<p>You <em>could</em> say IS-LM is too simplistic.</p>

<p>The fundamental discrete time consumption-based asset pricing equation says that if $ r_t $ is the risk-free interest rate at time $ t $, we have</p>

<p>$$ \beta (1+r_t) \frac{u'(C_{t+1})}{u'(C_t)} = 1 $$</p>

<p>(I'm dropping the expectation since I will be working with a perfect foresight model later on) where $ u $ is the utility function of <em>any</em> consumer, $ 0 &lt; \beta &lt; 1 $ is a discount factor which measures "impatience" of the same consumer, and $ c_t, c_{t+1} $ is the consumption of the same consumer in time $ t $ and $ t+1 $ respectively. The most elegant form of the relation is with the assumption of power utility $ u(c) = c^{1 - \gamma}/(1 - \gamma) $ with $ \gamma &gt; 0 $, in which case we get (after log-linearization)</p>

<p>$$ r_t = R + \gamma \Delta c_t $$</p>

<p>where $ \Delta c_t $ is log consumption growth and $ R = -\log(\beta) $ is the real interest rate consistent with flat consumption. The intuition is simple: people want to smooth consumption over time. If they expect their consumption in the future to be much higher than their consumption today, they will try to borrow against the better future and push real interest rates up, and vice versa. </p>

<p>Consider a baseline economy with no investment, perfect foresight, and a representative consumer first. (This is unrealistic, but IS-LM is even more unrealistic, so...) In this case, the resource constraint of the economy reads $ Y_t = C_t + G_t $ in every period, where $ G_t $ is government spending. If you assume consumers have utility that is additively separable over time (the asset pricing relation assumes this as well), then the no investment assumption turns their decision problem into a set of one-period decision problems. (This is a technical condition to ensure what I am doing is well-defined.) Define $ D_t $ as the level of output consistent at time $ t $ with $ G_t = 0 $, and let $ m_t $ be the "fiscal multiplier", defined by</p>

<p>$$ m_t = \frac{dY_t}{dG_t} $$</p>

<p>Most models won't give constant $ m $, but if necessary we may linearize the model around somewhere other than $ G_t = 0 $ and make this assumption locally. These assumptions imply $ Y_t = D_t + m_t G_t $ at any time $ t $, so we may express consumption as $ C_t = D_t + (m_t -1) G_t $. Plugging into the relation gives</p>

<p>$$ r_t \approx R + \gamma \Delta d_t + \gamma (m_t -1) \Delta \left( \frac{g_t}{1 - m_t g_t} \right) $$</p>

<p>where $ g_t = G_t/Y_t $ is the ratio of government spending to GDP and $ \Delta d_t = \Delta \log(D_t) $. (Ignore the pole of the denominator - we're working around $ g_t = 0 $.) The fraction $ \frac{g_t}{1 - m_t g_t} $ is increasing in $ g_t $ with $ m \geq 0 $, so we see that the real interest rate at time $ t $ depends on two things: the "natural" growth rate of the economy given by $ \Delta d_t $, and the consumption growth induced by changes in government spending rel to GDP $ g_t $. However, the <em>sign</em> of the effect of an anticipated increase in government spending depends on the value of the fiscal multiplier $ m $. If $ m &lt; 1 $, then government spending crowds out consumption, and <em>rising</em> government spending is associated with <em>low</em> real interest rates, i.e expectations of high spending in the future would push rates down today, and vice versa if $ m &gt; 1 $. Most everyone would agree that the fiscal multiplier in the US economy today is less than $ 1 $, so news of higher spending in the future should be associated with lower real interest rates now, if with anything at all.</p>

<p>It also matters, of course, if government spending affects $ \Delta d_t $ - if we're speaking of an infrastructure investment program, for instance, then we should expect that the change in government spending also leads to a change in the path of $ D_t $. The two effects for such a program act in opposite directions, so the net effect of a government investment program tomorrow on real interest rates today could go either way, even in this simple economy.</p>

