The Budget Lab Small Macro Model (BLSMM): Model Documentation

• A unit of time corresponds to one (fiscal) year, denoted by (\(t\)).
• Interest rates, inflation, growth rates, and the unemployment rate are specified in percent, e.g., 5.00%, not 0.05. Interest, inflation, and growth rates are at annual rates.
• The output gap (real GDP relative to potential real GDP) is specified in percent of potential real GDP.
• Budget quantities are expressed in nominal dollars and as ratios to nominal GDP and potential nominal GDP.
• Budget surpluses are positive; deficits are negative.
• GDP is expressed in both inflation-adjusted (real) and nominal terms; when necessary for clarity, a dollar sign ($) is used to distinguish nominal from real quantities, as in \(\mathrm{GDP\$}\) to represent nominal GDP.
• An asterisk (*) denotes a long-run anchor or potential level or growth rate of a variable. Such variables are generally exogenous in model simulation.
• A subscript 0, as in \(\mathrm{TP10}_0\) denotes an exogenous constant.
• Interest rates begin with \(\mathrm{R}\).
• Parameters are generally denoted with Greek letters; their values are listed separately.
• A variable followed by (\(L\)) indicates a finite polynomial in the lag operator, as in \(\sigma(L)\).  Individual elements of such polynomials are denoted by numerical subscripts, as in \(\sigma_1\).
• A lower case \(\varepsilon\) denotes a shock or residual term, which for model purposes is treated as exogenous, as in \(\varepsilon_{\mathrm{xgap}}(t)\).

Let \(\mathrm{GDP}\) denote real GDP and \(\mathrm{GDP\$}\) nominal GDP. Let \(g^*(t)\) denote growth in potential real GDP, in %, as indicated in (1).

\begin{equation}
   g^*(t) \equiv 100(\mathrm{GDP}^*(t) / \mathrm{GDP}^*(t-1) - 1). \tag{1}
\end{equation}

To decompose contributions to potential real GDP growth, we first note that real GDP can be expressed as the product of total employment and productivity, where employment reflects all employed persons (civilian employment) and productivity is measured as real GDP per employed person. Continuing to let potential quantities be denoted with an asterisk, we let \(\mathrm{CE}^*(t)\) denote potential civilian employment and \(\mathrm{LQ}^*(t)\) potential labor productivity. The latter is assumed to be exogenous, with a growth rate \(lq^*(t)\) as in

\begin{equation}
   \mathrm{LQ}^*(t) / \mathrm{LQ}^*(t-1) \equiv 1 + 0.01\,lq^*(t). \tag{2}
\end{equation}

The potential labor force \([\mathrm{LF}^*(t)]\), an important determinant of potential employment, grows at an exogenous rate denoted by \(lf^*(t)\), as in

\begin{equation}
   \mathrm{LF}^*(t) / \mathrm{LF}^*(t-1) \equiv 1 + 0.01\,lf^*(t). \tag{3}
\end{equation}
Potential employment \([\mathrm{CE}^*(t)]\) is equal to the potential labor force less potential/structural unemployment identified with the NAIRU \([un(t)]\), as in

\begin{equation}
   \mathrm{CE}^*(t) = \mathrm{LF}^*(t) \cdot (1- 0.01\,un(t)). \tag{4}
\end{equation}

The NAIRU is exogenous and expressed in percent of the labor force.

As described above, potential real GDP is equal to the product of potential productivity and potential employment, as in

\begin{equation}
   \mathrm{GDP}^*(t) = \mathrm{LQ}^*(t) \cdot \mathrm{CE}^*(t). \tag{5}
\end{equation}

Combining the two previous expressions,

\begin{equation}
   \mathrm{GDP}^*(t) = \mathrm{LQ}^*(t) \cdot \mathrm{LF}^*(t) \cdot (1- 0.01\,un(t)). \tag{6}
\end{equation}

From (6) we derive at an expression for potential GDP growth that reflects contributions from potential productivity growth, potential labor force growth, and the NAIRU,

\begin{multline}
   g^*(t) = 100[(1+0.01 \cdot lq^*(t))(1+0.01 \cdot lf^*(t))(1- 0.01\,un(t)) \\ / (1- 0.01\,un(t-1)) - 1]. \tag{7}
\end{multline}

For purposes of computation we use (7). A compact approximation useful for expository purposes is

\begin{equation}
   g^*(t) \approx lq^*(t) + lf^*(t) -[un(t)- un(t-1)]. \tag{8}
\end{equation}

