2.1 The municipal bond market

State and local governments in the USA rely heavily on debt markets to finance a wide range of activities, from covering budget shortfalls to infrastructure investment. When a government decides to raise funds through a debt issue, it issues municipal bonds through a financial broker, one of a number of bidders for the rights to issue the bonds. At the time of issue, the broker sells the bonds on a “primary market” investors and other broker-dealers.

After the primary market sale, municipal bonds are traded in OTC markets, rather than on a central exchange. The OTC nature of the municipal secondary market requires a specific buyer–seller match for a sale to take place. Harris and Piwowar (Reference Harris and Piwowar2006) and Green et al. (Reference Green, Hollifield and Schürhoff2007) show that buyer characteristics, namely size, influence prices on the secondary market, such that there is price dispersion even on the same day for a given bond; some investors may have better information and market power than others. Municipal bonds are not like treasuries in that they are tax-free and not entirely risk-free; additionally, the OTC market induces and amount of transaction costs into this market. The average municipal bond is traded every 10 days, suggesting that price adjustment may be slower in this market than in other markets.

There is a wide dispersion on the yields of municipal bonds, reflecting a high degree of heterogeneity in municipal borrowing costs. Municipal bond spreads—determined by tax advantage, risk, and liquidity—also vary substantially over time. Fig. 1 shows the evolution of the 10th, 50th, and 90th percentiles of spreads on municipal bond transactions at the daily frequency from 2005 to 2019.Footnote ⁴ Both the average level and disperison of municipal bond spreads are elevated during the financial crisis and subsequent years and vary at a higher frequency than local marginal tax rates, suggesting a meaningful role for risk and liquidity.

Figure 1. The distribution of muni spreads changes over time.

Note: Municipal bond spreads are reported in percentage points. Colored lines plot the indicated daily moments for all MSRB municipal bond trades in my cleaned sample; black lines are 1-month rolling averages of daily moments.

The increased variance of spreads during a time of monetary expansion also suggests that monetary policy may have important heterogeneous effects on municipal yields. Indeed, in this paper I show that there is also a heterogeneity over the degree to which muni prices respond to monetary shocks. This heterogeneity in response implies that different types of governments not only have varying borrowing costs over the long run, but will also encounter differing changes in borrowing costs after short-run shocks. As a result, the transmission of shocks through municipal borrowing costs should be expected to have both level and distributional effects, as governments are affected differentially by shocks, depending on factors such as trading costs and liquidity.

Finally, it is important to note that municipal bond yields do indeed have an effect on the behavior of state and local governments. The key result from the literature on this count is found in Adelino et al. (Reference Adelino, Cunha and Ferreira2017), who use exogenous variation in a municipal bond rating recalibration by Moody’s to identify “windfall” spending by local governments. Local governments whose bonds were upgraded (and therefore experienced lower yields) increased their spending in subsequent years by two to ten percent relative to local governments that did not experience a ratings upgrade. Appendix D.2 provides further supporting evidence for the effect of market muni yields on real government behavior. That market yields reflect real costs for state and local governments which incentivize public finance decisions is key to the logic of the monetary policy channel highlighted in this paper.

2.2 Time series evidence

2.2.1 Strategy

The main results of the paper are time series estimates of the response of municipal yield indices to monetary shocks. The exercise is in the spirit of Rosa (Reference Rosa2014), who studies the effect of monetary shocks on indices of AAA and AA bonds exclusively. I expand on Rosa’s work by considering indices representing a broader set of munis, as well as specific geographic and sectoral indices, reminiscent of Gilchrist et al. (Reference Gilchrist, Yue and Zakrajsek2019), who study the effects of US interest rate shocks on the sovereign bond yields of several small open economies. While my results for highly rated bonds are similar to what Rosa finds, my other results provide a fuller picture of the effect of monetary shocks on the municipal bond market, especially with regard to potential heterogeneity.

Estimation equation. For each of the indices in question, I am interested in estimating the equation

(1) \begin{align} \Delta y_t = \beta _0 + \beta _1m_t + X_t \varepsilon _t, \end{align}

where $m_t$ is the Bu et al. (Reference Bu, Rogers and Wu2021) monetary shock series and $\Delta y_t = y_{t+1} - y_{t-1}$ is the 2-day change in the yield to maturity of the asset around the FOMC meeting date.Footnote ^{5 ,} Footnote ⁶ The coefficient $\hat{\beta _1}$ reflects the number of basis points the muni index yield should increase for every 1 bp monetary shock and is estimated with heteroskedasticity-robust standard errors.

