Marginal Propensity to Consume and the Velocity of Money

April 1, 2017 1 comment

My last post pointed out my frustration with the treatment of the marginal propensity to consume and the spending multiplier in a traditional principles text.  I just want to briefly point out how this whole business fits nicely into my alternate paradigm where the aggregate demand curve is thought of as PY=MV (as opposed to Y=C+I+G+NX).

Here is how Mateer and Coppock approach the MPC (emphasis added):

When a person’s income rises, he or she might save some of this new income but might be just as likely to spend part of it too.  The marginal propensity to consume (MPC) is the portion of additional income that is spent on consumption.

The problem here is the dichotomy between consumption and saving.  As I have mentioned before, it matters what we mean exactly by “saving.”  Normally (outside of the Keynesian AS/AD model) we assume that saving means investment and investment is also in the CIG equation (it’s the I).  So if “saving” means that you lend part of your income to a firm who buys investment goods, then doesn’t that part of your income become the income of the producer of those investment goods and isn’t that exactly the same as the portion you spent on consumption goods?  (Yes.)  So then is this all just nonsense or is there some grain of truth in here that we are just saying in an idiotic way?

Well, there is one and only one way of saving that is not exactly the same as consuming in the sense that when you do it, the money you devote to it becomes another person’s income.  That way is holding money.  So what happens if we change the way we explain this slightly so that instead of saying that people consume a fraction of their income c and “save” the rest, we say that people spend a fraction of their income c and hold the rest as money and this spending may be on either consumption or investment goods.

Now you can ask yourself “what is the effect on total spending of a given increase in government expenditure.  Let’s assume that the government borrows $100 from the central bank and spends it on something and let’s assume that the marginal propensity to spend (MPS) is 0.9.  Then the person who gets that initial $100 will spend 90 and the person who gets that will spend 81 and so on.  Take that series out to infinity and you will get the total increase in spending of 100/(1-MPS) or the familiar spending multiplier.

But now notice that another way to see this is to notice that people are holding 10% of their income as money and we have increased the money supply by $100.  This means that total incomes (which must equal total spending) must rise by–you guessed it–1/(1-.9) times the change in the money supply.  Or, to put it another way, incomes have to rise enough that people become willing to hold the new money.

But now you are coming strikingly close to another similar concept.  If people are willing to hold 10% of their income as money, that means each dollar in the economy will have to change hands 10 times on average for people to become willing to hold all the money.  Or in other words, the velocity of money will be one over the average propensity to hold money.  (There is, of course, the possibility that the average and marginal propensities may not be the same but that’s not of particular importance to my point.)  And that is just 1-MPS.  So in other words, if you characterize this thing correctly (it’s about spending and not consuming) then the Keynesian spending multiplier just becomes the velocity of money.

Of course, if you look at it this way, you are forced to notice that the real cause of the increase in aggregate demand in this case is an increase in the money supply not something magical about government spending.  If they just dumped the money into the economy from helicopters, the effect would mostly be the same.  And that’s exactly why this way is better, and probably also why it isn’t what we do.

However, there is one magical thing about government spending which is that the government can choose a higher MPS (usually 1) and so can get more spending bang  per dollar of increase in M in the first round.  So the “helicopter multiple” (if you will) would be slightly less ((1/(1-MPS))-1).  And because of this, it becomes theoretically possible that the government could boost aggregate demand without increasing the money supply by borrowing or taxing.

If they increase taxes by $100, they reduce private consumption and/or investment by $100 times the helicopter multiple but then when they spend it they increase private consumption and/or investment by the same amount but with the government spending added on top (assuming that the propensity to spend is unaltered).  This, of course, is a standard fiscal policy implication (see textbook claim 2 in the previous post).  But if you think about it in the context of the equation of exchange, you can actually see what’s going on here.  Essentially, the government is just increasing the velocity of money by taking it away from people and spending it at a higher rate than they would have.

And there you find the special power of government to boost aggregate demand.  Just like the special power of government to do anything else, it boils down to their ability to force people to do a thing more or less.  In this case it’s the ability to force them to spend more money.  The same thing would happen if you put a horde of killer robots on the streets programmed to shoot missiles at anyone they ran across holding any money.  Or, for that matter if–in some crazy, hypothetical, sci-fi universe–all money were somehow electronic and it was possible for the government to charge you interest for holding it and they simply increased the rate of interest so that you would want to hold less money for any given income level.  This would also increase the velocity of money and boost aggregate demand.  I wonder though, would they call it fiscal or monetary policy? (And would they call the increase in interest rates tightening or loosening?)

