tag:blogger.com,1999:blog-59088308271350608522017-04-28T10:41:46.785-04:00Bond EconomicsBrian Romanchuk's commentary and books on bond market economics.Brian Romanchuknoreply@blogger.comBlogger55813BondEconomicshttps://feedburner.google.comtag:blogger.com,1999:blog-5908830827135060852.post-59232110105950230452017-04-27T14:12:00.000-04:002017-04-28T06:24:56.897-04:00Fun With Central Bank Calvinball<div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-PbnNLyn6-XU/VJN8rAfZ1uI/AAAAAAAABPg/rouADpT1fVIQdLRae3mum8Z_rf2GjbPJQCPcB/s1600/logo_DSGE.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="https://1.bp.blogspot.com/-PbnNLyn6-XU/VJN8rAfZ1uI/AAAAAAAABPg/rouADpT1fVIQdLRae3mum8Z_rf2GjbPJQCPcB/s1600/logo_DSGE.png" /></a></div>In the comments to "Does the Governmental Budget Constraint Exist?" Nick Edmonds reminded about another piece of mainstream macro logic that triggers me: the governmental budget constraint holds, because the inflation-targeting central bank says so. From the perspective of a mathematician, this is just a variation of "because I want to assume it to be true." In this article, I run through whatever logic appears to exist.<br /><br /><a name='more'></a>As a preamble, there might be an explanation for this that makes sense <i>somewhere</i>. I gave up after a couple of years of looking at this. I am sure some economics professor can try to explain this; the only question is whether it can be translated into mathematics.<br /><h2>Fundamentalist Bourbakianism</h2><br />As a preamble, I am a hardline member of the <a href="https://en.wikipedia.org/wiki/Nicolas_Bourbaki" target="_blank">Bourbaki faction: all mathematics is just set theory</a>. Philosophers of mathematics might beg to differ with on me, but that is not my problem.<br /><br />The view is simple: everything in mathematics is a set, or something related to a set. If it has nothing to do with a set, it isn't mathematics.<br /><br />If you are not familiar with this view, I will ask you to think about the following question: what is a function? (If you do not know the answer, I highly recommend that you think about it before running off to a search engine. The answer is at the bottom.)<br /><br />We can now turn to the issue of central banks in mainstream macro.<br /><h2>Position: The Budget Constraint Holds Because <i>X</i> Says So</h2>If one is attempting to decipher mainstream DSGE macro, you can run into statements that run roughly like this:<br /><div style="text-align: center;"><i>The budget constraint holds because of central bank policy [insert stuff about inflation targeting, or whatever].</i></div><br />If we translate this into mathematics:<br /><br /><div style="text-align: center;"><i>"The (budget constraint equation) holds because we assume X, and we assume that X implies that the budget constraint holds."</i></div><br />Any of the following could be inserted as <i>X</i> in that statement:<br /><br /><ul><li>inflation-targeting central banks;</li><li>the Easter Bunny says so;</li><li>Chuck Norris says so.</li></ul><div>This is just adding an extra level of indirection when compared to just saying: "We assume that the budget constraint holds."</div><h2>Position: The Central Bank Theory of the Price Level</h2><div>The next argument is that there is that the initial price level will always move to ensure that the budget constraint holds.</div><div><br /></div><div>From a mathematical perspective, this seems indistinguishable from the Fiscal Theory of the Price Level. However, it is not the private sector doing this; rather, the central bank can always set the price level at time zero. There is no reason within the mathematics why this is so: it just is.</div><div><br /></div><div>The question about the central bank's control over the price level appears open.</div><div><ul><li>If it can only set the price level to the level indicated by the Fiscal Theory of the Price Level, then it is just "the Fiscal Theory of the Price Level, because central banks."</li><li>If the central bank can set the initial price level at any level, that implies that it has arbitrary control over the evolution of the price level. We do not need economics models, we just ask the central bankers what price level they want that day. This position raises many philosophical questions about the study of economics.