Expressions (7) and (8) show that the growth rate of potential GDP is closely approximated by the sum of potential growth in productivity and the labor force, less the first-difference of the NAIRU, where growth rates are measured in annualized percentage rates and the NAIRU is measured in percent of the labor force. As a quantitative matter, estimates of the NAIRU typically change at most slightly from year to year, in which case growth in potential real GDP is approximately equal to growth in potential plus growth in the potential labor force: \(g^*(t) \approx lq^*(t) + lf^*(t)\).2

Letting \(\mathrm{PGDP}\) denote the GDP price level, which is indexed to 100 in the base year, we have expressions for potential nominal GDP and nominal GDP.3

\begin{equation} \mathrm{GDP\$}^*(t) = (\mathrm{PGDP}(t)/100)\,\mathrm{GDP}^*(t) \tag{9} \end{equation}

\begin{equation} \mathrm{GDP\$}(t) = (\mathrm{PGDP}(t)/100)\,\mathrm{GDP}(t). \tag{10} \end{equation}

The total budget balance of the federal government, expressed in nominal dollars, is denoted by \(\mathrm{BUD\$}(t)\). Positive values indicate surplus and negative values indicate deficit. The primary budget balance, which is equal to receipts less primary (non-interest) outlays, is denoted by \(\mathrm{BUDP\$}(t)\). Receipts and primary outlays are denoted by \(\mathrm{GFR}(t)\) and \(\mathrm{GFOP}(t)\), respectively.

We let \(\mathrm{rbudp}^*(t)\) denote the primary budget deficit (\(\mathrm{BUDP}\)) as a percent of potential nominal GDP, as in

\begin{equation}
   \mathrm{rbudp}^*(t) \equiv 100\,\mathrm{BUDP}(t)/\mathrm{GDP\$}^*(t). \tag{11}
\end{equation}

Similar expressions for the ratios of total receipts and primary outlays to potential nominal GDP are shown in the next two expressions, respectively.4

\begin{equation}
   \mathrm{rgfr}^*(t) \equiv 100\,\mathrm{GFR}(t)/\mathrm{GDP\$}^*(t). \tag{12}
\end{equation}

\begin{equation}
   \mathrm{rgfop}^*(t) \equiv 100\,\mathrm{GFOP}(t)/\mathrm{GDP\$}^*(t). \tag{13}
\end{equation}

Clearly, the following identity also holds among these ratios to potential nominal GDP:

\begin{equation}
   \mathrm{rbudp}^*(t) \equiv \mathrm{rgfr}^*(t) - \mathrm{rgfop}^*(t). \tag{14}
\end{equation}

Let \(\mathrm{xgap}(t)\) denote the output (real GDP) gap to potential, in percent:

\begin{equation}
   \mathrm{xgap}(t) \equiv 100\,(\mathrm{GDP}(t)/\mathrm{GDP}^*(t) - 1). \tag{15}
\end{equation}

The output gap evolves according to a simple linear relationship with impacts from the real bond yield and from the change in the ratio of the primary budget balance as a percentage of potential nominal gross domestic product.

The nominal bond yield is denoted with \(\mathrm{R10}(t)\) and expected inflation with \(\pi^e(t)\). The term \(\mathrm{r10bar}\) is the level of the real 10-year bond yield that would be consistent with no output gap in the absence of primary budget deficits or other shocks. It can also be interpreted as a constant that appears in a linear approximation of a more complex, nonlinear model of the output gap. Pre-multiplying by \(\sigma(L)\) allows it to be dimensioned in percentage points for ease of interpretation. The output gap is determined by (16).

\begin{multline}
   \mathrm{xgap}(t) = \eta\,\mathrm{xgap}(t-1) - \theta(L)\,\mathrm{rbudp}^*(t) - \sigma(L)(\mathrm{R10}(t) \\ - \pi^e(t) - \mathrm{r10bar}(t)) + \varepsilon_{\mathrm{xgap}}(t), \tag{16}
\end{multline}

where \(\sigma(L)\) and \(\theta(L)\) are functions in the lag operator, \(L\).5 The variable \(\varepsilon_{\mathrm{xgap}}(t)\) is an exogenous residual term.

A dynamic version of Okun's Law relates the unemployment gap to the current and lagged output gap, where \(u(t)\) denotes the unemployment rate (in %), \(un(t)\) denotes the NAIRU, \(\alpha(L)\) denotes a function in the lag operator, and \(\varepsilon_u(t)\) denotes a residual for this relationship.

\begin{equation}
   u(t) - un(t) = -\alpha(L)\,\mathrm{xgap}(t) + \varepsilon_u(t). \tag{17}
\end{equation}

Let \(\pi(t)\) denote inflation, let \(\pi^e(t)\) denote long-run expected inflation (over 10 years), and let \(\pi^*(t)\) denote the inflation target that influences monetary policy and expectations of monetary policy.