 

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Fiscal Policy I: The Spending Multiplier and the (False) Paradox of Thrift

December 30, 2016 Leave a comment

I’m going to write down some of my thoughts on fiscal policy here mostly for the sake of organizing my thinking.  There are a few issues to tackle here so I intend to take it in two or three bites.

As I have said in the past, I have some gripes with the way that AS/AD is taught.  I have more of a monetarist view of aggregate demand so I like to think of it as PY=MV instead of Y=C+I+G+NX.  This becomes especially important when considering fiscal policy.  The textbook treatment makes essentially two claims.

Textbook claim 1: Deficit spending boosts aggregate demand

When government spending increases it increases G and this shifts aggregate demand to the right.  When taxes increase it decreases C which shifts AD to the left.  So the government can increase aggregate demand by taxing less or spending more or some combination of both that results in a larger deficit.  Theoretically, this can be done in a counter-cyclical manner (deficits in recessions and surpluses during expansions) in order to smooth out the business cycle.  It’s pretty obvious that it doesn’t happen this way in practice but that’s a subject for another time.

Textbook claim 2: Even deficit-neutral spending can increase AD because of the spending multiplier.

Basically, because people don’t consume their entire income, if the government takes some of it and spends it, it will go back to them as income and they will spend just as much but we will get the government spending on top of the private consumption spending which means higher aggregate demand.  Or in other words, if we raise taxes and government spending by the same amount, the decrease in C will be smaller than the increase in G.

There were several things that bugged me about these claims when I was a student, and I’ve seen some of my more talented students grapple with the same issues.  The problem is that this Keynesian model contradicts much of the other stuff we tell them in a macro principles class.  The most obvious example of this is the “paradox of thrift.”

Savings=Destruction……or was it Investment?

We spend half the class telling students that saving equals investment.  This is very sensible.  In order for someone to invest, someone has to save.  This is one of those deep truths that economics is built on.  Problem is, as soon as we get to AS/AD we start saying silly things like “assume that people only consume x% of their income and they save the rest.” And from this we come to conclusions like “if the marginal propensity to consume increases, C increases and that increases AD.”  Or textbook claim #2 above.  But these conclusions only follow from the assumption that “saving” in this context is destruction.

A smart-Alec, teacher’s-pet type who has been paying attention and thinking carefully up to this point might ask the question: “Wait, if people want to consume more of their income and save less, isn’t that exactly the type of thing we just got done saying shifts the supply of loanable funds/goods to the left and causes a decrease in investment?  Does it really make sense to just ignore the effect on investment?  And if we don’t just ignore it, won’t it completely annihilate these claims we are making?  And doesn’t the answer to that last question imply an answer to the second-to-last question?  And which parts of this are going to be on the final?”

These are almost all fantastic questions.  So how do you answer them? Maybe smarter people who have been teaching this longer have good answers within the CIG framework.  However, I’m skeptical.  As far as I can see, that framework is not capable of dealing with stuff like this and that is a real problem!  These little contradictions lurking under the surface make it so that the more carefully you think about what you are being taught, the less sense it all makes and that’s super annoying.

So what can be done?  Well, you can re-frame aggregate demand in a monetary way.  Let aggregate demand be PY=MV.  This doesn’t necessarily change any implications of the model but it forces you to talk about the things that affect aggregate demand in a monetary way.  And this is appropriate because aggregate demand is a fundamentally monetary phenomenon.  And this is what the Keynesian perspective is overlooking.

So now what can we say about fiscal policy in this MV framework?  Well any affect on aggregate demand must be either from a change in M or a change in V.  We could get these.  But in order to figure out what is really going on we have to ask a couple questions.

Question 1: Is the spending funded by increased taxes?

If yes: OK, then consumption and/or investment probably went down.  So it depends how much they went down.  This depends on the marginal propensity to spend.  Notice, that I am not talking about saving now.  Most of what people save is investment.  So it’s not their propensity to consume or save that matters.  We can lump all households and firms together and talk about what is actually at issue here, namely the willingness to hold money.