</li></ul></div><h2>Position: Set Elements Magically Changing</h2><div>Yet another alternative is the following:</div><div><ul><li>The Treasury picks an exogenous sequence of primary fiscal surpluses, <i>s</i>.</li><li>If the budget constraint does not hold, the central bank says "you cannot do that." (How this translates into a set operation is unclear.)</li><li>Therefore, the Treasury is forced to pick a new sequence of primary surpluses <i>z</i>, which follows the budget constraint.</li></ul><div>Mathematically, this is the same thing as assuming the set of feasible primary surpluses are those that meet the budget constraint. Once again, there is no justification why this must hold. </div></div><h2>Position: Those Darned Infinite Rates</h2><div>The next position attempts to justify how the central bank can enforce the budget constraint: it will raise interest rates to infinity if it does not hold.</div><div><br /></div><div>This is equivalent to saying that the price of Treasury bills will clear at $0.</div><div><br /></div><div>Needless to say, there is no examination of the mathematics of market clearing.</div><div><ul><li>If the household sector inherited any money from the previous period, it can buy an infinite number of bills. In what sense has the market cleared?</li><li>If the household sector had no money, the size of the central bank's balance sheet is zero. How can entity without a balance sheet drive the price of an asset to zero?</li></ul><div>In the absence of a mathematical examination of the market clearing condition, and a specification of the limits of central bank balance sheets, this argument looks sketchy.</div></div><div><br /></div><div><i>And rates have to be infinite, </i>implying that there is no solution to the optimisation problem. Any other outcome just results in the exact same chain of real quantities that are in the budget accounting identities. None of the analysis I did in the previous post made any assumption about nominal rates being "low."</div><div><br /></div><div>(The zero bound is the only major issue for the previous development.)</div><div><br /></div><div>Furthermore, the "neo-Fisherian" effect will hold as soon as we have finite interest rates: the higher the nominal rate, the higher the level of expected inflation. Really high interest rates imply really high future inflation; it is hard to see how saying that "central banks target inflation" justifies this entire line of thought. Once again, if mainstream economists actually solved the models that they develop, we could resolve these issues.</div><div><br /></div><h2>Position: Because Equilibrium</h2><div>"The central bank chooses the equilibrium." Good luck with converting that description to mathematics. This seems to the consensus view on the topic, by the way.</div><h2>Concluding Remark</h2><div>How mainstream macro went this far down the rabbit hole is a complete mystery.<br /><h2>Appendix</h2>In my first lecture on linear systems theory at McGill (a Master's level course), Professor George Zames asked us the "What is a function?" question. <i>Nobody got it right. </i>People thrashed around with discussions of "mappings" or rules, or whatever nonsense they teach undergraduate electrical engineers. (I took real analysis, but later.) He gave us the answer the next day, and I entered the Bourbaki cult.<br /><br />The answer: a function is set of ordered pairs. That is <i>y = f(x)</i> is just a shorthand for <i>f ={(x, y)}</i>. (<i>Alex Douglas rapped my knuckles for my original answer; I deliberately ignored the extra conditions on the set.</i> The typical restriction is that if <i>(x,y)</i> and <i>(x,z)</i> are elements of <i>f</i>, then <i>y=z</i>. Depending on how you want to attack the issue, it could be set up slightly differently, I believe.) There's no little elves mapping an input to outputs; it's just a set.<br /><br /></div>(c) Brian Romanchuk 2017<img src="http://feeds.feedburner.com/~r/BondEconomics/~4/lqI3A9SriTE" height="1" width="1" alt=""/>Brian Romanchukhttps://plus.google.com/112203809109635910829noreply@blogger.com1http://www.bondeconomics.com/2017/04/fun-with-central-bank-calvinball.htmltag:blogger.com,1999:blog-5908830827135060852.post-79422727485327224652017-04-26T08:23:00.000-04:002017-04-26T08:23:23.192-04:00Does The Governmental Budget Constraint Exist?