Inflation evolves according to a Phillips curve that is vertical in the long run:

\begin{equation}
   \pi(t) = \gamma_1\,\pi(t-1) + (1-\gamma_1)\,\pi^e(t-1) + \gamma_2\,(un(t) - u(t)) + \varepsilon_\pi(t). \tag{18}
\end{equation}

Long-run inflation expectations respond gradually to actual inflation. The form of the expression allows for the flexibility to anchor long-run inflation expectations to the inflation target when \(\lambda_3 \in (0,1]\).

\begin{equation}
   \pi^e(t) = \lambda_1\,\pi^e(t-1) + \lambda_2\,\pi(t) + \lambda_3\,\pi^*(t) + \varepsilon_{\pi^e}(t). \tag{19}
\end{equation}

The weights on expected inflation, actual inflation, and the inflation target sum to one: \(\lambda_1 + \lambda_2 + \lambda_3 = 1\).

The price level (\(\mathrm{PGDP}\)) evolves according to inflation, as in

\begin{equation}
   \mathrm{PGDP}(t) \equiv \mathrm{PGDP}(t-1)\,(1+0.01\,\pi(t)). \tag{20}
\end{equation}

The short-term interest rate, \(\mathrm{RF}(t)\), (federal funds rate) is determined with a Taylor-type policy rule that includes a real “r-star” term appropriate to that interest rate, \(rf^*(t)\), and impacts from inflation and unemployment gaps relative to the inflation target and the NAIRU, respectively.

\begin{multline}
   \mathrm{RF}(t) = rf^*(t) + \pi(t) + \mu_1[\pi(t) - \pi^*(t)] + \mu_2[\pi^e(t) - \pi^*(t)] \\ + \mu_3[un(t) - u(t)] + \varepsilon_{\mathrm{RF}}(t). \tag{21}
\end{multline}

The inflation target, \(\pi^*(t)\), is set to 2.0 in the baseline to match the Fed's announced target.6

In simulation \(\mathrm{RF}\) may take on negative values (breech the zero lower bound), which may be interpreted as an indication that the Federal Reserve would consider non-traditional tools including large-scale asset purchases and forward guidance that are not present in the model.

The long-term interest rate in the model (\(\mathrm{R10}(t)\)) is identified with the 10-year Treasury Note yield, which appears in the expression for the output gap shown above. The 10-year yield is equal to the sum of contributions from the expected future short-term interest rate, \(\mathrm{MPE10}(t)\), and the term premium, \(\mathrm{TP10}(t)\):

\begin{equation}
   \mathrm{R10}(t) = \mathrm{MPE10}(t) + \mathrm{TP10}(t). \tag{22}
\end{equation}

The average expected future short-term interest rate over the 10-year life of the Treasury note, \(\mathrm{MPE10Y}\), is influenced by the current level of the overnight policy rate, \(\mathrm{RF}(t)\) and by its longer-run determinants, including \(rf^*(t)\) and expected inflation.

\begin{multline}
   \mathrm{MPE10}(t) = \phi_1\,\mathrm{RF}(t) + (1-\phi_1)\{rf^*(t) + \pi^e(t) \\ + \phi_2[\pi^e(t) - \pi^*(t)]\} + \varepsilon_{\mathrm{MPE\,10}}(t). \tag{23}
\end{multline}

At one extreme, if \(\phi_1\) were set to unity, the average expected future short-term interest rate would equal its current level. At the opposite extreme, if \(\phi_1\) were set to zero, the average expected short rate would be equal to \(rf^* + \pi^e\) plus an adjustment for the gap between long-run inflation expectations and the inflation target, \(\phi_2[\pi^e(t) - \pi^*(t)]\).

In a steady state—inflation and inflation expectations equal to target, no unemployment gap, and zero residual terms, \(\mathrm{RF}(t)\) and \(\mathrm{MPE10}(t)\)—both the current federal funds rate and the average expected federal funds rate will equal the sum of the inflation target and \(rf^*\). More generally, \(\mathrm{MPE10}(t) - \mathrm{RF}(t)\) can be either positive or negative. Combining the policy rule and the expression for \(\mathrm{MPE10}\),

\begin{multline}
   \mathrm{MPE10}(t) - \mathrm{RF}(t) = (1-\phi_1)\{(1+\phi_2 - \mu_2)[\pi^e(t) - \pi^*(t)] \\ - (1+\mu_1)[\pi(t) - \pi^*(t)] - \mu_3[un(t) - u(t)] - \varepsilon_{\mathrm{RF}}(t)\} + \varepsilon_{\mathrm{MPE\,10}}(t). \tag{24}
\end{multline}