Note that if I “save” half of my income by buying a corporate bond or a stock, that money goes to a firm and they spend most of it on investment.  But there is one–and only one–type of “saving” that is destruction in the sense commonly proposed.  This is holding money. (Of course this is still hyperbole…) It’s the willingness to hold money, both by firms and households that determines velocity and therefore aggregate demand.  So if everyone holds 10% of their income in the form of money and we take some of it, spend it, then it goes back to them and they hold 10% of it still, we can increase aggregate demand by increasing both taxes and spending together in nearly the exact way described by the spending multiplier because this will increase velocity. This can even be shown mathematically rather easily.  You just have to change “saving” to “holding money” and “marginal propensity to consume” into “marginal propensity to spend” and the spending multiplier basically becomes velocity.  I love it when a plan comes together!

Now, the smart-Alec, Keynesian is probably thinking: “so if it works out the same, what difference does it make?”  The difference is that now I have explained it without saying something idiotic that contradicts a bunch of stuff I was saying a week ago.

No, the government spending is funded by deficits: Proceed to question 2.

Question 2: Who are they borrowing from?

If the private sector: Then this must affect the market for loanable funds in the way described above.  Specifically, the demand must increase  which will cause an increase in real interest rates and crowd out some private consumption as well as some private investment.  Does this change aggregate demand?  Maybe.  A first approximation would be to say that the sum of the private consumption and investment crowded out would be exactly equal to the amount of government spending.  This is what you would see if you just look at a partial equilibrium in the loanable funds market and assume the private supply and demand are unchanged.  However, disturbing this market may very well affect the willingness to hold money which will affect velocity and may have some effect on velocity and therefore aggregate demand.  However, this is much more suble than just saying “G increases and C stays the same.”

If the central bank: Ok, let’s say you have the government borrow money but instead of dipping into the private market and pulling loanable funds away from consumers and investors, their spending is actually financed by the creation of additional money by the central bank.  Now we will almost certainly see an increase in aggregate demand as we will see a pure increase in the desire to purchase goods and services without any direct offsetting decrease in some other sector (much like what is commonly assumed).  But in our MV model, this is easy to see.  It’s just an increase in M.  Unless it causes a corresponding decrease in V for some reason it will be an increase in aggregate demand.  But then is it actually fiscal policy or is it monetary policy?

In my opinion this last case is the most important one and is the source of a considerable amount of bickering about the efficacy of “fiscal policy” especially in the presence of a “liquidity trap” or the zero lower bound.  I will leave these issues for later but I think the paradigm in the principles texts makes it very difficult to sort these things out.  Mine is better.

[By the way, note that if the central bank is targeting an interest  rate, then an increase in demand for loanable funds caused by an increase in government spending may very well cause an increase in the money supply in a way that would appear automatic in the sense that no explicit action would be observed on the part of the central bank.  In other words, the same interest rate target might become more expansionary monetary policy in the presence of deficit spending.  In this case was fiscal policy effective?  Was it necessary?  It depends.  Are you a Keynesian or a monetarist?]

 

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A Fiat Money Origin Story

April 3, 2016 13 comments

Nick Rowe has a recent post about Chartalism which got me thinking about the fundamental explanation for the value of money again.  He calls this an “origin story” and seems to be of the opinion that the origin doesn’t really matter, once it gets going you can remove the original source of fundamental value and it just stays up.  Kinda like Wile E. Coyote running off a cliff.  As long as nobody looks down, we’re fine.  I personally think this is nuts.  But I figure, why not try going through the explanation like an “origin story” from primitive commodity money to modern fiat money.  Maybe that will help?  I have mostly tried to avoid all of that because it seems unnecessarily confusing and I usually want to distill the story down to its most fundamental point as much as possible.  However, I think maybe this leaves people too much room to fall back on little misconceptions that are deeply lodged their thinking about this.  So why not start from the beginning and try to hammer out all the points (or most of them) in one try?  For the record, this is not a historical work.  It’s a made-up history that I think is fairly consistent with reality as it unfolded in the western world.  Whether or not the Chinese had some type of script that was linked to taxes thousands of years ago or there were some hunter-gatherers somewhere with a credit-based economy before commodity money became prevalent is not relevant to my point. Read more…

LOL

So I’ve been away for a while and I was looking through a few comments I missed in the last few months and someone linked my post about Austrian economics and libertarianism right in the middle of posts by HuffPo, Slate, and Daily Kos.  I can only assume that the author didn’t actually read my post because I wasn’t conflating the two things at all.  My whole point was that Austrian economics is making libertarians look bad.  Clearly, that wouldn’t make sense if I thought they were the same thing….Right?  At any rate, I consider it an honor to have turned up in a hastily executed google search of “Austrian economics” and “libertarians” along with those fine paragons of this perpetual food fight we call the internet.  So I’m providing a reciprocal link.  I suspect it will get him about as many hits as his link got me.

