its value at a particular time $t$ is $x(t)$. (This follows the convention of systems theory, which is preferable to using subscripts to denote the time indexing, as we use subscripts to distinguish variables.)

The models refers to a central government, and another sector that encapsulates all other economic entities. For ease of presentation, the central government will be referred to as the government, and the non-central government sector as the private sector. If we extend this to the real world, this is somewhat awkward, as sub-sovereigns or foreign governments are lumped in with the "private sector." If one is building a model with such entities, one would presumable need to use the rather clumsy name of "non-central government sector."

We assume that the government issues a brand new currency at time point 0. All prices are to be expressed in terms of this currency unit. We denote the private sector holdings at the beginning of time $t$ as $M(t)$. Since the currency is created at time 0, we know that $M(0) = 0.$. Furthermore, the government will not lend to the private sector, and so $M(t) \geq 0$ for all $t$.

The government imposes taxes that are fixed nominal amounts, with a total tax bill $T$ on the private sector. Since there are no transfer payments, $T(t) \geq 0.$

Finally, the government wishes to procure a commodity or service (e.g., labour hours). We will refer to this as a good, denoted $g$. It fixes the price it will pay for this good $P_g(t)$ (we assume that $P_g(t) > 0$), and it will have a quantity $Q_g(t)$ offered to it by the private sector, where $Q_g(t) \geq 0$.

The accounting identity for the money supply is:
M(t+1) = M(t) + P_g(t)Q_g(t) - T(t).

Model Solution

The solution of the model is straightforward.

At $t=0$, the accounting identity is:
M(1) = P_g(0)Q_g(0) - T(0).
Since $M(1) \geq 0$,
Q_g(0) \geq \frac{T(0)}{P_g(0)}.
We see that the government can guarantee a minimum quantity supplied at $t=0$ by the appropriate choice of price paid and the fixed tax. More can be supplied, as other entities may wish to hold a strictly positive money balance at the beginning of time period 1. (And as noted by Tcherneva, the quantity supplied is likely quantised -- an integer value -- and so other entities may need to supply extra goods to the government in order for the end-of-period money balance to be non-negative.)

In later periods, the outcome is slightly more complicated.

If we assume that the quantity supplied is strictly positive,
P_g(t) Q_g(t) = T(t) + M(t+1) - M(t),
Q_g(t) = \frac{T(t) + M(t+1) - M(t)}{P_g(t)}.
Without any behavioural assumptions, there are no obvious implications we can draw about the properties of $Q_g$. We would need a model of the business cycle to advance further.

There is one special case of note. If the quantity supplied is to be zero, we see that:
M(t+1) = M(t) - T(t).
In order for $M(t+1)$ to be non-negative, supplying goods is only possible if the existing money stock is greater than the single period tax bill. So long as the tax bill has a lower bound, the private sector can only supply zero goods for a finite time period (until the money stock is fully run down).

The reason why this case is of interest is that if the private sector is not supplying any of the good to the government, the requisition price of that good has no effect on anything. Under typical behavioural assumptions, the requisition price could be set to any value that is below some threshold that triggers non-zero supply, and the solution of a broader model is unchanged. As will be discussed elsewhere, the idea is that the requisition price is used as a control variable to drive the price level, but if the quantity supplied is zero, it loses its effect on the private sector economy.

Interpretation in a Fictional Country

Imagine a government is set up in a country where there is a pre-existing society, possibly with its own currency (from a failed government, or a foreign currency, or a commodity currency).  It wants to create its own sovereign (new) currency. Colonial regimes fit this description (more or less), and they used brutal methods to impose new currency regimes.

Let us assume that the governing elites are concentrated in one area, but have the capacity to enforce taxes. (The infrastructure needed to impose taxes is not explicitly modeled; if necessary, one could imagine troops and police are being provided by the colonial master country.) In addition to creating a currency, it wishes to requisition labour hours from the populace to prepare fancy feasts for the elite.

 It can kick-start the currency by doing the following.
  • Impose a household head tax of $20/week.
  • Pay $\$$10/hour for labour (preparing meals).
(Note that "$" refers to the brand-new currency.)