<p>I won't work out the model with investment (allowing consumers to invest) here, but the effect of investment is to allow consumers to trade consumption now for consumption in the future in aggregate; so it dampens the impact of a change in government spending on real interest rates (since now consumers have a tool for smoothing consumption in the face of government expenditure shocks). </p>

<p>It's also true that the announcement in the State of the Union address probably carried less information than an announcement in this model, since market participants were already aware of such an infrastructure investment plan for months (maybe a year?), but with a fiscal multiplier less than $ 1 $ it's not easy to say anything about the direction of the real interest rate response. </p>

CC BY-SA 3.0
2018-02-01T06:33:54.437Z
2018-02-01T11:12:44.910Z
2018-02-01T11:12:44.910Z
Ege Erdil
<p>Your answer hits a lot of points, some theoretical, some practical.</p>

<ol>
<li>Most versions of IS/LM that I am aware of have only one interest rate, and if we want to be strict, cannot be applied to a yield curve. (Is “the” interest rate the short rate set by the central bank, or the market-determined bond yield?) There are standard ways of thinking about this, but I never look at IS/LM, so I will not comment.</li>
<li>If you look at any modern textbook on fixed income pricing, you will see that long-term bond yields are equal to the expected path of the short rate plus a term premium. If we ignore the term premium, this means that a 10-year bond is roughly equal to the expected average of the policy rate over the next 10 years.</li>
<li>During a rate hike cycle, the typical pattern is for the yield curve to flatten. This can be seen by looking at data on the FRED database: <a href="https://fred.stlouisfed.org/series/T10Y2Y" rel="nofollow noreferrer">Link to FRED 2/10 year slope</a>. This is because the market generally expects the policy rate to stop rising after a few years; and so the average policy rate during any maturity converges towards the terminal rate. (Term premia etc. will affect that.) </li>
<li>Under most economic models, increased spending - “all else equal” - would at least raise nominal GDP in the short run (or at the minimum, force the central bank to raise the policy rate, which is what matters here). Whether that increase represents higher real GDP, or greate inflation, becomes a controversial question. (One could argue that an inflation-targeting central bank can cancel out the effect of fiscal policy, but that then requires the central bank to correctly forecast how to tighten monetary policy to exactly offset fiscal policy. Such a move might not be considered holding everything “all else equal.”) If market participants believe that the central bank’s policy rate is approximated by a Taylor Rule, then this would presumably lead to higher policy rates. However, it is possible that there could be spare capacity in the economy, and interest rates could be unchanged, and inflation remains near the central bank’s implicit target.</li>
<li>Market participants will react to new information about growth and the policy rate. This can be seen by looking at yield changes on the the dates of important data releases, auch as Nonfarm Payrolls. A lack of reaction to the State if the Union speech may just tell us that it had no new information. </li>
</ol>

<p>There are a number of reasons that the announcement could be ignored. </p>

<ul>
<li>The market may have priced in a loosening of fiscal policy much earlier, such as after the election result was announced.</li>
<li>There is no guarantee that any plan can get through Congress. Also, even if it is passed, it might be offset by cuts to other programs.</li>
<li>Finally, its size needs to be taken in context relative to the volatility of growth in the private sector. Infrastructure spending 10 years from now is a lot less important for interest rates than the potential for a recession in the next few years.</li>
</ul>

