If R < G, Dsge Model Assumptions Intermission Down
in gild to capitalise on the popularity of a recent book – it has null to do alongside inequality.)
This article discusses an “endowment” economic scheme (economic output is fixed, without whatever labour input trade-offs) which was described inwards to a greater extent than exceptional inwards an before article. I volition i time once again summarise the situation:
In gild to pivot downward the cost level, I impose an supposition that velocity is constant (discussed farther here, in my article on “monetary frictions”). For example, if velocity is fixed at 100, as well as the identify produces 100 apples that yr as well as it holds $1 inwards money, the size of the economic scheme is $100 (implying the cost of an apple tree is $1). I hash out the constant velocity supposition farther inwards the Appendix, equally it tin hold out relaxed.
At the initial fourth dimension point, the identify calculates the economical trajectory that optimises its utility, which is based on consumption over time. The shape of the utility business office does non affair also much, other than the fact that time to come consumption is discounted at a for sure rate.
For simplicity, I volition assume that prices are flexible (future prices tin hold out laid independently of the electrical current price), but prices convey to abide by the velocity constraint. When prices are flexible, DSGE model assumptions forcefulness the existent charge per unit of measurement of render on Treasury Bills to equal the (real) discount charge per unit of measurement inwards the utility function. This agency that if the charge per unit of measurement of nominal involvement is laid to whatever value, (expected) inflation volition hold out equal to the nominal involvement charge per unit of measurement less the fixed existent rate. For simplicity, I volition assume that the authorities are targeting the cost level, as well as so the expected inflation charge per unit of measurement is zero. In this case, the expected nominal involvement charge per unit of measurement equals the existent discount charge per unit of measurement for all time.
We tin catch forthwith that this exceptional model framework does non check existent the world economical data. In recent years, key banks inwards a publish of developed countries convey kept existent policy involvement rates negative, which would hold out impossible inwards this framework (the existent discount charge per unit of measurement is assumed to hold out positive). The fact that DSGE models brand no useful predictions most the existent the world should come upwardly equally no surprise; most marketplace position practitioners figured that out years ago. I am only detailing these problems to explicate why I volition hold out ignoring the “state of the art” inquiry that is beingness cranked out past times academics inwards my time to come articles discussing financial policy.
I volition at nowadays assume that the publish of apples produced grows each yr past times some factor. Since I quest to aspect at the identify sector inwards aggregate, this increase charge per unit of measurement includes the population increase charge per unit of measurement (typically denoted n), equally good equally per-household productivity increase (typically denoted g).
It should hold out noted that the representative identify framework breaks downward inwards the instance of a growing population. If every unmarried identify shrinks its holdings of authorities liabilities past times 0.5% per year, but the publish of households grows past times 1% per year, the aggregate increase rates of authorities liabilities is positive, non negative. Very simply, the representative identify supposition makes real picayune mathematical or economical sense.
Take my model economy, as well as assume that nosotros convey cost grade stability. We as well as so assume that the publish of apples produces grows past times 2% per year, but the existent discount charge per unit of measurement is only 1%.
The velocity human relationship tells us that the money stock has to grow past times 2% per year. However, the aggregate involvement rates on authorities liabilities is less than or equal to 1%, since at that spot is no involvement paid on money. Even if involvement were paid on “money” (it would convey to hold out “reserves” at the key bank), the charge per unit of measurement of increase of authorities liabilities due to involvement compounding is 1%. The only way the desired economical trajectory could hold out achieved is that the authorities has to run a (growing) principal deficit every year, so that the money stock would grow inwards draw of piece of work alongside nominal GDP.
This appears unremarkable. But this solution violates the so-called governmental budget constraint, which states that the discounted trajectory of principal financial surpluses equals the stock of initial debt. Since the authorities never runs a principal surplus, this “constraint” is manifestly non respected.
