<p>I have a question about martingale convergence in some macro models. In these models, it is usually the case that we confirm an almost-sure limit exists using Doob's forwards convergence, and then we make an argument that this limit must be a particular value given some inequality restriction. My question is about the latter part, as I don't understand how we can determine the limit when only given an inequality in the settings I have seen.</p>
<hr />
<p>Let me lay out two settings where I have seen this type of argument.</p>
<p>Suppose we have a household with utility over consumption streams, <span class="math-container">$c$</span>, given by <span class="math-container">$U(c)=\sum_{t=0}^{\infty}\beta^t u(c_t)$</span>, where <span class="math-container">$u'&gt;0,u''&lt;0,$</span> and <span class="math-container">$u'''&gt;0$</span>. Suppose there are i.i.d. income shocks <span class="math-container">$y_t\in \{y_1,\dots ,y_N\}$</span> each period. The realisation of income shocks is derived from the realisations of some underlying 'states of the world', <span class="math-container">$\{s_1,s_2,\dots ,s_N\}$</span>. A history up till time <span class="math-container">$t$</span>, denoted <span class="math-container">$s^t$</span>, is a length <span class="math-container">$t$</span> sequence of these states. Let <span class="math-container">$\pi(s^t)$</span> be the probability of history <span class="math-container">$s^t$</span> occurring given no information and <span class="math-container">$\pi(s^t\mid s^k)$</span> be the probability given <span class="math-container">$s^k$</span> information. The household has access to a risk-free asset in each period, <span class="math-container">$a_t$</span>, with price normalised to <span class="math-container">$1$</span>, and which returns <span class="math-container">$Ra$</span> in the next period. Write <span class="math-container">$c_t(s^t)$</span>, <span class="math-container">$a_t(s^t)$</span>, and <span class="math-container">$y_t(s^t)$</span> as the consumption choice, asset choice, and income realisation given history <span class="math-container">$s^t$</span>. The household problem is therefore,</p>
<p><span class="math-container">$$\max_{\{c_t,a_t\}_{t=0}^{\infty}}\mathbb{E}_0 \sum_{t=0}^{\infty}\sum_{s^t}\pi(s^t)\beta^t u(c_t(s^t))$$</span> <span class="math-container">$$\text{st. } \qquad c_t(s^t)+a_{t+1}(s^t)\leq y_t(s^t)+Ra(s^{t-1}) \qquad \forall s^t= s^{t-1}\cup s_t$$</span></p>
<p>The first-order conditions, and imposing <span class="math-container">$R\beta=1$</span>, give that marginal utility follows a martingale process <span class="math-container">$$u'(c_t(s^t))=\mathbb{E}[u'(c_{t+1}(s^{t+1}))\mid s^t] \qquad (\dagger)$$</span> Since <span class="math-container">$u'''&gt;0$</span>, Jensen's implies that <span class="math-container">$$u'(c_t(s^t))&gt;u'(\mathbb{E}[c_{t+1}(s^{t+1})\mid s^t]) \qquad (\ddagger)$$</span> By Doob's forwards convergence theorem, as <span class="math-container">$u'&gt;0$</span>, <span class="math-container">$(\dagger)$</span> implies that the marginal utilities converge almost surely to a constant value. My professor claims that given Jensen's inequality, <span class="math-container">$(\ddagger)$</span>, this limit must be <span class="math-container">$0$</span>. Why does <span class="math-container">$(\ddagger)$</span> imply the limit must be <span class="math-container">$0$</span>? In my mind, all <span class="math-container">$(\ddagger)$</span> implies is that consumption is a submartingale, but doesn't say anything about the size of the increment. Why can it not be the case that consumption asymptotically converges upwards to some positive (but finite) value <span class="math-container">$M$</span> with the incremental change in consumption going to <span class="math-container">$0$</span>?</p>
<hr />
<p>A second example comes from a model with private information frictions. Suppose we have the same model as above. Agents can only borrow or lend from a risk-neutral insurance company at rate <span class="math-container">$1+r=\frac{1}{\beta}$</span> rather than purchasing state-contingent assets. Moreover, the income realisation of each consumer is private information to them only and the insurance company cannot credibly verify these. Therefore, the insurance company must pick an incentive compatible mechanism, amounting to a state-contingent transfer <span class="math-container">$\tau(s^t)$</span>.</p>
<p>After simplifying the incentive compatibility constraints of the consumers to only local downwards ones, the insurer's problem (the dual of the consumer's) is <span class="math-container">$$v(w)=\max_{\{\tau_n,w_n\}_{n=1}^N} \sum_{n=1}^N\pi_n[-\tau_n+\beta(v(w_n))]$$</span> <span class="math-container">$$\text{st. } \begin{cases} \sum_{n=1}^N\pi_n[u(y_n+\tau_n)+\beta w_n]\geq w \\ 
u(y_n+\tau_n)+\beta w_n\geq u(y_n+\tau_{n-1})+\beta w_{n-1} &amp; \forall n\in \{2,3,\dots ,N\}\end{cases}$$</span>  Where <span class="math-container">$w_n$</span> is the 'utility promise' of future utils to the consumer after income shock <span class="math-container">$y_n$</span> is observed. Explicitly, <span class="math-container">$$w(s^k)=\sum_{t=k}^{\infty} \sum_{s^t}\beta^{t-k}\pi(s^k\mid s^t)u(y(s^t)+\tau(s^t))$$</span> One can prove that the value function <span class="math-container">$v$</span> is strictly decreasing, strictly concave, and differentiable. The first-order conditions of the problem give that the marginal value follows a martingale process <span class="math-container">$$v'(w)=\mathbb{E}[v'(w')] \qquad (\sim)$$</span> With a bit of manipulation of the incentive compatibility and FOCs, one can prove that the utility promises spread out, <span class="math-container">$$w_1'&lt;w&lt;w_n' \qquad (\approx)$$</span> Here, again, my professor states that <span class="math-container">$(\sim)$</span> gets us almost-sure convergence and <span class="math-container">$(\approx)$</span> implies (since <span class="math-container">$v'&lt;0$</span> and <span class="math-container">$v''&lt;0$</span>) that the only possible limit for <span class="math-container">$v'(w)$</span> is <span class="math-container">$-\infty$</span>. Why does this final point follow? I cannot see why the difference between <span class="math-container">$w_1'$</span> and <span class="math-container">$w_N'$</span> cannot tend to <span class="math-container">$0$</span> so that <span class="math-container">$v'(w)$</span> tends to some finite negative number.</p>
<hr />
<p>Is my misunderstanding coming from not fully getting Doob's forwards convergence (it has been a while since I properly thought about it), or do I not understand how the models are working?</p>