CC BY-SA 3.0
2018-02-01T14:00:05.567Z
You can do IS-LM in a dynamic setting (you can find something like this in Blanchard-Fischer, for instance). It's not true that under most economic models increased government spending leads to increased NGDP, this depends crucially on the monetary policy regime in place. With inflation targeting, higher government spending only leads to higher NGDP growth insofar as it leads to higher RGDP growth (monetary offset). I would like to complain more about the "most economic models" part of the answer, but I'm running out of characters...
Ege Erdil
CC BY-SA 3.0
2018-02-01T13:17:49.793Z
2018-02-01T18:46:14.237Z
2018-02-01T18:46:14.237Z
Brian Romanchuk
<p>At the last State of the Union address, the President of the United States has reaffirmed his intentions to undertake a massive <a href="https://www.cbsnews.com/news/state-of-the-union-trump-infrastructure/" rel="nofollow noreferrer">1.5 trillion dollar infrastructure spending</a> project over the coming years. If you believe simple macroeconomic models, like the <a href="https://en.wikipedia.org/wiki/IS%E2%80%93LM_model" rel="nofollow noreferrer">IS-LM model</a>, this kind of government spending should lead to higher real GDP and higher interest rates. I am assuming that, in the real world, this would correspond to a steepening of the yield curve, i.e. medium and long term yields go up. However, looking at <a href="https://www.treasury.gov/resource-center/data-chart-center/interest-rates/Pages/TextView.aspx?data=yield" rel="nofollow noreferrer">yield curve data</a>, I cannot discern any significant uptick in long term yields DoD. Does this mean that either:</p>

<ol>
<li>the yield curve should not actually respond the way I assumed?</li>
<li>this announcement carried no real information that would have
changed market participants' expectations about the future?</li>
<li>the IS-LM model is too simplistic to even adequately predict the directional
impact of government spending on interest rates?</li>
</ol>

<p>I realize that I may be abusing the IS-LM model here, as it probably takes a more long term, qualitative view. However, I would think, due to <a href="https://en.wikipedia.org/wiki/Efficient-market_hypothesis" rel="nofollow noreferrer">market efficiency</a>, its predictions should be priced into the yield curve immediately.</p>

CC BY-SA 3.0
2018-02-01T03:35:03.063Z
Did you look at the 7 year rate?
Hot Licks
CC BY-SA 3.0
2018-02-01T03:38:51.687Z
7y look up insignificantly, i.e. definitely within one standard deviation DoD.
braaterAfrikaaner
CC BY-SA 3.0
2018-02-01T03:47:01.183Z
It was 2.22 on the first of November, began steadily climbing shortly after that.  This was about the time it became clear that the tax cut would be pushed through.
Hot Licks
CC BY-SA 3.0
2018-02-01T12:18:13.407Z
DoD = Day on Day? (1 day change)?
Brian Romanchuk
CC BY-SA 3.0
2018-02-05T03:02:25.833Z
@Brian, yes Day on Day
braaterAfrikaaner
<macroeconomics><interest-rate><government-spending>
Impact on interest rates of massive government spending
braaterAfrikaaner
# Impact on interest rates of massive government spending

(Asked by: username_0 on 2018-02-01)

At the last State of the Union address, the President of the United States has reaffirmed his intentions to undertake a massive [1.5 trillion dollar infrastructure spending](https://www.cbsnews.com/news/state-of-the-union-trump-infrastructure/) project over the coming years. If you believe simple macroeconomic models, like the [IS-LM model](https://en.wikipedia.org/wiki/IS%E2%80%93LM_model), this kind of government spending should lead to higher real GDP and higher interest rates. I am assuming that, in the real world, this would correspond to a steepening of the yield curve, i.e. medium and long term yields go up. However, looking at [yield curve data](https://www.treasury.gov/resource-center/data-chart-center/interest-rates/Pages/TextView.aspx?data=yield), I cannot discern any significant uptick in long term yields DoD. Does this mean that either:

1. the yield curve should not actually respond the way I assumed?
2. this announcement carried no real information that would have
   changed market participants' expectations about the future?
3. the IS-LM model is too simplistic to even adequately predict the directional
   impact of government spending on interest rates?

I realize that I may be abusing the IS-LM model here, as it probably takes a more long term, qualitative view. However, I would think, due to [market efficiency](https://en.wikipedia.org/wiki/Efficient-market_hypothesis), its predictions should be priced into the yield curve immediately.

