# Impact on interest rates of massive government spending

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 project over the coming years. If you believe simple macroeconomic models, like the IS-LM 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, 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, its predictions should be priced into the yield curve immediately.

• Did you look at the 7 year rate? – Hot Licks Feb 1 '18 at 3:35
• 7y look up insignificantly, i.e. definitely within one standard deviation DoD. – braaterAfrikaaner Feb 1 '18 at 3:38
• 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 Feb 1 '18 at 3:47
• DoD = Day on Day? (1 day change)? – Brian Romanchuk Feb 1 '18 at 12:18
• @Brian, yes Day on Day – braaterAfrikaaner Feb 5 '18 at 3:02

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.