When the residuals (the \(\varepsilon\)'s) are zero, the sign of \(\mathrm{MPE10} - \mathrm{RF}\) is equal to the sign of \((1+\phi_2 - \mu_2)[\pi^e(t) - \pi^*(t)] - (1+\mu_1)[\pi(t) - \pi^*(t)] - \mu_3[un(t) - u(t)]\). As an example, \(\mathrm{MPE10}\) will be lower than \(\mathrm{RF}\) when actual inflation is sufficiently high relative to expected inflation and/or when the unemployment rate is sufficiently far below the NAIRU.

The other contribution to the long-term interest rate is its term premium, \(\mathrm{TP10}(t)\), which is set equal to the sum of its long run level, \(\mathrm{TP10}_0\), and an exogenous residual.

\begin{equation}
   \mathrm{TP10}(t) = \mathrm{TP10}_0(t) + \varepsilon_{\mathrm{TP10}}(t). \tag{25}
\end{equation}

Baseline assumptions

When we pre-populate the workbook with 10-year macroeconomic and budget projections, we do so with assumptions for the neutral level of the real federal funds rate, \(rf^*\), and the term premium, \(\mathrm{TP}(t)\), to be consistent with baseline projected paths of interest rates, inflation, and the output gap. In the baseline, the real neutral federal funds rate is exogenous, as is the reference level for the real 10-year yield, \(\mathrm{r10bar}(t)\) that appears in the expression for the output gap, (13). Baseline assumptions for both are chosen to be consistent with baseline projections for interest rates informed by the most recently published set of baseline economic and budget projections from the Congressional Budget Office (CBO).

Updates to reflect User deltas

Feedback onto the primary budget balance from user assumptions for potential growth of productivity and the labor force

User assumptions that alter potential GDP growth are allowed to influence primary budget ratios to nominal GDP, loosely based on budgetary feedback channels estimated by the Congressional Budget Office.7 In CBO’s modeling, increases in potential productivity raise the primary budget balance ratio to nominal GDP (smaller deficits as a % of nominal GDP), as do increases in the potential labor force. The impacts on the primary budget ratio reflect decreases in primary outlays as a percent of nominal GDP; the ratio of government receipts to nominal GDP is little affected by changes in potential productivity and labor force growth in CBO’s modeling.

When users input alternative assumptions for potential labor force and productivity growth, those assumptions are allowed to influence the ratio of primary outlays to nominal GDP and hence the primary budget balance as a ratio to nominal GDP as described in this section. For simplicity and because the effects are not quantitatively substantial according to CBO’s modeling, BLSMM does not automatically update the revenue ratio to GDP. However, users have the option of directly altering both the receipts and primary outlays ratios through respective user deltas.

Let superscripts denote the baseline scenario (B) and a user scenario (U), respectively. With this notation \(lf^{*U}(t) - lf^{*B}(t)\) represents the user delta for potential growth of the labor force in year \(t\). Similarly, \(lg^{*U}(t) - lg^{*B}(t)\) represents the user delta for potential productivity growth. Let \(\psi_1\) and \(\psi_2\) denote coefficients that represent the feedback of these user deltas onto the ratio of primary outlays to nominal GDP in percentage points. The feedback onto the primary outlays ratio accumulates according to

\begin{multline}
   \mathrm{rgfop}^{*U}(t) - \mathrm{rgfop}^{*B}(t) = [\mathrm{rgfop}^{*U}(t-1) - \mathrm{rgfop}^{*B}(t-1)] \\
   + \psi_1[lf^{*U}(t) - lf^{*U}(t)] + \psi_2[lq^{*U}(t) - lq^{*U}(t)] \tag{26}
\end{multline}

Based on CBO's modeling, we set \(\psi_1 = -0.13\) and \(\psi_2 = -0.23\).

As an example, a permanent increase in potential productivity growth (user delta) of 0.1 percentage point relative to baseline would lower the ratio of primary outlays to nominal GDP by \(0.1 \times (-0.23) = -0.023\) percentage point in the first year, and by an additional \(-0.023\) percentage point each year thereafter, so that after 10 years the ratio of primary outlays to nominal GDP would be 0.23 percentage point below its baseline value, all else equal. According to CBO, the decline in the primary outlays ratio to nominal GDP mostly reflects a higher trajectory for the latter, slightly offset by small increases (relative to baseline) in nominal primary outlays.