**Tags:** macroeconomics, interest-rate, government-spending

> **Comments:**
> **username_1**: Did you look at the 7 year rate?  
> **username_0**: 7y look up insignificantly, i.e. definitely within one standard deviation DoD.  
> **username_1**: It was 2.22 on the first of November, began steadily climbing shortly after that.  This was about the time it became clear that the tax cut would be pushed through.  
> **username_2**: DoD = Day on Day? (1 day change)?  
> **username_0**: @Brian, yes Day on Day  

---

## 2 answers

### Answer by username_3 on 2018-02-01. Score: 1 | Accepted answer

You *could* say IS-LM is too simplistic.

The fundamental discrete time consumption-based asset pricing equation says that if $ r\_t $ is the risk-free interest rate at time $ t $, we have

$$ \beta (1+r\_t) \frac{u'(C\_{t+1})}{u'(C\_t)} = 1 $$

(I'm dropping the expectation since I will be working with a perfect foresight model later on) where $ u $ is the utility function of *any* consumer, $ 0 < \beta < 1 $ is a discount factor which measures "impatience" of the same consumer, and $ c\_t, c\_{t+1} $ is the consumption of the same consumer in time $ t $ and $ t+1 $ respectively. The most elegant form of the relation is with the assumption of power utility $ u(c) = c^{1 - \gamma}/(1 - \gamma) $ with $ \gamma > 0 $, in which case we get (after log-linearization)

$$ r\_t = R + \gamma \Delta c\_t $$

where $ \Delta c\_t $ is log consumption growth and $ R = -\log(\beta) $ is the real interest rate consistent with flat consumption. The intuition is simple: people want to smooth consumption over time. If they expect their consumption in the future to be much higher than their consumption today, they will try to borrow against the better future and push real interest rates up, and vice versa.

Consider a baseline economy with no investment, perfect foresight, and a representative consumer first. (This is unrealistic, but IS-LM is even more unrealistic, so...) In this case, the resource constraint of the economy reads $ Y\_t = C\_t + G\_t $ in every period, where $ G\_t $ is government spending. If you assume consumers have utility that is additively separable over time (the asset pricing relation assumes this as well), then the no investment assumption turns their decision problem into a set of one-period decision problems. (This is a technical condition to ensure what I am doing is well-defined.) Define $ D\_t $ as the level of output consistent at time $ t $ with $ G\_t = 0 $, and let $ m\_t $ be the "fiscal multiplier", defined by

$$ m\_t = \frac{dY\_t}{dG\_t} $$

Most models won't give constant $ m $, but if necessary we may linearize the model around somewhere other than $ G\_t = 0 $ and make this assumption locally. These assumptions imply $ Y\_t = D\_t + m\_t G\_t $ at any time $ t $, so we may express consumption as $ C\_t = D\_t + (m\_t -1) G\_t $. Plugging into the relation gives

$$ r\_t \approx R + \gamma \Delta d\_t + \gamma (m\_t -1) \Delta \left( \frac{g\_t}{1 - m\_t g\_t} \right) $$

where $ g\_t = G\_t/Y\_t $ is the ratio of government spending to GDP and $ \Delta d\_t = \Delta \log(D\_t) $. (Ignore the pole of the denominator - we're working around $ g\_t = 0 $.) The fraction $ \frac{g\_t}{1 - m\_t g\_t} $ is increasing in $ g\_t $ with $ m \geq 0 $, so we see that the real interest rate at time $ t $ depends on two things: the "natural" growth rate of the economy given by $ \Delta d\_t $, and the consumption growth induced by changes in government spending rel to GDP $ g\_t $. However, the *sign* of the effect of an anticipated increase in government spending depends on the value of the fiscal multiplier $ m $. If $ m < 1 $, then government spending crowds out consumption, and *rising* government spending is associated with *low* real interest rates, i.e expectations of high spending in the future would push rates down today, and vice versa if $ m > 1 $. Most everyone would agree that the fiscal multiplier in the US economy today is less than $ 1 $, so news of higher spending in the future should be associated with lower real interest rates now, if with anything at all.