CC BY-SA 4.0
2024-12-21T12:10:02.183Z
Hi Joseph: I didn't read the second part I don't know the answer to your first question ( it's interesting ) but  jensen's inequality states that  $E(f(x)) >= f(E(x))$ if $f(x)$ is convex so should there not be an expected value on the first term, $u^{\prime}(c_{t}(s^{t}))$ ? in your statement regarding Jensen's inequality. Not that it makes anything crystal clear to me anyway regarding your question but maybe that will make things clearer in the question ?
mark leeds
CC BY-SA 4.0
2024-12-22T07:24:55.207Z
@markleeds Hi Mark. Along the optimal stochastic consumption path $\{c_t(s^t)\}$, FOCs give us that $u'(c_t(s^t))=\mathbb{E}[u'(c_{t+1}(s^{t+1})\mid s^t]$. I apply (conditional) Jensen's to the right-hand side to obtain $u'(c_t(s^t))=\mathbb{E}[u'(c_{t+1}(s^{t+1})\mid s^t]> u'(\mathbb{E}[c_{t+1}(s^{t+1})\mid s^t])$, so that $u'(c_t(s^t))>u'(\mathbb{E}[c_{t+1}(s^{t+1})\mid s^t])$. Is this what you are referring to?
Joseph Basford
CC BY-SA 4.0
2024-12-23T14:49:39.203Z
Thanks Joseph. I understand the greater than sign but I forgot that you also have a martingale so you can just use $u^\prime(c_{t}(s^t))$. Thanks and my apologies for noise.  I've been reading about martingales lately so, even though I can't help, if you get an answer and it's not from econ.stackexchange, can you post it ? Thanks. Also, one possible suggestion is to put it on math.se but maybe not with all the notation and economics. More along the lines of, "if I have a martingale such as (and then something simpler ), and it obeys jensen's inequality, does it converge to anything ?
mark leeds
CC BY-SA 4.0
2024-12-24T08:18:59.237Z
@markleeds No problem at all. After thinking about it some more, I am pretty certain it has to do with the assumption of i.i.d. income shocks, which I think precludes decaying increments to $0$. But hopefully someone can spell out this point more clearly as I'm still a bit muddled up about it.
Joseph Basford
CC BY-SA 4.0
2024-12-25T12:45:50.693Z
Hi Joseph: you're a better math person than me but I remember recently reading that with probability 1, brownian motions go to $-\infty$ or $+\infty$. maybe there's a similar type theorem that adds conditions and makes a related kind of statement for martingales ? The lemma is on the bottom of page 4 of the first volume of williams and rogers. I didn't follow the proof there but I found a better proof is on the math.stackexchange site.
mark leeds
CC BY-SA 4.0
2024-12-25T12:54:37.880Z
This is not the proof I found previously but still a decent proof by Eldredge.As it stands, the result is not helpful for your question anyway. It's somewhat of an odd result ( if you ask me ) which is why I was thinking it could be related to what you need. https://math.stackexchange.com/questions/3392744/question-about-the-textitsup-b-t-infty-of-a-brownian-motion
mark leeds
CC BY-SA 4.0
2024-12-25T13:18:10.827Z
Joseph: The actual lemma in williams and rogers on page 4 is $ P(sup~B(t) = +\infty, ~~~inf ~ (B_t) = -\infty ) = 1 $, where $B_t$ is a brownian motion.
mark leeds
CC BY-SA 4.0
2024-12-25T13:59:00.400Z
Here is the one I was trying to find. https://math.stackexchange.com/questions/3081905/supremum-of-brownian-motion
mark leeds
<macroeconomics><mathematical-economics><mechanism-design><asymmetric-information><martingales>
Martingale Convergence and Doob's Convergence
Joseph Basford
# Martingale Convergence and Doob's Convergence

