Comparing the model debt to GDP forecast to the CBO forecast

[jdebacker]

The CBO Long-Term Budget Outlook provides forecasts of the debt-to-GDP ratio (D/Y) 30 years out. The current forecast is for D/Y (using debt held by the public as the metric) to reach 166 percent by 2054.

Under most calibrations, OG-USA seems to overshoot this path. For example, a recent calibration has givenD/Y values (in percentages) of:

Year	CBO	OG-USA
2024	99	99
2034	116	142
2044	139	186
2054	166	223

While there may be many issues why D/Y differs in the two models (assumptions about the path of government spending, the openness of financial markets, the differential between private interest rates and those on government debt, etc.), I want to focus this discussion on a particular measurement issue; comparing a simulated world without inflation to the forecast of a world with positive inflation.

OG-USA is a real model, which means everything is measured in units of GDP and there is no inflation. When comparing units of real variables (e.g., government spending and GDP) between OG-USA and the CBO forecasts, this doesn't pose a problem. However, the stock of government debt, D, is different since it is measured in nominal terms. Thus, without an increase in the nominal value of government debt, the ratio of the real value of D to real GDP will decline in relation to the rate of inflation. This is noted by Aizenman and Marion (2009), among others.

The problem of comparing the CBO and OG-USA forecasts relate to this. In particular, I think OG-USA will always forecast a higher D/Y ratio (with an increasing difference over time) than a forecast of a world with positive inflation since OG-USA does not account for the way in which inflation eats away at the real value of the the stock of debt over time.

If this is the case, one way to make OG-USA more consistent with CBO would be to include (exogenous) inflation in the law of motion for the stock of government debt.

Not that such an adjustment would be needed for stocks of nominal variables (like D), but not stocks of real variables (like K), since inflation does not reduce the quantity of real variables.

CC @rickecon

[jdebacker]

Here's a sketch of the issue:

Let the law of motion for the stock of debt, $D_t$, be given by:

$$ D_{t+1} = D_t + Outlays_t - Rev_t $$

To put these into real terms, we divide through by the price level, $P_t$:

$$ \begin{split} \frac{D_{t+1}}{P_{t+1}} &= \frac{D_t + Outlays_t - Rev_t}{P_{t+1}} \\ &= \frac{D_t + Outlays_t - Rev_t}{P_{t}(1+\pi_{t})} \end{split} $$

Where $\pi_{t}$ is the inflation rate from period $t$ to $t+1$. Letting a "hat" denote variables in real terms, we have:

$$ \hat{D}_{t+1} = \frac{\hat{D}_t + \hat{Outlays}_t - \hat{Rev}_t}{(1+\pi_t)} $$

So inflation will eat away at the real value of debt, even in a law of motion where all right hand side variables are in real terms.

Note that with other dynamic equations representing a law of motion for a stock variable, such as the capital stock, the adjustment for inflation is unnecessary since those equations are written in real terms. e.g., We specified the law of motion for the capital stock in real terms as:

$$ \hat{K}_{t+1} = (1 - \delta)\hat{K}_t + \hat{I}_t $$

[jdebacker]

Danny Yagan outlines this issue nicely in the paper "Excess Interest and the Growth Dividend"

cc @rickecon