In percentage-point terms, the impact onto the primary outlays ratio from labor force growth is about one-half as large as the impact from productivity growth because of a smaller impact on GDP from the former than from the latter.

Feedback from growth and debt onto the real neutral interest rate

User assumptions that determine potential GDP growth and the evolution of government debt influence the real neutral interest rate, \(rf^*(t)\). Let lowercase d represent debt as a % of nominal GDP: \(d(t) \equiv 100*D(t)/\mathrm{GDP\$}(t)\). The term that appears below in the user adjustment to the real neutral funds rate, \([\mathrm{dhat}^U(t) - d^B(t)]\), is a model-based proxy for the change in the debt/GDP ratio (in \%) for the user scenario.8 With that notation, \(rf^*\) is adjusted to reflect assumptions that determine potential growth and debt/GDP according to the following expression.

\begin{multline}
   rf^{*U}(t) = rf^{*B}(t) + \kappa_1[lf^{*U}(t) - lf^{*B}(t)] + \kappa_2[lg^{*U}(t) - lg^{*B}(t)] \\ + \kappa_3[\mathrm{dhat}^U(t) - d^B(t)]. \tag{27}
\end{multline}

The parameters \(\kappa_1\), \(\kappa_2\) and \(\kappa_3\) are set to 0.667, 0.667 and 0.02, respectively. Those values imply that the neutral federal funds rate will adjust up or down by two-thirds of the sum of user deltas for potential growth of the labor force and labor productivity, and by 2 basis points for each percentage point of debt as a % of GDP implied by the user's assumption that determines the path of the primary budget balance ratio, rbudp\(^{*u}\).

Adjustments are made to \(\mathrm{r10bar}^U\) to reflect user deltas for potential growth of real GDP and for changes in the debt ratio that result from user deltas that influence potential growth of nominal GDP. The same parameter values for \(\kappa_{1,}\kappa_2\) and \(\kappa_3\) are employed to update \(\mathrm{r10bar}^U\).

The fiscal block includes expressions that govern the evolution of government debt, total and primary budget balances, net interest, and the average effective interest rate on publicly held federal debt. Debt, budget balances, and net interest are expressed in current dollars and as percentages of nominal GDP.

The average effective interest rate on publicly held federal debt responds gradually to market interest rates, recognizing that a fraction of government debt matures in a typical year and is refinanced or is newly issued.

\begin{equation}
   \mathrm{RG}(t) = \delta_1\,\mathrm{RG}(t-1) + (1-\delta_1)[\delta_2\,\mathrm{RF}(t) + (1-\delta_2)\,\mathrm{R10}(t)]. \tag{28}
\end{equation}

We set \(\delta_1\) at 0.8333 (5/6), implying that approximately one-sixth of the existing stock of debt is either newly issued or refinanced each year.

Net interest payments by the federal government are at the average effective interest rate applied to average level of publicly held debt during the year, proxied by the average of its beginning- and end-of-year levels

\begin{equation}
   \mathrm{NI}(t) = 0.5*(\mathrm{D}(t) + \mathrm{D}(t-1))*(\mathrm{RG}(t)/100). \tag{29}
\end{equation}

The level of primary budget surplus or deficit, BUDP(t) can be computed by unwinding the identity for the ratio to potential nominal GDP, as in

\begin{equation}
   \mathrm{BUDP}(t) = (\mathrm{rbudp}^*(t)/100)*\mathrm{GDP\$}^*(t). \tag{30}
\end{equation}

Nominal receipts and nominal primary outlays can be uncovered by unwinding the relevant identities as shown in the next two expressions.

\begin{equation}
   \mathrm{GFR}(t) = (\mathrm{rgfr}^*(t)/100)*\mathrm{GDP\$}^*(t). \tag{31}
\end{equation}

\begin{equation}
   \mathrm{GFOP}(t) = (\mathrm{rgfop}^*(t)/100)*\mathrm{GDP\$}^*(t). \tag{32}
\end{equation}

The total budget balance is denoted by BUD(t).

\begin{equation}
   \mathrm{BUD}(t) = \mathrm{BUDP}(t) - \mathrm{NI}(t). \tag{33}
\end{equation}

Debt accumulates according to the total budget surplus or deficit,9

\begin{equation}
   \mathrm{D}(t) = \mathrm{D}(t-1) - \mathrm{BUD}(t). \tag{34}
\end{equation}

In a post-simultaneous block, various budget ratios to actual nominal GDP are computed, such as \(\mathrm{rbudp}(t) \equiv 100\,\mathrm{BUDP}(t)/\mathrm{GDP\$}(t)\).