Municipal bond indices. The dependent variable in all specifications of (1) is a yield on one of several muni indices constructed by S&P.Footnote ⁷ S&P constructs these indices using a broad selection of bonds issued by state, local, and regional government entities in the USA, which are not subject to income taxes. The bonds must have been issued in 2010 or later and must have a minimum of 2 million US dollars par value on the market. The indices are constructed as value weighted averages of the constituent bonds.Footnote ⁸

Monetary Shocks. In this analysis, I use the strategy of Bu et al. (Reference Bu, Rogers and Wu2021) to identify monetary shocks on FOMC announcement dates, the full details of which are in Appendix C.2. The BRW method uses the movements of prices of zero-coupon US treasury bonds with maturities $i \in \{1,2,\ldots, 30\}$ on FOMC announcement dates to back out the implied monetary shocks on each date. This is accomplished in a Fama-Macbeth-style (Fama and MacBeth (Reference Fama and MacBeth1973)) procedure, which starts by making a standardizing assumption that defines the monetary shock as having a one-to-one effect on the 10-year treasury yield. The procedure then estimates the time series relationship between each of the zero-coupon bonds and the 10-year treasury yield. To back out the implied shock $m_t$ on each FOMC date, it then regresses the yields of the zero-coupon bonds on the “loadings” estimated in the first step on each FOMC date.

This monetary shock series has a number of desirable characteristics. First, it relies completely on publicly available data; the treasury yield is taken from the Federal Reserve website, and the estimated zero-coupon yields as estimated by Gürkaynak et al. (Reference Gürkaynak, Sack and Wright2006) are found at https://www.federalreserve.gov/pubs/feds/2006/200628/200628abs.html. The publicly available nature of the data allows the shocks to be constructed at no cost, and the series is quite easy to update through the current date. Furthermore, the authors argue in the paper that this series is robust to the information critique of Nakamura and Steinsson (Reference Nakamura and Steinsson2018), but is nevertheless highly correlated with existing estimates of monetary policy shocks. Finally, these shocks are able to incorporate well the unconventional nature of monetary policy in the aftermath of the financial crisis; this is an especially important feature for this paper, as my sample only begins in 2005.

2.2.2 Baseline results

This section summarizes the average response of municipal bond yields to monetary shocks. Table 1 gives the estimated coefficients of monetary shocks on yields for three indices of GO municipal bonds. These indices group all municipals, state governments, and local governments, respectively, with an index for bonds from S&P 500 firms for comparison.Footnote ⁹ I choose to focus on GO bonds, which are backed by the full taxing power of the issuer rather than a specific revenue source, in order to more closely match the budget situation of the government in the model. These bonds can be reasonably thought of as representing the borrowing cost situations of their local governments.

Table 1. Baseline time series results

Note: An observation corresponds to 1 day, around which a window is constructed from the previous day’s price and the price at a given horizon. Each column refers to a time series regression of a municipal bond index on monetary shocks. Heteroskedasticity-robust standard errors are reported in parentheses.

This exercise closely mirrors that of Rosa (Reference Rosa2014), though my results exhibit slightly higher responses to monetary shocks than his due to the inclusion of a wider set of bonds and the shock corresponding to the 10-year rate. Nevertheless, the coefficients are quite dampened relative to models in which governments can borrow at the risk-free rate: municipal bond yields only increase (decrease) by 26 basis points in response to a 100 point change in the risk free rate, which is less than half of the response of corporate bonds, and far less than treasuries. This dampened response cannot be fully explained by the tax-free nature of municipals, and therefore must be composed of illiquidity and/or risk effects. Models that do not take this dampened response into account will tend to overestimate the effects of monetary policy on local fiscal policy.

Interestingly, there does not seem to be much difference, on average, between the responses of state bonds and local bonds to monetary shocks.Footnote ¹⁰ This result is somewhat surprising, given the quite different tax and spending obligations between these two types of governments. Instead, it seems heterogeneity shows up in other ways, which I show in the next sections.

2.2.3 Heterogeneous responses

By state. A natural place to look for heterogeneity across localities is in the presence of geographic variation. For this section, I estimate (1) separately for indices of GO bonds originating in US states, for which these indices exist.Footnote ¹¹ Fig. 2 summarizes the estimates, highlighting visually the heterogeneity of responses across the USA.