<div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-PbnNLyn6-XU/VJN8rAfZ1uI/AAAAAAAABPg/rouADpT1fVIQdLRae3mum8Z_rf2GjbPJQCPcB/s1600/logo_DSGE.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="https://1.bp.blogspot.com/-PbnNLyn6-XU/VJN8rAfZ1uI/AAAAAAAABPg/rouADpT1fVIQdLRae3mum8Z_rf2GjbPJQCPcB/s1600/logo_DSGE.png" /></a></div>This article wraps up my discussion of the transversality condition and the governmental budget constraint. In summary, the governmental budget constraint used within mainstream macro has very serious flaws. I would have liked to use the title "The Governmental Budget Constraint Does Not Exist," but we need to take into account the rather curious Fiscal Theory of the Price Level. Furthermore, there are unsettling implications for the entire Dynamic Stochastic General Equilibrium (DSGE) model approach that relies upon optimisation. I abandoned looking at these models for this reason, and this article suggests why I believe the problems run much deeper. Unlike the previous articles, this article is largely free of mathematics, but I start out listing the various equations I refer to. <br /><br /><a name='more'></a><i>(I have been involved in discussions on this topic on Twitter, starting from an initial contact by Alex Douglas. I have just run across the work of C Trombley, who has written an article on similar lines here -- <a href="http://stochastictalk.blogspot.ca/2017/04/i-am-very-model-of-modern-macro-textbook.html" target="_blank">I Am The Very Model Of A Modern Macro Textbook</a>. It's interesting, but I had been stuck writing out my own chain of logic, and could not respond to his points.)</i><br /><h2>Can't Keep Your Equations Straight Without A Program</h2><script type="text/x-mathjax-config"> MathJax.Hub.Config({tex2jax: {inlineMath: [['$','$'], ['\\(','\\)']]}}); </script> <script type="text/x-mathjax-config">MathJax.Hub.Config({ TeX: { equationNumbers: { autoNumber: "AMS" } } }); </script><script src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML" type="text/javascript"> </script> This section lists the various equations discussed here, and how to interpret them. They all refer to simplified mathematical frameworks that are seen in a variety of textbook DSGE models.<br /><br /><a href="http://www.bondeconomics.com/2017/04/mathematics-of-budget-constraint-again.html" target="_blank">The notation is defined in the previous article</a>. In all cases, the variable $b(t)$ refers to the stock of government debt in real terms, $r$ is the real interest rate, and $s(t)$ is the primary fiscal surplus.<br /><br />What I refer to as the governmental budget constraint is the following equation.<br />$$\begin{equation}<br />b(0) = \sum_{i=1}^{\infty} \frac{s(i)}{(1+r)^i} \label{eq:summation}<br />\end{equation}$$<br />This equation says that the summation of the discounted real primary surpluses is finite, and equals the initial stock of debt. The validity of this expression is what we are interested in.<br /><br /><div>The next equation is the <i>1-period accounting identity</i>:</div><br />$$\begin{equation}<br />b(t+1) = (1+r) b(t) - s(t+1). \label{eq:accountident}<br />\end{equation}$$<br /><br />This just says that the debt at time $t+1$ is equal to $(1+r)$ times the debt level at time $t$, minus the surplus in the current period ($t+1$). That is, debt will compound by the discount rate, less the primary surplus. <i>As an accounting identity, this has to hold.*</i> One annoying habit of some mainstream economists is to also refer to this as a "governmental budget constraint," conflating the dubious ($\ref{eq:summation}$) with the true-by-definition ($\ref{eq:accountident}$).<br /><br />The next equation is the <i>transversality condition:</i><br /> $$\begin{equation}<br />\lim_{t-> \infty} \frac{b(t)}{(1+r)^t} = 0. \label{eq:limit}<br />\end{equation}$$<br />This equation says that the stock of debt outstanding cannot grow faster than the discount rate $r$. Transversality is a term that comes from optimisation theory. Since we do not actually apply it, the exact definition does not matter.<br /><br />Finally, there are couple of workhorse accounting identities that relate the stock of debt at time $0$ and time $t$.<br /><br />The reasonable-looking forward relation, which tells us the level of future debt based on the initial debt level, and intervening primary surpluses.