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You Heard Me!

September 16, 2015 Leave a comment

I’ve been out of blogging for a while with other stuff to do but with commodities tanking and the Fed seemingly poised to raise short-term rates in the near future, and some commentators remarking that “QE risks becoming a semi-permanent feature,” I feel like the world is largely conforming to my view.  However, this is an easy feeling to have erroneously and I kind of wish I had made more official predictions in the past, even though that’s sort of a dangerous business to get into.  No doubt, if I had them, I would wish I hadn’t made them but at least then I would be humble.

So just to get myself on the record, partly for fun, and partly so that if they happen, I can point to them and say “neener, neener, I told you so” to the world, I want to take a page from my blogging hero and do an economic version of “You Heard Me.”  The premise behind this game is to make bold predictions which would make someone who hears them remark something along the lines of “wait, what did you just say?” to which you reply “you heard me!”  The point is not necessarily to nail the prediction exactly but to boldly state a general feeling about something, be it a football player or an economy.  Basically this is one prediction but I will try to add a little bit of flesh to the bones.

  1. If you watch CNBC or any show like that, in almost every discussion about the FED, someone will say “look, we know rates are going up eventually” and everyone agrees with them.  That’s wrong.  Rates are never going up again.  You heard me!

Of course, it’s possible that rates could go up a little.  Short-term rates may indeed be raised this week.  I don’t claim to know what the Fed will do in two days.  But there is a meme of “normalization” going around which is wrong.  Somehow people have gotten the idea that it is “normal” for short term rates to be something like 2-4% and this “low rate environment” must be only temporary.  It’s not.  Maybe they get rates to 1% for a while, maybe not, I don’t know (my guess is not).  But they will be doing QE again before short-term rates ever hit 2%.  By “they” I mean the FED but basically this applies to the whole world.

10 year treasury

There is the 10-year treasury rate since about 1980.  See if you can spot the “normal” level.

2. Inflation will continue to run below the FED’s target and will intermittently cause prices of commodities, stocks, real estate, etc. to nosedive requiring the FED to take action to stave off a full-on deflation.

3. QE will become the new standard policy tool.  You heard me!

The “low rate environment” is not because of temporary exogenous shocks, it is the deterministic result of our monetary system (not policy, system. You heard me!)  This system requires either: people to become continually more leveraged, governments/central banks to buy continually more stuff, or lots of people to go bankrupt.  For the last 35 years or so, we have been exhausting our ability to induce people to become continually more leveraged by continually lowering interest rates.  When we can’t do that, we need either the government (“fiscal policy) or the central bank (“quantitative easing”) to just start buying stuff.  Because we are not going to significantly lift off of the lower bound, and because remaining at the lower bound will not be sufficient to stave off a wave of defaults forever, we will eventually need to reinstate QE.  The next time, they will drop the pretext of a temporary support measure and just start adjusting it up or down every month and that will become the policy tool of choice to smooth out the path of the economy.

4. QE will tend to get bigger over time relative to the rest of the economy.  This is for the same reason described above.

5. QE will spill over into other assets, like stocks.  You heard me!

This is already going on in Japan and China.  Note that 3, 4, and 5 will tend to require bouts of 3 in order to get done because they will be viewed as “extraordinary.”  This means that they will not dare to do these things until it gets bad enough that people are jumping up and down shouting “do something!”

6.  Keynesians will constantly complain that we need more fiscal stimulus to get off the lower bound.  They will be half right.

Okay, that’s not very bold but how many years do we have to hear that line before we start to wonder if it’s not just a matter of smoothing out short-run fluctuations in the real economy?  My prediction: a lot.  You heard me!

7. Market monetarists will insist that our problems are all because central banks are too tight.  They will be about 3/4 right.