In particular, I volition render to the Bond Valuation Formula (part of the Fiscal Theory of the Price Level) which I depict inwards this before post. The Bond Valuation Formula is a straight upshot of the governmental budget constraint; if you lot believe i holds, the other does equally well. I aspect at this formula because it is i of the few references I convey constitute where the mathematics of the budget constraint is genuinely fleshed out. Once again, formula is:
(The formula, when translated into English, says that the existent value of authorities debt is the discounted value of the path of existent principal surpluses.) The formula is manifestly violated past times my instance economy. How is this possible, when it was supposedly derived using actual mathematics? We quest to aspect at the proof inwards the Appendix of the Cochrane working paper to catch where the difference occurs. After some algebra, the unmarried menses accounting identity for authorities finance was rearranged to:
You volition quest to read the newspaper for the description of the notation, but the work lies is inwards the bit term. Translating the mathematics into English, the term is the discounted value of the existent value of authorities debt outstanding equally fourth dimension goes to infinity. John Cochrane as well as so states “I impose the green status that the final term is zero.” (In the literature, this status is known equally the “transversality condition”.) This “usual condition” is what is violated inwards my example; the existent value of authorities debt is growing past times 2% per year, but the discount charge per unit of measurement is only 1%. Therefore, the bit term does non converge (loosely, “it goes to infinity”). (UPDATE) Note that this instance is a worst-case error; if r=g, then the bit term may converge to a non-zero value, creating an "error" inwards the cost grade predicted past times the Fiscal Theory of the Price Level. (Previous judgement added to brand this to a greater extent than general; thank you lot to "srini" for pointing this out.)
John Cochrane argues that this status must hold out imposed inwards gild to come across the atmospheric condition of identify optimality, which is a criterion argument. In my view, this declaration is incorrect. I convey a tidings inwards some other article – A Contradiction At The Heart Of DSGE Models – which (partially) explains my logic. (I volition add together to a greater extent than comments inwards a afterwards article.) Instead, I volition destination this postal service discussing the implications of this disagreement.
Even if the reader is unconvinced alongside my critique of the Transversality condition, the province of affairs for DSGE modelling is yet non real satisfying. If nosotros impose the Transversality Condition on my model economy, nosotros teach the province of affairs that is impossible (for some reason) for the authorities to target cost grade stability. It has to forcefulness the ratio of the money stock to gross domestic product to shrink past times (slightly to a greater extent than than) 1% per yr inwards gild for the term to converge to zero. This would imply a policy of deliberate deflation of (at least) 1% per year.
In the existent world, nosotros do non catch whatever style for the ratio of authorities liabilities to gross domestic product to become to zero, which is what the transversality status says must happen.
If the authorities does non pay involvement on money, nosotros tin uncovering a solution inwards which at that spot are no surpluses, fifty-fifty if the existent charge per unit of measurement of involvement is greater than the existent increase rate. This is because the increase charge per unit of measurement of authorities liabilities due to involvement volition hold out lower than the charge per unit of measurement of involvement if money holdings are non-zero. For example, if the nominal charge per unit of measurement of involvement on Treasury Bills is 4%, as well as one-half of authorities liabilities are inwards the shape of money, the nominal stock of authorities liabilities is only growing at 2% per year.
By increasing the weighting of money inwards identify portfolios, it is ever possible to uncovering a solution to the work inwards which no principal financial surpluses are ever run. (If existent economical output is shrinking, it is necessary to convey an inflation charge per unit of measurement that keeps nominal gross domestic product increase positive.) Once again, this volition violate the governmental budget constraint (and the Bond Valuation Formula).
If nosotros do non impose monetary frictions of some sort, my logic breaks down. In this case, at that spot is null to pivot downward the initial cost level. One could utilisation the Bond Valuation Formula to create upwardly one's heed the initial cost grade (which implies that the governmental budget constraint holds). But since the cost grade is essentially arbitrary inwards this case, it could hold out laid to anything as well as nosotros tin yet uncovering a solution to the identify optimisation problem. Those solutions could violate the governmental budget constraint (and thence the bond valuation formula). There is no ground to prefer i solution over the other, other than ideological prejudices.
This is what happens if you lot piece of work alongside mathematical models which are heavily under-determined: their solutions do non brand a lot of sense.
If nosotros tin uncovering an instance model economic scheme where the “governmental budget constraint” is violated, it is non inwards fact a constraint. It is as well as so a human relationship which may or may non hold, which is an only useless slice of trivia. Dropping the authorities budget constraint from the DSGE framework is non easily done, equally it is tied to a constraint for identify consumption over time. The trajectory of the economic scheme is supposed to hold out the optimal solution for all fourth dimension going forward; but it nosotros lack a constraint on time to come behaviour, nosotros cannot solve backwards to teach the solution inwards the present.
As seen above, the crux of my declaration involves the transversality condition. I volition render to transversality inwards a afterwards article, adding to my before critique.