<br />$$\begin{equation}<br />b(t) = (1+r)^t b(0) - \sum_{i=1}^t (1+r)^{t-i} s(i). \label{eq:fwdsum}<br />\end{equation}$$<br /><br />There is also the unusual backward relation, which tells us the current level of debt based on a future level. This equation is just an algebraic restatement of the previous. (My first article had a deliberately obtuse attempt to decipher this equation.)<br />$$\begin{equation}<br />b(0) = \sum_{i=1}^t \frac{s(i)}{(1+r)^i} + \frac{b(t)}{(1+r)^t}. \label{eq:bkwdsum}<br />\end{equation}$$<br /><h2>Attempting to Follow Mainstream Logic</h2>For a variety of reasons, mainstream economists want to use the governmental budget constraint ($\ref{eq:summation}$). Within models, households face a budget constraint, and it would be unfair if governments did not have one (beyond the accounting identity ($\ref{eq:accountident}$)). As I discussed in "<a href="http://www.bondeconomics.com/2017/04/on-being-pelted-by-peanuts-part-i.html" target="_blank">On Being Pelted With Peanuts: Part I</a>," they could just assume it to be true.<br /><br />Of course, just assuming something to be true is not too useful. Mexico will pay for that wall, if we assume that they will do so. As such, mainstream economists searched for a reason for it to be true, The usual logic appears to work as follows. (I have never seen a coherent description of the logic behind this, so I had to use guesswork.)<br /><ol><li>Starting with the backward relation ($\ref{eq:bkwdsum}$), we can manipulate equations to show that the transversality condition ($\ref{eq:limit}$) implies the accounting identity ($\ref{eq:summation}$). (<a href="http://www.bondeconomics.com/2017/04/mathematics-of-budget-constraint-again.html" target="_blank">I did the proof in the previous article.</a>)</li><li>When households search for the optimal solution to their utility maximisation problem in the DSGE model, the optimal solution (allegedly) displays the transversality condition ($\ref{eq:limit}$).</li><li>Therefore, household optimisation preferences will imply that ($\ref{eq:summation}$) holds.</li></ol><div>The mainstream economists then go on to wave their hands about future surpluses cancelling out the effect of "debt-financed" fiscal stimulus, and so fiscal policy is ineffective, etc.</div><div><br /></div><div>Not so fast.</div><div><br /></div><div>If the backward relation ($\ref{eq:bkwdsum}$) holds, so does the forward relation ($\ref{eq:fwdsum}$). For the sake of argument, assume that the initial real stock of debt is fixed. (We return to this assumption later.) If we look at that equation, we see that the future debt level is pinned down by an accounting identity: the household sector in aggregate cannot alter the future trajectory of the debt by one (real) penny, no matter what optimisation choices it takes. (The models we are discussing feature fiscal policy that is completely unrelated to the state of the economy.) That is, the premises behind logical steps 1 and 2 above are inconsistent. The confusion in "Being Pelted With Peanuts" was the result of the inconsistency in the logic being used.</div><div><br /></div><div>The next line of defence is to argue that since households have future money ("at infinity") that they will not need, they will spend it now. In other words, they cannot stop the chain of single-period accounting identities which determine the ratio of future debt to the initial level, they can (somehow) change the starting point.</div><div><br /></div><div>Although that might work for an individual household, that cannot work in aggregate. All purchases made by households in the crippled DSGE modelling frameworks flow right back to the household sector, and there is no way of the household sector reducing its aggregate holdings of government-issued liabilities (other than voluntarily destroying money or bill holdings, which is not optimising behaviour). After all, we were only able to argue that household preferences influenced government debt outstanding because the only sector that held government debt in the model was the household sector. Furthermore, since we are assuming that all households act the same ("representative household"), they would all try to buy at the same time, without any extra supply forthcoming. There is no way of affecting the nominal debt level at time zero.