It will be true that if central banks were looser, NGDP would grow faster and inflation expectations and nominal rates would pick up.  It’s just a question of what they would have to do to “ease” further.  The MM line seems to be just talking about being looser (and I guess, to be fair, holding interest rates lower for longer) will get us back to normalcy.  The problem with this is that it still relies on the myth of “normalization.”  In order to ease more, in the long run, the FED would eventually have to do more of the things I mention above, not just change their language.  It’s true that if they just did them without waiting for recessions and financial crises to force their hand, those things might be avoided but the MMers haven’t quite come around to accepting the ultimate consequences of this.  (Bonus prediction: we won’t adopt NGDP futures targeting, even though it is a way better idea than what we are doing.)

8.  Scott Sumner will continue to make the occasional snide remark about the Central Bank buying up the whole world in order to argue that monetary policy can always get looser.  He will be 100% right.  At some point it will stop being funny….You heard me.

 

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The Backwards Brain Bicycle

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The RG/MV Model

In my last post, I discussed my complaints with the standard approach to teaching aggregate demand in an intro class.  I have been trying to come up with a better way of doing it and that has spilled over into IS/LM.  I think I have a better model for that too.  I will try to describe it here.  For the record, I’m not saying IS/LM is “wrong” exactly, just that it is misleading and is not a very clear representation of the relationships it is meant to explain.  My model can be turned into the standard IS/LM model with a few assumptions.  But forcing you to make those assumptions, I think, helps greatly to understand what is going on.

The point of aggregate demand is to connect the real economy to the monetary economy.  All short-run deviations from the long-run equilibrium are due to the monetary mechanism not functioning perfectly.  This is what we are trying to model.  The threads which connect these two things are the price level and the interest rate.  Much like the traditional model, I will divide the demand side of the economy into two sectors: the real goods market and the money market.  For the sake of simplicity, I am assuming a closed economy with no taxes or government spending. These other things can be added but it is a little more complicated than just adding a G onto aggregate demand (which, remember, is the whole point).  I will leave that for another time.  We will start with real goods.

The RG Curve

Assume that people have some preferences over consumption now and future real wealth (you can say future consumption if you prefer, but I like this way of saying it better).  They also have a budget constraint.  The budget constraint depends on income and the real interest rate.  Solving the maximization problem is a simple microeconomic exercise.  You will get a function for consumption and one for investment, which depend on income Y and the real rate r.

Now, at this point, you can recognize that these functions could have a lot of different properties, but let’s assume that they have the following.

IY(Y,r)>0

Ir(Y,r)>0

The response of consumption to changes in either variable is not of fundamental importance. Note that you may assume that the consumption function is linear in Y and independent of r, but this is a restriction on consumer preferences and not necessary. (This eliminates one of my pet peeves about IS/LM.)

Now let this function I(Y,r) be the supply of loanable goods and let the demand for loanable goods be determined by the investment opportunities faced by firms such that the marginal product of investment is always equal to 1+r. Equilibrium in the market for loanable goods, then requires the quantity supplied to be equal to the quantity demanded.

Figure 1

RG

This will determine an equilibrium real interest rate for any level of income. Note that expenditure (C(Y,r)+I(Y,r))is always equal to income. This is just the budget constraint from the consumer maximization problem. So equilibrium in the real goods market (which is made up of equilibrium in the loanable goods market as well as the consumption goods market since, from the consumer maximization we always have I(Y,r)+C(Y,r)=Y) implies a relationship between income and real rates.

Now if we assume that income equals output (Y), then we have a relationship between output and real rates. Note that this assumption, along with the consumer budget constraint, essentially represent the “Keynesian cross” from IS/LM. When income is higher, the supply of loanable goods will increase and the real interest rate will be lower. We can then write a function representing all combinations of income (Y) and real rates (r) which are an equilibrium in the real goods market. Let’s call this the “RG curve” for “real goods.” This will be downward sloping and essentially equivalent to the IS curve.

The MV Curve

Now consider the role that money plays. Notice that we did not include the price level or the nominal interest rate in the real goods markets. Also, remember that aggregate demand can be written as PY=MV. Let M be exogenous. Then our goal is to explain V.

Let L(i) be the fraction of total expenditure that people are willing to hold as money. This will depend on the interest rate. The higher the rate, the less cash they will be willing to hold, since the nominal rate is the price of holding money. If people are holding more money than desired, they will try to spend it, either on additional consumption or investment (probably by lending it) and this will have to increase either the price level or total output and therefore reduce velocity (and vice-versa if they are holding less than desired).