It also matters, of course, if government spending affects $ \Delta d\_t $ - if we're speaking of an infrastructure investment program, for instance, then we should expect that the change in government spending also leads to a change in the path of $ D\_t $. The two effects for such a program act in opposite directions, so the net effect of a government investment program tomorrow on real interest rates today could go either way, even in this simple economy.

I won't work out the model with investment (allowing consumers to invest) here, but the effect of investment is to allow consumers to trade consumption now for consumption in the future in aggregate; so it dampens the impact of a change in government spending on real interest rates (since now consumers have a tool for smoothing consumption in the face of government expenditure shocks).

It's also true that the announcement in the State of the Union address probably carried less information than an announcement in this model, since market participants were already aware of such an infrastructure investment plan for months (maybe a year?), but with a fiscal multiplier less than $ 1 $ it's not easy to say anything about the direction of the real interest rate response.


---

### Answer by username_2 on 2018-02-01. Score: 1

Your answer hits a lot of points, some theoretical, some practical.

1. Most versions of IS/LM that I am aware of have only one interest rate, and if we want to be strict, cannot be applied to a yield curve. (Is “the” interest rate the short rate set by the central bank, or the market-determined bond yield?) There are standard ways of thinking about this, but I never look at IS/LM, so I will not comment.
2. If you look at any modern textbook on fixed income pricing, you will see that long-term bond yields are equal to the expected path of the short rate plus a term premium. If we ignore the term premium, this means that a 10-year bond is roughly equal to the expected average of the policy rate over the next 10 years.
3. During a rate hike cycle, the typical pattern is for the yield curve to flatten. This can be seen by looking at data on the FRED database: [Link to FRED 2/10 year slope](https://fred.stlouisfed.org/series/T10Y2Y). This is because the market generally expects the policy rate to stop rising after a few years; and so the average policy rate during any maturity converges towards the terminal rate. (Term premia etc. will affect that.)
4. Under most economic models, increased spending - “all else equal” - would at least raise nominal GDP in the short run (or at the minimum, force the central bank to raise the policy rate, which is what matters here). Whether that increase represents higher real GDP, or greate inflation, becomes a controversial question. (One could argue that an inflation-targeting central bank can cancel out the effect of fiscal policy, but that then requires the central bank to correctly forecast how to tighten monetary policy to exactly offset fiscal policy. Such a move might not be considered holding everything “all else equal.”) If market participants believe that the central bank’s policy rate is approximated by a Taylor Rule, then this would presumably lead to higher policy rates. However, it is possible that there could be spare capacity in the economy, and interest rates could be unchanged, and inflation remains near the central bank’s implicit target.
5. Market participants will react to new information about growth and the policy rate. This can be seen by looking at yield changes on the the dates of important data releases, auch as Nonfarm Payrolls. A lack of reaction to the State if the Union speech may just tell us that it had no new information.

There are a number of reasons that the announcement could be ignored.

* The market may have priced in a loosening of fiscal policy much earlier, such as after the election result was announced.
* There is no guarantee that any plan can get through Congress. Also, even if it is passed, it might be offset by cuts to other programs.
* Finally, its size needs to be taken in context relative to the volatility of growth in the private sector. Infrastructure spending 10 years from now is a lot less important for interest rates than the potential for a recession in the next few years.

> **Comments:**
> **username_3**: You can do IS-LM in a dynamic setting (you can find something like this in Blanchard-Fischer, for instance). It's not true that under most economic models increased government spending leads to increased NGDP, this depends crucially on the monetary policy regime in place. With inflation targeting, higher government spending only leads to higher NGDP growth insofar as it leads to higher RGDP growth (monetary offset). I would like to complain more about the "most economic models" part of the answer, but I'm running out of characters...  

---