</div><div><br /></div><div><i>The only thing that can adjust is the price level at time zero. </i>This is not "inflation," as that is the rise in the price level in future periods versus time zero. Instead, the entire price level has to shift instantly, which destroys the real value of financial assets held during the previous period. (Raising interest rates in time zero does nothing to stop this, as this only protects the real value of government bills against the inflation from time period $0$ to $1$.) This is how the assumption that the initial real debt level is fixed is relaxed: by changing the initial price level (which is the only thing that can move).<br /><br />This adjustment mechanism appears implausible, but it reflects the general under-determination of the price level at $t=0$. Almost all attention is paid on the relative price between current prices and the future, but there is little discussion why the initial price level has to be at any particular level. The only variables with nominal scaling are the inherited financial assets from the previous period. If those debt ratios are "too high," we just scale nominal GDP instantly so that the ratio hits the correct level. (Calvo pricing does not help; firms that are unable to adjust prices to the new starting point get squashed like bugs.)</div><div><br /></div><div>This effect is the <a href="http://www.bondeconomics.com/2014/12/monetary-impotence-and-triumph-of.html" target="_blank">Fiscal Theory of the Price Level</a> (FTPL). The implications of the FTPL are stark: the price level is entirely driven by the state of expectations about fiscal policy. The price level at $t=0$ is entirely determined by fiscal policy expectations at $t=0$. The price level at $t=1$ is entirely determined by fiscal policy expectations at $t=1$. <i>This means that monetary policy settings at $t=0$ is utterly irrelevant for the level of inflation at $t=0$.</i></div><br />The FTPL justifies the governmental budget constraint by saying that the private sector will raise (or lower) the price level -- changing the real value of existing debt -- if it ever looks like the budget constraint will not hold. This has nothing to do with "transversality" in optimisations. However, it is once again an assumption about economic behaviour that holds only because we assume that is true. If we assume that other factors influence the initial determination of the price level. the governmental budget constraint disappears.<br /><br />The FTPL appears to be the only internally coherent class of representative household DSGE models. Unfortunately, the models are fairly degenerate, in that nothing else will really matter for inflation. Furthermore, it seems that their empirical usefulness is highly questionable. (Do we have plausible infinite horizon fiscal forecasts?)<br /><h2>Further Optimisation Ugliness</h2>Even if we want to ignore the Fiscal Theory of the Price Level, the interaction between the macro constraints and household constraints are worrisome. I have not wasted much of my time looking at microeconomics, but I have severe doubts about how its "laws" have been applied to DSGE macro. We cannot assume that households are "infinitely small"; we need to model $N$ households, and see how they interact with an aggregate macro budget identity. In other words, we cannot use theorem statements that are cherry-picked from micro textbooks without ensuring that all the conditions required by those theorems apply to the model in question.<br /><br />The household sector's financial assets are held in a vise, which is the governmental fiscal surplus. There is no action that can be taken to change the end-of-period holdings, no matter what level of production takes place.<br /><br />If your financial asset holdings is completely outside of your control, how do they matter in an optimisation? It seems that the optimal strategy is to completely ignore financial asset holdings in the utility maximisation problem.**<br /><br />However, such a step destroys the entire premise of inter-temporal optimisation. Unless the model features real investment, there is nothing (other than the irrelevant financial balances) that link period $t$ and $t+1$. This suggests that the optimal solution to these problems is just to pick the naive point-in-time utility maximisation at every time point. (Insert the production function into the 1-period component of the utility function; find the maximising output.) The fact that optimisation is carried out on an infinite horizon is just a smoke screen; what happens in period 1 has no effect on the solution in period 0.