(UPDATE) The newspaper "Interest Rates as well as Fiscal Sustainability" past times Scott Fullwiler covers a lot of this ground, but inwards to a greater extent than historical depth. The Fullwiler newspaper notes that this increase charge per unit of measurement status goes dorsum to the Domar 1944 newspaper "The 'Burden of the Debt' as well as the National Income". The ground why that before piece of work is ignored inwards the modern DSGE literature appears to hold out the fact that the transversality status is assumed to hold, as well as so what happens to the debt/GDP ratio is completely ignored. The before literature on sustainability was based on how the debt/GDP ratio evolved. (h/t Michael Sankowski at Mike Norman Economics.)
In my previous post, I hash out broader monetary frictions than exactly the constant velocity constraint than I utilisation inside this article. To hold out clear, I do non recall velocity is constant inwards the existent world. But some shape of monetary friction has to hold out imposed inwards gild for DSGE models to brand fifty-fifty a picayune fighting of sense.
If i uses a to a greater extent than complex monetary friction, such equally velocity that varies based on other variables, or money inwards the utility function, you lot could derive similar results to my example. The only requirement is that velocity has an upper jump equally good equally a lower jump inwards a higher identify zero. Such a constraint seems reasonable.
I believe that a constant velocity makes a lot of feel inwards a steady nation increase model. The key is non adopting the logical fallacy that Monetarists cruel for. If the economic scheme is assumed to be growing at a steady rate, it makes feel that the key banking concern needs to grow the monetary base of operations inwards draw of piece of work alongside that increase charge per unit of measurement inwards gild to avoid distortions. But the contrary direction of causality – attempting to drive steady gross domestic product increase past times growing the monetary base of operations at a fixed charge per unit of measurement – makes picayune sense. If nosotros motility away from the “steady state” condition, the residual of forces that led to a exceptional observed value of velocity would probable change.
(c) Brian Romanchuk 2014
Introduction
This article discusses an “endowment” economic scheme (economic output is fixed, without whatever labour input trade-offs) which was described inwards to a greater extent than exceptional inwards an before article. I volition i time once again summarise the situation:
- The economic scheme consists of a unmarried “representative” identify that receives a publish of apples each yr (without labour).
- The identify starts off alongside an initial stock of authorities liabilities (money or 1-year Treasury Bills).
- The authorities taxes as well as spends via transfer payments, but does non buy apples for authorities consumption.
In gild to pivot downward the cost level, I impose an supposition that velocity is constant (discussed farther here, in my article on “monetary frictions”). For example, if velocity is fixed at 100, as well as the identify produces 100 apples that yr as well as it holds $1 inwards money, the size of the economic scheme is $100 (implying the cost of an apple tree is $1). I hash out the constant velocity supposition farther inwards the Appendix, equally it tin hold out relaxed.
At the initial fourth dimension point, the identify calculates the economical trajectory that optimises its utility, which is based on consumption over time. The shape of the utility business office does non affair also much, other than the fact that time to come consumption is discounted at a for sure rate.
Real Rates And Inflation
For simplicity, I volition assume that prices are flexible (future prices tin hold out laid independently of the electrical current price), but prices convey to abide by the velocity constraint. When prices are flexible, DSGE model assumptions forcefulness the existent charge per unit of measurement of render on Treasury Bills to equal the (real) discount charge per unit of measurement inwards the utility function. This agency that if the charge per unit of measurement of nominal involvement is laid to whatever value, (expected) inflation volition hold out equal to the nominal involvement charge per unit of measurement less the fixed existent rate. For simplicity, I volition assume that the authorities are targeting the cost level, as well as so the expected inflation charge per unit of measurement is zero. In this case, the expected nominal involvement charge per unit of measurement equals the existent discount charge per unit of measurement for all time.
We tin catch forthwith that this exceptional model framework does non check existent the world economical data. In recent years, key banks inwards a publish of developed countries convey kept existent policy involvement rates negative, which would hold out impossible inwards this framework (the existent discount charge per unit of measurement is assumed to hold out positive). The fact that DSGE models brand no useful predictions most the existent the world should come upwardly equally no surprise; most marketplace position practitioners figured that out years ago. I am only detailing these problems to explicate why I volition hold out ignoring the “state of the art” inquiry that is beingness cranked out past times academics inwards my time to come articles discussing financial policy.