Under the assumption that the government can make the head tax stick (which is standard in these models), and the populace starts out with $0 in money holdings, there has to be at least 2 hours of service provided per household in the first week. More could be supplied, but that means that the households would have extra dollars lying around for the next week. Labour supplied could be lower if those excess dollars are used to pay the tax. However, this is only temporary, as the taxes will be eliminating the dollar holdings.

The model is only discussing the transactions undertaken by the government. It does not attempt to say what is happening among the households -- we need more information to see what happens. 

For example, the populace could take the demand as being similar to feudal arrangements, and everyone shows up for two hours per week to discharge their obligations.

However, if there was an existing monetary economy, the new currency (dollars) could be traded versus the existing goods and services and money. We could easily see specialisation, where one household shows up to do 20 hours work, gaining $\$$200. That household will pay $\$$20 in taxes, and will trade the remaining $\$$180 to other households in exchange for goods and services produced by the private sector. The households that provided those goods and services uses the dollars to discharge their tax obligation (without providing labour themselves to the government).

If we assumed that households acted like their counterparts in classical models, the $20 should have a market value equal to the market value of working for two hours for the government. If we assumed that wages for all jobs are equal (an obviously incorrect simplifying assumption), the government wage would have an exchange value to the existing private sector currency that is pinned down by wage arbitrage. (Since that assumption is too strong, we need some kind of trade-off function between the requisitioned labour position and other types of labour.)

In summary, the real value of labour helps generate a relationship between the requisition price (which is an arbitrary price administered by the government) and the price level of the whole economy.

Initial Implications

The model given is a toy model, but it has many implications for the discussion of MMT. One extremely common complaint about MMT coming from economists from other backgrounds is that "MMT has no mathematical models." In my view, that is a misunderstanding; proponents of MMT are part of the post-Keynesian tradition, which is very cynical about the usefulness of mathematical models. Complaining that MMT does not approach economics in the same way as neoclassical economics (e.g., referring to so-called canonical models) is eyebrow raising, when one considers that post-Keynesians have been arguing for decades that the neo-classicals have been doing practically everything incorrectly. If a group is operating incorrectly, one should expect other groups to act differently.

Neo-classicals quite reasonably argue that those complaints are somewhat tedious and/or behind the times, but at the same time, if one is bored with post-Keynesian complaints about neo-classical economics, one presumably should be aware that the post-Keynesians question the mathematical frameworks used, so one should expect a different mode of exposition. (Somehow, some people seem to be bored by arguments that they have not even looked at.)

However, if I want to translate the situation into terms that are more familiar, this monetary monopoly model is a toy model that captures key aspects of what proponents of MMT are discussing. More complex models are needed to discuss the business cycle in an industrial capitalist society, but those models should have some theoretical similarities to this toy model.

I will need to jump to discussing more complex models to flesh out the previous statements. But we can start off with a few observations.
  • A key point to this model is that the central government is acting like a monopolist. I believe it is a standard story in undergraduate textbooks (none of which I own, so no citations for this assertion) that a monopolist can either set the price, or the quantity. (It can set quantity by auctioning off goods.) What is distinctive about this model is that the government is setting a price, not the quantity.
  • As noted in the interpretation, the government is setting the price of one good that it is requisitioning. If we want a model that offers a plausible story about the determination of the price level in an industrial economy, we need other goods, and then have a means of pinning down a relationship between that good and other traded commodities (including labour costs). The requisition price cannot be completely decoupled from other prices, other than in the case where the quantity supplied is zero.
  • The fact that taxes are fixed ("lump sum") is a key factor for price level determination in the first period. However, fixed taxes are generally rare in the developed countries. (For example, property taxes are relatively fixed, but are collected by sub-sovereigns in Canada and the United States.) For a central government, taxes are better described as being as a percentage of nominal activity variables (incomes, or sales). Given the observed stickiness of prices in the real world, we can sort-of pretend that fixed taxes can be used to simulate them. However, taxes as a percentage of nominal incomes helps eliminate risks of large jumps in the price level, since the tax burden would explode.
  • The clear-cut determination of the (lower bound of the) value of the currency within this model is in stark contrast to the difficulty of determining the value of an unbacked private sector currency, such as the Bitcoin crypto-currency. The high price volatility of Bitcoin (when expressed in either units of hard fiat currencies, or commodity units) reflects the lack of a real-world price anchor. (Some crypto-currencies -- the "stable coins" -- are essentially unregulated securities with a peg arrangement, and can be priced as such.)
  • Real-world governmental requisition appears to be closer to price-taking than price-setting. In other words, one can model this by having the government set the quantity of goods to requisition, and let the price float. The difference between price taking and price setting is a big topic of discussion, that will be deferred. For now, I will firstly note that there is still considerable stickiness in the prices paid by government. Secondly, the decision to be a price taker was an ideological decision. That is, believing that the government ought to be a price taker is a normative statement, and the side effects of that decision need to be analysed. However, a Job Guarantee provides a case where the government is undoubtedly a price setter. As such, a proper analysis of the Job Guarantee means that there has to be some similarities to this model. A a result, the easiest complex models to implement that are true to the monetary monopoly model template are those that include a Job Guarantee.