<br /><br />Finally, since financial asset balances do not matter in these models, the rate of interest does not matter. Once again, the actions of the central bank are entirely irrelevant to the model outcome.<br /><h2>Concluding Remarks</h2>It is unacceptable that we have to speculate about the solutions to these models. The usefulness of mathematics is that it forces you to think clearly, and crystallise your logic in equations. However, once we lose the discipline of properly solving the equations, we are back to literary speculation.<br /><br /><b>Footnotes:</b><br /><br />* In some treatments of the topic, this accounting identity has been turned into an inequality, based on logic from financial mathematics. You know a field has completely lost any shred of common sense when the only things that we know hold with equality are turned into inequalities.<br /><br />** If money does not appear in the utility function, the construction appears straightforward. Assume we have a feasible trajectory $x$, with utility $U(x)$. We then construct $x^*$ with all state variables equal to $x$, other than the primary surplus sequence $s$ and the affected financial asset holdings (normally bills and money). The only restriction on $s^*$ is that it does not somehow bind the household sector's financial constraint by running too large surpluses. This set is non-empty; the time series $s^*(t) = s(t) - 1$ is one such primary surplus series. Importantly, the series $s^*$ is created by only adjusting the taxes imposed; real government consumption is fixed. Since the financial constraints would never bind, $x^*$ is also feasible. Moreover, $U(x^*) = U(x)$. Therefore, we can see that the optimal trajectory is (somewhat) indifferent to the path of financial variables. Of course, there is the problem of attaining a maximum of an optimisation where the set of feasible solutions is not closed and finite, <a href="https://medium.com/@alexanderdouglas/infinite-peanut-policy-reply-to-romanchuk-3502cff8f8b8" target="_blank">see this article by Alex Douglas</a>. If money appears in the utility function, the set of exogenous primary surplus sequences that we can use is limited to the set that allow the household sector to match the optimal money balance in each period (assuming that we do not allow the household sector to run negative bill holdings, or borrowings from the government).<br /><br />(c) Brian Romanchuk 2017<img src="http://feeds.feedburner.com/~r/BondEconomics/~4/UX4iflJUjjI" height="1" width="1" alt=""/>Brian Romanchukhttps://plus.google.com/112203809109635910829noreply@blogger.com13http://www.bondeconomics.com/2017/04/does-governmental-budget-constraint.htmltag:blogger.com,1999:blog-5908830827135060852.post-30507055624676641682017-04-25T08:37:00.000-04:002017-04-25T12:06:53.248-04:00Mathematics Of The Budget Constraint (Again)<div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-ByBqBSERMJk/VJN8r_rvyZI/AAAAAAAABQY/Y-iBIdhWkqEQhFI0zQvkUw_YrUKW8exugCPcB/s1600/logo_fiscal.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="https://4.bp.blogspot.com/-ByBqBSERMJk/VJN8r_rvyZI/AAAAAAAABQY/Y-iBIdhWkqEQhFI0zQvkUw_YrUKW8exugCPcB/s1600/logo_fiscal.png" /></a></div>This article attempts to give a simpler mathematical discussion of the governmental budget constraint and transversality. After throwing my hands up in the air in my previous article, I run through the basic mathematics of the accounting identity for governments, and we can see that what is called "transversality" is just equivalent to making the assumption that the discounted primary surpluses converge to be equal to the initial stock of debt. However, household sector optimisation is nowhere in sight, which raises the question why it comes up in discussion of this topic in the first place.<br /><br />Once again, the math-phobic may as well stay clear.<a href="https://medium.com/@alexanderdouglas/infinite-peanut-policy-reply-to-romanchuk-3502cff8f8b8" target="_blank"> I would also draw your attention to this article by Alex Douglas</a>; he is jumping ahead to an extremely point about optimisations (in general, we have no reason to believe that the optimum exists when the set of solutions is not closed and finite).