Bringing In Economic Growth
I volition at nowadays assume that the publish of apples produced grows each yr past times some factor. Since I quest to aspect at the identify sector inwards aggregate, this increase charge per unit of measurement includes the population increase charge per unit of measurement (typically denoted n), equally good equally per-household productivity increase (typically denoted g).
It should hold out noted that the representative identify framework breaks downward inwards the instance of a growing population. If every unmarried identify shrinks its holdings of authorities liabilities past times 0.5% per year, but the publish of households grows past times 1% per year, the aggregate increase rates of authorities liabilities is positive, non negative. Very simply, the representative identify supposition makes real picayune mathematical or economical sense.
Influenza A virus subtype H5N1 Counterexample
Take my model economy, as well as assume that nosotros convey cost grade stability. We as well as so assume that the publish of apples produces grows past times 2% per year, but the existent discount charge per unit of measurement is only 1%.
The velocity human relationship tells us that the money stock has to grow past times 2% per year. However, the aggregate involvement rates on authorities liabilities is less than or equal to 1%, since at that spot is no involvement paid on money. Even if involvement were paid on “money” (it would convey to hold out “reserves” at the key bank), the charge per unit of measurement of increase of authorities liabilities due to involvement compounding is 1%. The only way the desired economical trajectory could hold out achieved is that the authorities has to run a (growing) principal deficit every year, so that the money stock would grow inwards draw of piece of work alongside nominal GDP.
This appears unremarkable. But this solution violates the so-called governmental budget constraint, which states that the discounted trajectory of principal financial surpluses equals the stock of initial debt. Since the authorities never runs a principal surplus, this “constraint” is manifestly non respected.
In particular, I volition render to the Bond Valuation Formula (part of the Fiscal Theory of the Price Level) which I depict inwards this before post. The Bond Valuation Formula is a straight upshot of the governmental budget constraint; if you lot believe i holds, the other does equally well. I aspect at this formula because it is i of the few references I convey constitute where the mathematics of the budget constraint is genuinely fleshed out. Once again, formula is:
John Cochrane argues that this status must hold out imposed inwards gild to come across the atmospheric condition of identify optimality, which is a criterion argument. In my view, this declaration is incorrect. I convey a tidings inwards some other article – A Contradiction At The Heart Of DSGE Models – which (partially) explains my logic. (I volition add together to a greater extent than comments inwards a afterwards article.) Instead, I volition destination this postal service discussing the implications of this disagreement.
What If The Transversality Condition Holds?
Even if the reader is unconvinced alongside my critique of the Transversality condition, the province of affairs for DSGE modelling is yet non real satisfying. If nosotros impose the Transversality Condition on my model economy, nosotros teach the province of affairs that is impossible (for some reason) for the authorities to target cost grade stability. It has to forcefulness the ratio of the money stock to gross domestic product to shrink past times (slightly to a greater extent than than) 1% per yr inwards gild for the term to converge to zero. This would imply a policy of deliberate deflation of (at least) 1% per year.
In the existent world, nosotros do non catch whatever style for the ratio of authorities liabilities to gross domestic product to become to zero, which is what the transversality status says must happen.
What If r>g?
If the authorities does non pay involvement on money, nosotros tin uncovering a solution inwards which at that spot are no surpluses, fifty-fifty if the existent charge per unit of measurement of involvement is greater than the existent increase rate. This is because the increase charge per unit of measurement of authorities liabilities due to involvement volition hold out lower than the charge per unit of measurement of involvement if money holdings are non-zero. For example, if the nominal charge per unit of measurement of involvement on Treasury Bills is 4%, as well as one-half of authorities liabilities are inwards the shape of money, the nominal stock of authorities liabilities is only growing at 2% per year.
By increasing the weighting of money inwards identify portfolios, it is ever possible to uncovering a solution to the work inwards which no principal financial surpluses are ever run. (If existent economical output is shrinking, it is necessary to convey an inflation charge per unit of measurement that keeps nominal gross domestic product increase positive.) Once again, this volition violate the governmental budget constraint (and the Bond Valuation Formula).