Equilibrium in the money market requires velocity to be 1/L(i) and this must, by definition, be equal to PY/M. This can be seen on a graph of V and Y for a given P and i.

Figure 2

MV

This gives us a level of output which is consistent with the demand for money for any given price level and nominal interest rate. This will be increasing in i (since Li<0), decreasing in P and increasing in M. Call this function the “MV curve.”

RG/MV

At this point we have to deal with inflation expectations. One key feature of this model is that it is very explicit about which interest rates it is talking about where and so you have to deal with inflation expectations explicitly as well. This eliminates another one of my pet peeves about IS/LM (the more important one). The simplest way to do this is to assume that they are exogenous, and don’t depend on any of the other exogenous variables. Then, in equilibrium, the Fisher equation must hold.

i=r+π^e

Then we can rewrite the RG curve as RG(i-π^e). Then we have a nice downward-sloping RG curve and upward-sloping MV curve in i/Y space. Equilibrium in both the real goods market and the money market will determine a quantity of total output demanded and a nominal interest rate for any given price level. (Consumption, investment, velocity, and the real rate can all be easily recovered from this.)

Figure 3

RG-MV

Aggregate Demand

The aggregate demand curve is then a function giving all combinations of output and price which make up an equilibrium in both the real goods market and the money market. It can easily be seen that this is downward-sloping in price. If the price is higher, PY/M will shift to the left in figure 2 which will cause the MV curve to shift to the left and imply a lower aggregate quantity demanded. Furthermore, we can deduce how the curve will shift when the following exogenous variables change.

Money supply

If the money supply increases, this will shift PY/M to the right for any given P, which will cause the MV curve to also shift to the right and cause the aggregate quantity demanded for any given P to be higher. Note that for a given price level, and inflation expectation, this will lower the nominal rate, increase investment and decrease velocity. The degree to which the increase in money is absorbed by a decrease in velocity vs. an increase in aggregate quantity demanded depends on the slope of L(i).

Inflation expectations

If inflation expectations increase, this will cause RG(i-π^e) to shift to the right since any level of i will represent a lower level of r. This will cause the aggregate quantity demanded to be higher for any given P (an increase in aggregate demand). Note that this will cause an increase in i, but a decrease in r since the RG curve will shift up by exactly the amount of the increase in inflation expectations but it will move along the MV curve. This will mean an increase in output, investment and velocity.

Time preferences

If people become more impatient and want to consume more today, this will cause a decrease in investment supply and higher real rate for any given level of income which will cause the RG curve to shift to the right. This will cause the nominal rate, velocity and aggregate quantity demanded to increase for any given P.

Investment opportunities

If a new technology is discovered which increases the marginal product of investment, the demand for loanable goods will increase.  This will cause the RG curve to shift to the right and the AD curve to do the same, which will increase investment, output and real and nominal interest rates.  Note, that this may crowd out some consumption, depending on the shape of the indifference curves.

“Animal spirits”

If people decide they want to hold less money—L(i) decreases—then 1/L(i) will be higher for any given i which means that the MV curve will shift to the right and the nominal and real rates will be lower, aggregate quantity demanded, velocity, and investment will be higher.

Remarks

This model has two main benefits compared to IS/LM. One is that it has a bit more “micro foundations” in that it explicitly incorporates consumer preferences and demand for investment by firms. The other is that it carefully distinguishes between real rates and nominal rates which makes the transmission mechanism for monetary policy much more clear in my opinion. It also fits better with my framework of thinking about AD as Y=MV/P and dividing things into their effects on M and V rather than thinking in terms of C+I+G+NX and dividing things into their effects on those respectively (although note that you can still do that with this model).  I see this as a sort of monetarist version of IS/LM, though I don’t know if people with monetarist street cred would agree with me or not.

So far I haven’t explicitly tried to incorporate fiscal policy. To do this would require you to make some assumptions about how it affects peoples’ preferences and where the money comes from. (Is it from taxes or borrowing? If the latter, is it borrowed from the private market or from the central bank increasing the money supply?) Note however, that these questions are central to determining the effect of such policy.  You probably could just slap it onto AD if you wanted to assume that there were no effect on preferences (no crowding out) and you could change the household budget constraint to Y-T.  If the government borrows, you could add their demand to the demand for loanable goods.

Sorry the figures look like crap.  I need to figure out a better way to get them into wordpress

 

 

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