<br /><br /><a name='more'></a>The equations are being generated by MathJax, and they might not be rendered on some browsers. If you can read LaTex, you might be able to follow the argument anyway. Please note that I went nuts with the equations here, and so it may take some time for the equations to render. I will be reverting to low math content in the future, but I just wanted to underline how cumbersome it is to deal with infinite summations, a point that is glossed over in a lot of treatments I see.<br /><br />As a disclaimer, this was relatively rushed; there's probably a few typos in here.<br /><h2>Preliminaries</h2>Let $\cal T$ be the set of time series defined on $\mathbb Z_+$. That is, if $x \in {\cal T}$, then $x(t) \in {\mathbb R}$ for all $t \in {\mathbb Z_+}.$ (The set ${\mathbb Z}_+$ is the set of positive integers greater than or equal to $0$.)<br /><br />(Note: I am unsure what is the formal name for $\cal T$; I have a bad feeling about its properties.)<br /><br />Definitions associated with infinite sums and limits are <a href="http://www.bondeconomics.com/2017/04/on-being-pelted-by-peanuts-part-i.html#more" target="_blank">described in the previous article</a>.<br /><br /><b>Assumptions</b><br /><ul><li>Money holdings are zero at all times; the only government liabilities are 1-period bills. (If we allow money to be held, we then get terms associated with money creation in the formulae. These added complexities offer little value-added.)</li><li>We are starting at time $0$ for notational simplicity.</li><li>The (expected) real discount rate is equal to $r$ for all times. (There is some embedded assumptions about deflation as a result of this. The household can get whatever real interest rate it wishes on money balances if there is sufficient deflation. This technicality is typically ignored elsewhere; I am following that assumption so that my equations align with the usual textbook ones. Otherwise, we need to start tracking nominal balances and the price level as well, and my treatment would bear no resemblance to what we see elsewhere.)</li><li>Realised variables are equal to expectations at time $0$. (If perfect foresight is bothersome, pretend this is a simulation at $t=0$.) </li></ul><div><b>Variable definitions:</b></div><ul><li>Denote the real market value of government bills outstanding at time $t$ as $b(t)$. (That is, $b \in {\cal T}$.) The initial value of $b$ ($b(0)$) is a positive number.</li><li>The primary fiscal surplus at time $t$ is $s(t)$. The variable $s$ is a fixed member of $\cal T$; that is, it is an exogenous variable. The initial value is fixed: $s(0) = 0.$</li></ul><div><br /></div><div><b>Definition</b> The <i>1-period government accounting identity</i> is given by (for $t>0$):</div><br /><script type="text/x-mathjax-config"> MathJax.Hub.Config({tex2jax: {inlineMath: [['$','$'], ['\\(','\\)']]}}); </script> <script type="text/x-mathjax-config">MathJax.Hub.Config({ TeX: { equationNumbers: { autoNumber: "AMS" } } }); </script><script src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML" type="text/javascript"> </script> $$\begin{equation}<br />b(t+1) = (1+r) b(t) - s(t+1). \label{eq:accountident}<br />\end{equation}$$<br /><br /><b>Lemma </b>We can relate $b(t)$ and $b(0)$ as follows, for all $t \in {\mathbb Z}_+$:<br />$$\begin{equation}<br />b(t) = (1+r)^t b(0) - \sum_{i=1}^t (1+r)^{t-i} s(i). \label{eq:fwdsum}<br />\end{equation}$$<br /><b>Proof </b>Use induction.<br /><ul><li>Equation ($\ref{eq:fwdsum}$) is true by inspection for $t=0$, and by applying ($\ref{eq:accountident}$) for $t=1$.</li><li>Assume true for $t$.</li><li>Validate for $t+1$. Apply ($\ref{eq:accountident}$) and the induction assumption, we get:</li></ul><div>$$\begin{eqnarray}<br />b(t+1) & = & (1+r)b(t) - s(t+1), \\<br />& = & (1+r) \left( (1+r)^t b(0) - \sum_{i=1}^t (1+r)^{t-i} s(i) \right) - s(t+1), \\<br />&= & (1+r)^{t+1} b(0) - \sum_{i=1}^{t+1} (1+r)^{(t+1)-i} s(t).