(Asked by: username_0 on 2024-12-20)

I have a question about martingale convergence in some macro models. In these models, it is usually the case that we confirm an almost-sure limit exists using Doob's forwards convergence, and then we make an argument that this limit must be a particular value given some inequality restriction. My question is about the latter part, as I don't understand how we can determine the limit when only given an inequality in the settings I have seen.

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Let me lay out two settings where I have seen this type of argument.

Suppose we have a household with utility over consumption streams, $c$, given by $U(c)=\sum\_{t=0}^{\infty}\beta^t u(c\_t)$, where $u'>0,u''<0,$ and $u'''>0$. Suppose there are i.i.d. income shocks $y\_t\in \{y\_1,\dots ,y\_N\}$ each period. The realisation of income shocks is derived from the realisations of some underlying 'states of the world', $\{s\_1,s\_2,\dots ,s\_N\}$. A history up till time $t$, denoted $s^t$, is a length $t$ sequence of these states. Let $\pi(s^t)$ be the probability of history $s^t$ occurring given no information and $\pi(s^t\mid s^k)$ be the probability given $s^k$ information. The household has access to a risk-free asset in each period, $a\_t$, with price normalised to $1$, and which returns $Ra$ in the next period. Write $c\_t(s^t)$, $a\_t(s^t)$, and $y\_t(s^t)$ as the consumption choice, asset choice, and income realisation given history $s^t$. The household problem is therefore,

$$\max\_{\{c\_t,a\_t\}\_{t=0}^{\infty}}\mathbb{E}\_0 \sum\_{t=0}^{\infty}\sum\_{s^t}\pi(s^t)\beta^t u(c\_t(s^t))$$ $$\text{st. } \qquad c\_t(s^t)+a\_{t+1}(s^t)\leq y\_t(s^t)+Ra(s^{t-1}) \qquad \forall s^t= s^{t-1}\cup s\_t$$

The first-order conditions, and imposing $R\beta=1$, give that marginal utility follows a martingale process $$u'(c\_t(s^t))=\mathbb{E}[u'(c\_{t+1}(s^{t+1}))\mid s^t] \qquad (\dagger)$$ Since $u'''>0$, Jensen's implies that $$u'(c\_t(s^t))>u'(\mathbb{E}[c\_{t+1}(s^{t+1})\mid s^t]) \qquad (\ddagger)$$ By Doob's forwards convergence theorem, as $u'>0$, $(\dagger)$ implies that the marginal utilities converge almost surely to a constant value. My professor claims that given Jensen's inequality, $(\ddagger)$, this limit must be $0$. Why does $(\ddagger)$ imply the limit must be $0$? In my mind, all $(\ddagger)$ implies is that consumption is a submartingale, but doesn't say anything about the size of the increment. Why can it not be the case that consumption asymptotically converges upwards to some positive (but finite) value $M$ with the incremental change in consumption going to $0$?

---

A second example comes from a model with private information frictions. Suppose we have the same model as above. Agents can only borrow or lend from a risk-neutral insurance company at rate $1+r=\frac{1}{\beta}$ rather than purchasing state-contingent assets. Moreover, the income realisation of each consumer is private information to them only and the insurance company cannot credibly verify these. Therefore, the insurance company must pick an incentive compatible mechanism, amounting to a state-contingent transfer $\tau(s^t)$.