Table 2. Time series results by S&P rating

Note: An observation corresponds to 1 day, around which a window is constructed from the previous day’s price and the price at a given horizon. Each column refers to a time series regression of a municipal bond index on monetary shocks. Heteroskedasticity-robust standard errors are reported in parentheses.

Figure 2. Responses of muni yields to monetary shocks.

Note: Darker color indicates a stronger response of municipal yields issued in the state to monetary shocks. Missing states are those for which an index of GO bonds is not available. Maximum value: 43.85 (KS); minimum value: 22.19 (NY).

A few notes on these results are worth highlighting. First, the range of coefficients runs from 22.19 for New York to 43.85 for Kansas.Footnote ¹² Second, no immediately obvious patterns emerge, save the apparent tendency for high-population areas to have muni bonds which response more weakly to monetary shocks; this result appears again in Section 2.3.2. In any case, the results suggest that heterogeneity exists across space in the USA in the response of municipal bond yields to monetary shocks, and monetary policy may affect different areas of the USA differently.

By credit rating. I also investigate heterogeneity in yield responses for muni indices broken out by credit rating. Lower rated and unrated bonds seem to respond more strongly to monetary shocks than highly rated bonds, providing an explanation for why the magnitudes in this paper might differ somewhat from Rosa (Reference Rosa2014). There does not seem to be much explanatory power in examining differences in municipal bonds broken out by sector.

Table 2 shows the coefficient estimates of (1) run separately for indices of bonds in various S&P rating categories, at the 2-day horizon. Although some of the coefficient estimates are not statistically significant, there is a clear upward trajectory, that is, the coefficients on riskier bonds are higher. This is consistent with a world in which expansionary monetary policy–for example–lowers default risk for risky bonds.Footnote ¹³ Furthermore, the coefficients on AAA and AA bonds are more consistent with the findings of Rosa (Reference Rosa2014), who only looks at low-risk indices. My baseline estimates, therefore, are higher than his in part because the GO index is not made up entirely of AAA and AA bonds.

This section documented the presence of heterogeneity in municipal bond responses to monetary shocks, which are dampened on average relative to US treasuries and corporate bonds. The type of government does not seem to make much difference, though smaller and riskier governments have greater responses to monetary shocks.Footnote ¹⁴ In the next section, I move from time series to panel data in order to obtain more evidence on the potential sources of this heterogeneity.

2.3 Panel evidence: MSRB data

Monetary shocks have a dampened effect on municipal bonds, with heterogeneity across different types of bonds. There is evidence that risk drives a portion of this, but other contributing factors may also be at play. Namely, information about the public finances of specific local governments may influence the responses of munis to monetary shocks. Knowing that there is heterogeneity in these responses is important for the quantitative exercise below, but knowing the reasons for this heterogeneity may be necessary to begin drawing out policy implications from the model.

To investigate more fully the effects of public finances in heterogeneity in monetary passthrough, I use a transaction-level dataset of municipal bond trades from the Municipal Securities Rulemaking Board, hereafter MSRB. These data are available through Wharton Research Data Services and are available from 2005 onward. I end the sample on December 31, 2019 to avoid entanglements with the tumultuous nature of the muni market in early 2020. I restrict the sample to GO bonds issued by general governments (as defined by Bloomberg). Appendix C.1 provides more details on the dataset construction.

In this section I estimate the equation

(2) \begin{align} \Delta y_{it} = \beta _0 + \beta _1m_t + \Gamma _0X_{it} + \Gamma _1X_{it}m_t + \varepsilon _{it}, \end{align}

where $m_t$ is the same monetary shock as before and $X_{it}$ reflect bond-specific characteristics that might influence a bond’s response to monetary shocks. Here, I allow a longer adjustment period for yields (and spreads) $y_{it}$ , owing to the sparse nature of municipal bond trades. My baseline time period of adjustment is two weeks;Footnote ¹⁵ furthermore, I assign a bond’s yield as its most recent daily yield, provided the trade happened within the last week. Standard errors are clustered at the time level, owing to the grouped nature of the shock $m_t$ .

Table 3. Baseline panel estimates

Note: An observation corresponds to an FOMC date-bond pair from the cleaned MSRB municipal bond sample. Each column refers to a separate regression. Standard errors are reported in parentheses, and are clustered at the date level.