Concluding Remarks

Most people are interested in models of industrial capitalism, and not how to pin down the value of a brand-new currency unit. We need to bolt on equations describing the private sector to this model in order to accomplish this.

Given the difficulty of anyone to build legitimate forecasting mathematical models that capture all structural relationships within the economy, we should expect that attempts to model private sector behaviour will have flaws. The discussion of those flaws will derail any attempt to evaluate the usefulness of Modern Monetary Theory, which is why I do not recommend running straight to those models.

Link to the next article in this sequence: MMT and policy variables.


Although my fictional example is a nod to historical colonial events described in the MMT literature, it is obviously my hypothetical fable that literally implements the mathematical model. I will add more references to the literature, as well as the textual MMT documents that discuss the concept of a monetary monopoly to give more bibliographic information for readers so inclined. I have a few references to work with (and could dig some out of the Tcherneva article), but I would be very happy if readers pointed out what they view as the most important articles in the literature in this area (either via comments here on Twitter).

My belief is that most of my readers will only want a paragraph or two on the historical developments, and I only need to point them to the literature if they are so inclined to pursue that angle. My objective is to keep the primer short (although it might have reprinted blog articles documenting my comments on the "online MMT wars" as a second part to the book, which can be skipped), and so I am focusing on what a more advanced reader would want to know about MMT, along with bibliographic information.


Oeconomicus, Volume V, Winter 2002

(c) Brian Romanchuk 2020


  1. "The accounting identity for the money supply is:

    My math is very rusty so my comment may be off-base--but I think this is not quite correct.

    The money supply held, if any, is incremented by the sum of the two time series recorded in the current time span. Hence, at the end of the current time span, the money supply should be the carry-over from previous period plus current period change.

    To me, that would result in notation like this:


    If I am correct, the change messes up your later argument because M(t) would exist even in the beginning year.


    1. I wrote that M refers to beginning of period balances. It is done exactly this way to avoid having time series exist at t=-1, which mathematically is not part of the system of equations. Your M refers to the end of period balance, and doing so breaks the notation.

    2. I can't get my mind around your notation. Using your notation, the time series "Pg(t)Qg(t)−T(t)" becomes known at the same point in time as M(t+1) but carries a different locating time label. We seem to have two time locating systems in one equation.

    3. My notation is standard for discrete-time systems. All time series are defined on the time axis 0,1,2,...

      Using your notation (which is common), only one series is defined for t=-1, other series are defined on 0,1,2,... One wants to avoid creating special cases in notation.

      If someone wants to use your notation (it is entirely possible I used that notation in my sfc_models code), all variables need to have a value inserted for t=-1. This can create other problems.

    4. I can accept what you are saying. However:

      I wonder if this is the source of considerable economic math confusion. "Pg(t)Qg(t)−T(t)" is the sum of time series playing out over the interval between points x(t) and x(t+1) OR x(t) and x(t-1). Your notation uses the first choice, which seems to me to be a valid option.

    5. This is discrete time. From a mathematical standpoint, there are no other notions of time. To interpret, imagine that t=0 corresponds to January 2010. The P(0) is assumed to be a fixed price for the month of January, and Q(0) is a flow that happened in that calendar month.

      What is unusual about my notation is that the value used is the value at the beginning of the month, not the end.

    6. "...imagine that t=0 corresponds to January 2010."

      The problems come at the transition points which are discrete points, not time-spans. So, using flow Q(0), the value increases systematically during the time span, revealing itself accurately only at the discrete transition point (to become fixed forever).