The Case Of Pure Price Flexibility
If nosotros do non impose monetary frictions of some sort, my logic breaks down. In this case, at that spot is null to pivot downward the initial cost level. One could utilisation the Bond Valuation Formula to create upwardly one's heed the initial cost grade (which implies that the governmental budget constraint holds). But since the cost grade is essentially arbitrary inwards this case, it could hold out laid to anything as well as nosotros tin yet uncovering a solution to the identify optimisation problem. Those solutions could violate the governmental budget constraint (and thence the bond valuation formula). There is no ground to prefer i solution over the other, other than ideological prejudices.
This is what happens if you lot piece of work alongside mathematical models which are heavily under-determined: their solutions do non brand a lot of sense.
Concluding Remarks
If nosotros tin uncovering an instance model economic scheme where the “governmental budget constraint” is violated, it is non inwards fact a constraint. It is as well as so a human relationship which may or may non hold, which is an only useless slice of trivia. Dropping the authorities budget constraint from the DSGE framework is non easily done, equally it is tied to a constraint for identify consumption over time. The trajectory of the economic scheme is supposed to hold out the optimal solution for all fourth dimension going forward; but it nosotros lack a constraint on time to come behaviour, nosotros cannot solve backwards to teach the solution inwards the present.
As seen above, the crux of my declaration involves the transversality condition. I volition render to transversality inwards a afterwards article, adding to my before critique.
(UPDATE) The newspaper "Interest Rates as well as Fiscal Sustainability" past times Scott Fullwiler covers a lot of this ground, but inwards to a greater extent than historical depth. The Fullwiler newspaper notes that this increase charge per unit of measurement status goes dorsum to the Domar 1944 newspaper "The 'Burden of the Debt' as well as the National Income". The ground why that before piece of work is ignored inwards the modern DSGE literature appears to hold out the fact that the transversality status is assumed to hold, as well as so what happens to the debt/GDP ratio is completely ignored. The before literature on sustainability was based on how the debt/GDP ratio evolved. (h/t Michael Sankowski at Mike Norman Economics.)
Appendix: The Constant Velocity Assumption
In my previous post, I hash out broader monetary frictions than exactly the constant velocity constraint than I utilisation inside this article. To hold out clear, I do non recall velocity is constant inwards the existent world. But some shape of monetary friction has to hold out imposed inwards gild for DSGE models to brand fifty-fifty a picayune fighting of sense.
If i uses a to a greater extent than complex monetary friction, such equally velocity that varies based on other variables, or money inwards the utility function, you lot could derive similar results to my example. The only requirement is that velocity has an upper jump equally good equally a lower jump inwards a higher identify zero. Such a constraint seems reasonable.
- If velocity cruel arbitrarily unopen to zero, the implication is that people would convey money holdings that are arbitrarily large relative to their nominal incomes. We catch real few people who earn $50,000 per yr walking around alongside billions of dollars inwards their pockets.
- If velocity became arbitrarily large, the economic scheme would convey to somehow business office fifty-fifty though people would convey almost no money inwards notes as well as coins or banking concern accounts (deposits at the key banking concern – reserves – would hold out an arbitrarily modest part of their residual sheets). Although this province of affairs appears feasible, but I debate it would hold out untenable inwards practice, equally the mortal sector would lack liquid authorities liabilities to dorsum upwardly mortal sector short-term liabilities. The relative lack of authorities liabilities would brand paying taxes extremely hard (since gross taxes paid would presumably rising inwards draw of piece of work alongside GDP).
I believe that a constant velocity makes a lot of feel inwards a steady nation increase model. The key is non adopting the logical fallacy that Monetarists cruel for. If the economic scheme is assumed to be growing at a steady rate, it makes feel that the key banking concern needs to grow the monetary base of operations inwards draw of piece of work alongside that increase charge per unit of measurement inwards gild to avoid distortions. But the contrary direction of causality – attempting to drive steady gross domestic product increase past times growing the monetary base of operations at a fixed charge per unit of measurement – makes picayune sense. If nosotros motility away from the “steady state” condition, the residual of forces that led to a exceptional observed value of velocity would probable change.
See Also:
- My summary article where I render an overview of my analysis of DSGE models.
- (UPDATE) See the newspaper "Interest Rates as well as Fiscal Sustainability" past times Scott Fullwiler for a longer treatment. (h/t Michael Sankowski at Mike Norman Economics.)
- Michael Sankowski wrote a serial of articles most this at Trader's Crucible; here is a key article in which he notes that nosotros cannot uncovering inwards existent fourth dimension whether or non the constraint holds..
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