2.3.2 Public finance variables

The key margin of heterogeneity I investigate involves a series of finance variables for municipalities. To obtain these data, I use the Census/Survey of Governments data from the US Census. This survey obtains hundreds of balance sheet variables for state and local governments in the USA, taking a representative sample annually and a full population sample every 5 years. Following Schwert (Reference Schwert2017), I select the local governments in the USA with annual revenues of over $50 million. I then obtain the 6-digit CUSIP codes for these government issuers from the Bloomberg Terminal and connect them to my panel dataset.

Table 4. Panel estimates: interactions with public finance variables

Note: An observation corresponds to an FOMC date-bond pair from the cleaned MSRB municipal bond sample, merged with Census of Governments public finance data. Each row corresponds to a regression with the specified variable interacted with monetary shocks, as in (2). The reported coefficient is the coefficient on the interaction term. Standard errors are reported in parentheses, and are clustered at the date level. $N = 9225$ for all regressions.

I estimate a series of regressions for (2) using numerous different public finance indicators as independent variables in separate regressions to explain municipal yield responses. Results are given in Table 4, from which a consistent theme emerges: larger governments are associated with less responsive municipal bond yields. Whether on the margin of revenues, expenditures, or debt, the estimated coefficients are negative across the board, and many are statistically significant. It is unclear why government size correlates with risk; one possibility is that larger governments are seen as having a more reliable tax base, resulting in better credit ratings and lower responses to monetary shocks. In any event, the association with size does seem to correspond with the state-level time series heterogeneity.

2.4 Empirical takeaways

This empirical section has studied, from a number of angles, the effect of monetary shocks on municipal bond markets. On average, the yields on an index of GO municipal bonds respond by 26 basis points to a 100 basis point shock to the risk-free rate. This dampened effect is consistent with the lower volatility of municipal yields in general relative to treasuries. Furthermore, while the “level” of the issuer does not seem to matter, the location of the issuer does. In other words, I document heterogeneity across the USA in the response of municipal yields to monetary shocks. This heterogeneity maps to the model from Section 3.

I then investigate potential sources of this heterogeneity. There is some limited evidence that the heterogeneity may arise from differences in bond ratings.Footnote ¹⁶ There may be an association as well between the size of a locality and its borrowing costs’ response to monetary shocks, but the data on government finances is limited, especially for smaller governments. In the main quantitative results below, I do not take a stance on the source of heterogeneous responses, but the various options may carry different policy implications.

3.1 Environment

The locality is modeled as a small open economy with a representative household and representative government, each of which maximizes the utility of the representative consumer. The household is made up of $1 - \kappa$ traditional consumers and $\kappa$ “hand to mouth,” or HTM, consumers. The traditional, or “Ricardian,” optimizers in the household choose a consumption bundle comprised of two types of final goods, tradable ( $c^T$ ) and non-tradable ( $c^N$ ), as well as labor supply $h$ and debt $d$ . The government chooses government purchases of non-tradable goods $g$ and municipal debt $d^G$ , given a tax rate $\tau ^G$ on the exogenous stream of tradable goods for the economy, $y^T$ . HTM consumers simply choose labor $h^H$ and consume exactly their labor income in every period.

Tradable goods and bonds of both agents are traded with the rest of the world, where the locality is endowed with an exogenous income of the tradable good in every period. Non-tradable goods are produced using domestically supplied labor by monopolistically competitive firms within the region to satisfy demand for non-tradable consumption and government purchases of public goods. Inflation is induced by Calvo-style price setting on the part of these monopolistic competitors, who maximize expected future profits subject to household and government demand, as well as the expected constraints on price changes.

The aggregate risk-free interest rate in the economy is determined by a central authority and is exogenous with respect to local variables. Additionally, both the household and the government are subject to their own “proprietary” interest rates, $r^H$ and $r^G$ , which depend on the aggregate rate, deviation of debt from steady state, and parameters determining the relationship between monetary shocks and the actual interest rate paid by either the household or the government. Of particular interest in this project is the response of the government’s borrowing costs to monetary shocks. The strength of this response will affect passthrough of interest rate shocks to households and presents a potential source of heterogeneity of monetary passthrough in the US economy.