      I would argue that the value for Q(0), M(0), and T(0) is not established until time-span(0) ends (which is the beginning of time-span(1)).

      Looking at P(0), I would agree that this value could be established by government at the beginning of time-span(0),

      What if government tries to establish Q(0), M(0) and T(0) by fiat at the beginning of time-span(0)? I think the assumed decision-making role of the private sector would determine the form of the mathematical expression.

    7. Look at a data release for monthly data. There are values for months. There are no intermediate time points. A discrete time model determines relationships between such data.

      The government sets a price and tax for the month. The supplied goods (Q) will be offered during the month, but there is nothing in the model which tells us exactly when. All we know is the final total of the monthly flow.

      If the government changed the price during the month, the value of P would reflect the weighted average for the month, so that the accounting will work.

    8. This comment has been removed by the author.

    9. I think this is a great post, an important one for MMT. The problem is that I remain unconvinced that the notation is consistent throughout.

      Could we try this:

      You write "Since the currency is created at time 0, we know that M(0)=0.."

      I would record a value of zero at position 0 on our time scale.

      Having recorded 0, money is introduced and variables Qg(x), Pg(x), M(x) and T(x) begin to play out. How do we notate the variables that we record next on our time scale?

      My follow-up question would be if the same system is used consistently throughout the post?

    10. The notation is consistent. It’s the beginning of period money balance. I might switch to end of period, but then that would force me to re-write everything to be consistent with that change. I would only do that when it goes into a book.

  2. Define series of discrete time steps with interval 1:

    t = 0,1,2,...,N

    Specify initial condition:

    M(0) = 0

    Change in money supply:

    M(t) - M(t-1) = Pg(t)Qg(t)-T(t)

    If the government or private firms issue credit against assets then there is a second market psychology driven mechanism for setting prices in the society. The government would have to choose what price to set as a price floor when the credit system tries to shrink its balance sheet or destroy money. The government would have to regulate the credit system to prevent rampant inflation when certain positive price feedbacks are driven by balance sheet expansion in the private credit markets.

    1. If you look at the definitions used, adding private sector monetary instruments has no effect on the equations. (You decided to redefine M to be an end-of-period balance, which causes severe difficulties for notation.)

      I wrote multiple times that if we add more complicated behaviour to the private sector, more behavioural equations are needed.

    2. I take your point on the notation. Does the model as defined abstract away from systems with market based or government based credit and money creation? If so what happens when credit is added to the model?

    3. Money in this model - M - is only government issued money. As discussed in the text, whether other monetary instruments exist is outside the model.

  3. "Since M(1)≥0,


    Sorry to keep bugging you, but:

    Because the time series "Pg(t)Qg(t)−T(t)" is backward looking, at the end of period t=0 it should have the value M(1). You correctly write "M(1)=Pg(0)Qg(0)−T(0)".

    While M(1) could be zero, I don't think we should leave it out of the equation to come to any conclusion about Qg(0). Therefore, I would rearrange to find

    (Pg(0))(Qg(0)) = M(1) + T(0).

    All I gain from this equation is recognition that government has given the private sector a sum of money plus taxes marked paid in exchange for a priced bundle of goods.

    1. The objective is to find a mathematical statement about the quantity supplied. Leaving M(1) in the equation eliminates our ability to say much about Q. Since P, T are control variables set by the government, the statement Q(0) > T(0)/P(0) gives a relationship defined only by variables under the government’s control.

      The government gave a sum of money for goods its requisitioned, taxes were imposed, and M is what is left over. The government did not give M(1) or T(0).

  4. How does your approach in general compare with the Fiscal Theory of the Price Level?

    1. I covered this in

      Although they sound similar, FTPL relies on infinite horizon expectations, while this model relies on transactions in the spot market. The FTPL is essentially non-falsifiable, and appears to predict that the price level ought to have big jumps (that do not happen).

  5. "At t=0, the accounting identity is:

    The careful reader can figure out that you want the M(1) value to be the final sum of span-zero activity. It would make a lot more sense to me if you wrote

    At t=0, the accounting identity is:
    Pg(0)Qg(0)−T(0) = M(1).

    1. That is literally the same equation. Doing it that way is bad, since it reverses the standard for notation. We are writing down an expression for M(1), and we write it down as the sum of terms that affect money holdings.

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