<br />\end{eqnarray}$$<br />Validating the induction assumption. $\fbox{}$</div><br /><b>Lemma </b>The following relationship holds:<br />$$\begin{equation}<br />b(0) = \sum_{i=1}^t \frac{s(i)}{(1+r)^i} + \frac{b(t)}{(1+r)^t}. \label{eq:bkwdsum}<br />\end{equation}$$<br /><b>Proof </b>By inspection (apply ($\ref{eq:fwdsum}$)). $\fbox{}$<br /><br /><b>Theorem</b> The equation<br />$$\begin{equation}<br />b(0) = \sum_{i=1}^{\infty} \frac{s(i)}{(1+r)^i} \label{eq:summation}<br />\end{equation}$$<br />is well defined if and only if<br />$$\begin{equation}<br />\lim_{t-> \infty} \frac{b(t)}{(1+r)^t} = 0. \label{eq:limit}<br />\end{equation}$$<br /><b>Proof: </b>We first prove that ($\ref{eq:limit}$) implies ($\ref{eq:summation}$). Rearrange terms of ($\ref{eq:bkwdsum}$) to give:<br />\[<br />b(0) - \sum_{i=1}^t \frac{s(i)}{(1+r)^i} = \frac{b(t)}{(1+r)^t}.<br />\]<br />This implies that<br />$$\begin{equation}<br />\left| b(0) - \sum_{i=1}^t \frac{s(i)}{(1+r)^i} \right|= \left| \frac{b(t)}{(1+r)^t} \right|.<br />\label{eq:absval} \end{equation}$$<br />Fix any $\epsilon > 0$. By applying the definition of ($\ref{eq:limit}$), there exists an $M$ such that<br />\[<br />\left| \frac{b(t)}{(1+r)^n} \right| < \epsilon, \forall n \geq M.<br />\]<br />Apply to ($\ref{eq:absval}$):<br />\[<br />\left| b(0) - \sum_{i=1}^n \frac{s(i)}{(1+r)^i} \right| < \epsilon, \forall n \geq M<br />\]<br />We then apply the definition of an infinite summation to see that ($\ref{eq:summation}$) holds.<br /><br />To validate that ($\ref{eq:summation}$) being well-posed implies ($\ref{eq:limit}$), we rearrange ($\ref{eq:bkwdsum}$) to give:<br />\[<br /> \frac{b(t)}{(1+r)^t} = b(0) - \sum_{i=1}^t \frac{s(i)}{(1+r)^i}.<br />\]<br />Fix any $\epsilon > 0$. By applying ($\ref{eq:summation}$), there exists an $M$ such that the right-hand side hand side has modulus less than $\epsilon$ for all $t \geq M$. We then apply the definition of the limit to see that the left-hand side converges to zero. $\fbox{}$<br /><br /><b>Remark</b> This proof is plodding, but there still might be issues that it glosses over. In a journal article in applied mathematics, nobody would bother with the $\epsilon$ arguments (unless it was much more difficult). However, the proof text would have to be careful to indicate why the various summations and limits exist, That is, it is unacceptable to write down infinite summations and use them in other manipulations without ensuring that the summations exist.<br /><h2>Discussion</h2>The theorem provided tells us that the condition that is called the "transversality condition" is a necessary and sufficient condition for the condition on the discounted sum of primary surpluses (equation ($\ref{eq:summation}$)).<br /><br />This is what is asserted in various DSGE macro papers, and <a href="http://www.bondeconomics.com/2017/04/on-being-pelted-by-peanuts-part-i.html" target="_blank">which caused me agony in my previous article</a>. Since it is actually straightforward, why complain?<br /><br />My complaint is this: this derivation was driven entirely by straightforward application of the 1-period accounting identity. There is no optimisation involved at any point during the derivation (the notion of transversality comes from optimisation theory). Very simply, the household sector has no choice with respect to this result, therefore it makes no sense to pretend that it is the result of microfoundations.<br /><br />In other words, since the initial stock of household debt holdings ($b(0)$) is fixed, and the path of the primary surpluses was assumed to be exogenous (a crazy assumption, but standard for simple DSGE models), the future path of debt holdings ($b(t)$) is deterministic, and not the result of any optimisation result. This raises the obvious corollary: if household wealth is determined entirely by fiscal policy, in what sense does it even matter for the optimisation problem?<br /><br />Correspondingly, there is no reason to believe that the condition <i>must</i> hold; it either holds or it does not. Since the nominal discount rate is quite often below the nominal growth rate of the economy, the expectation is that it will in general not hold.<br /><br />I will return to the economic discussions in a later article (with less equations).<br /><br />(c) Brian Romanchuk 2017<img src="http://feeds.feedburner.com/~r/BondEconomics/~4/zrgChCQykVw" height="1" width="1" alt=""/>Brian Romanchukhttps://plus.google.com/112203809109635910829noreply@blogger.com2http://www.bondeconomics.com/2017/04/mathematics-of-budget-constraint-again.html