After simplifying the incentive compatibility constraints of the consumers to only local downwards ones, the insurer's problem (the dual of the consumer's) is $$v(w)=\max\_{\{\tau\_n,w\_n\}\_{n=1}^N} \sum\_{n=1}^N\pi\_n[-\tau\_n+\beta(v(w\_n))]$$ $$\text{st. } \begin{cases} \sum\_{n=1}^N\pi\_n[u(y\_n+\tau\_n)+\beta w\_n]\geq w \\
u(y\_n+\tau\_n)+\beta w\_n\geq u(y\_n+\tau\_{n-1})+\beta w\_{n-1} & \forall n\in \{2,3,\dots ,N\}\end{cases}$$ Where $w\_n$ is the 'utility promise' of future utils to the consumer after income shock $y\_n$ is observed. Explicitly, $$w(s^k)=\sum\_{t=k}^{\infty} \sum\_{s^t}\beta^{t-k}\pi(s^k\mid s^t)u(y(s^t)+\tau(s^t))$$ One can prove that the value function $v$ is strictly decreasing, strictly concave, and differentiable. The first-order conditions of the problem give that the marginal value follows a martingale process $$v'(w)=\mathbb{E}[v'(w')] \qquad (\sim)$$ With a bit of manipulation of the incentive compatibility and FOCs, one can prove that the utility promises spread out, $$w\_1' Here, again, my professor states that $(\sim)$ gets us almost-sure convergence and $(\approx)$ implies (since $v'<0$ and $v''<0$) that the only possible limit for $v'(w)$ is $-\infty$. Why does this final point follow? I cannot see why the difference between $w\_1'$ and $w\_N'$ cannot tend to $0$ so that $v'(w)$ tends to some finite negative number.

---

Is my misunderstanding coming from not fully getting Doob's forwards convergence (it has been a while since I properly thought about it), or do I not understand how the models are working?

**Tags:** macroeconomics, mathematical-economics, mechanism-design, asymmetric-information, martingales
> **Comments:**
> **username_1**: Hi Joseph: I didn't read the second part I don't know the answer to your first question ( it's interesting ) but  jensen's inequality states that  $E(f(x)) >= f(E(x))$ if $f(x)$ is convex so should there not be an expected value on the first term, $u^{\prime}(c_{t}(s^{t}))$ ? in your statement regarding Jensen's inequality. Not that it makes anything crystal clear to me anyway regarding your question but maybe that will make things clearer in the question ?  
> **username_0**: @markleeds Hi Mark. Along the optimal stochastic consumption path $\{c_t(s^t)\}$, FOCs give us that $u'(c_t(s^t))=\mathbb{E}[u'(c_{t+1}(s^{t+1})\mid s^t]$. I apply (conditional) Jensen's to the right-hand side to obtain $u'(c_t(s^t))=\mathbb{E}[u'(c_{t+1}(s^{t+1})\mid s^t]> u'(\mathbb{E}[c_{t+1}(s^{t+1})\mid s^t])$, so that $u'(c_t(s^t))>u'(\mathbb{E}[c_{t+1}(s^{t+1})\mid s^t])$. Is this what you are referring to?  
> **username_1**: Thanks Joseph. I understand the greater than sign but I forgot that you also have a martingale so you can just use $u^\prime(c_{t}(s^t))$. Thanks and my apologies for noise.  I've been reading about martingales lately so, even though I can't help, if you get an answer and it's not from econ.stackexchange, can you post it ? Thanks. Also, one possible suggestion is to put it on math.se but maybe not with all the notation and economics. More along the lines of, "if I have a martingale such as (and then something simpler ), and it obeys jensen's inequality, does it converge to anything ?  
> **username_0**: @markleeds No problem at all. After thinking about it some more, I am pretty certain it has to do with the assumption of i.i.d. income shocks, which I think precludes decaying increments to $0$. But hopefully someone can spell out this point more clearly as I'm still a bit muddled up about it.  
> **username_1**: Hi Joseph: you're a better math person than me but I remember recently reading that with probability 1, brownian motions go to $-\infty$ or $+\infty$. maybe there's a similar type theorem that adds conditions and makes a related kind of statement for martingales ? The lemma is on the bottom of page 4 of the first volume of williams and rogers. I didn't follow the proof there but I found a better proof is on the math.stackexchange site.  
> **username_1**: This is not the proof I found previously but still a decent proof by Eldredge.As it stands, the result is not helpful for your question anyway. It's somewhat of an odd result ( if you ask me ) which is why I was thinking it could be related to what you need. https://math.stackexchange.com/questions/3392744/question-about-the-textitsup-b-t-infty-of-a-brownian-motion  
> **username_1**: Joseph: The actual lemma in williams and rogers on page 4 is $ P(sup~B(t) = +\infty, ~~~inf ~ (B_t) = -\infty ) = 1 $, where $B_t$ is a brownian motion.  
> **username_1**: Here is the one I was trying to find. https://math.stackexchange.com/questions/3081905/supremum-of-brownian-motion  

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## 0 answers