3.2 Household

3.2.1 Basic problem

The household in the small economy is made up of $1 - \kappa$ “Ricardian” agents and $\kappa$ “hand-to-mouth” agents. The representative Ricardian optimizer solves

(3) \begin{align} \max _{\left\{c_t^T,c_t^N,h_t,d_{t+1}\right\}} \mathbb{E}_0\sum _{t=0}^{\infty }\beta ^t\left [U(c_t) - V(h_t) + W(g_t) \right ], \end{align}

where

(4) \begin{align} c_t = A\left(c_t^T,c_t^N\right) \end{align}

(5) \begin{align} c_t^T + p_tc_t^N + d_t = y_t^T\left(1 - \tau ^G\right) + w_th_t + \frac{d_{t+1}}{1+r_t^H} + T_t. \end{align}

Here, tradable consumption and debt are denominated in terms of the “national” price, which is normalized to unity. Prices $p_t$ and $w_t$ are prices—of non-tradables and labor, respectively—relative to the price of the tradable good. $\tau ^G$ is the tax on exogenous tradable good allocation, and $T_t$ is the lump-sum transfers to households from firm profits. Quantities are in per-person terms, such that total labor supply from optimizers is given by $(1 - \kappa )h_t$ .

3.2.2 Optimality conditions

Accordingly, the household’s first-order conditions are given as follows:

(6) \begin{align} \lambda _t = U^{\prime}(c_t)A_1\left(c_t^T,c_t^N\right) \end{align}

(7) \begin{align} \frac{A_2\left(c_t^T,c_t^N\right)}{A_1\left(c_t^T,c_t^N\right)} = p_t \end{align}

(8) \begin{align} \frac{\lambda _t}{1+r_t^H} = \beta \mathbb{E}_t\lambda _{t+1} \end{align}

(9) \begin{align} V^{\prime}(h_t) = \lambda _tw_t. \end{align}

The equation in (6) is the first-order condition for tradable goods consumption, defining the shadow value of income denominated in the tradable goods price. The condition for non-tradable consumption, when plugged into (6), yields (7), which allocates consumption according to the relative price of the two goods. (8) trades off the benefits of borrowing today with the costs of paying it back tomorrow, and (9) equates marginal costs and benefits of labor supply.

3.2.3 Hand-to-mouth agents

The remaining $\kappa$ agents in the economy behave in a hand-to-mouth manner. These agents supply per-capita labor $h_t^H$ and use labor income to consume tradable and non-tradable goods:

(10) \begin{align} c_t^{T,H} + p_tc_t^{N,H} = w_th_t^H. \end{align}

Here, as above, consumption is aggregated according to $c_t^H = A\left(c_t^{T,H},c_t^{N,H}\right)$ .

The first-order conditions for these consumers mirror those of the traditional agents, without the intertemporal condition:

(11) \begin{align} \lambda _t^H = U^{\prime}\left(c_t^H\right)A_1\left(c_t^{T,H},c_t^{N,H}\right) \end{align}

(12) \begin{align} \frac{A_2\left(c_t^{T,H},c_t^{N,H}\right)}{A_1\left(c_t^{T,H},c_t^{N,H}\right)} = p_t \end{align}

(13) \begin{align} V^{\prime}\left(h_t^H\right) = \lambda _t^Hw_t. \end{align}

The total amount of non-tradable consumption in the economy, then, is the sum $(1 - \kappa )c_t^N + \kappa c_t^{N,H}$ , and likewise with tradable consumption $(1 - \kappa )c_t^T + \kappa c_t^{T,H}$ and total labor $(1 - \kappa )h_t + \kappa h_t^H$ . This departure from Ricardian equivalence in the model allows a realistic local government spending multiplier to be obtained.

3.3 Local government

The representative local government uses local fiscal policy to solve the problem (3)Footnote ¹⁸ by choosing a borrowing level $d_{t+1}^G$ subject to

(14) \begin{align} p_tg_t = \tau ^Gy_t^T + \frac{d_{t+1}^G}{1 + r_t^G} - d_t^G, \end{align}

Other than its choice variables,Footnote ¹⁹ another key difference emerges for the local government: it exerts full market power over its own debt when it issues in the financial market. In other words, while the representative household is a price taker with respect to borrowing costs because it is subject to the aggregate interest rate, the local government can be thought of as a singular agent and the only issuer of its asset. The government, therefore, must take into account the effect of its debt issues on its borrowing costs, both in the current period and in the future. The first-order condition with respect to debt purchases, then, is given by:

(15) \begin{align} W^{\prime}\left(g_t\right)\frac{1 + r_t^G - d_{t+1}^G\frac{\partial r_t^G}{\partial d_{t+1}^G}}{\left(1 + r_t^G\right)^2} = \beta \mathbb{E}W^{\prime}(g_{t+1}) \left( -1 + \frac{ - d_{t+2}^G\frac{\partial r_{t+1}^G}{\partial d_{t+1}^G}}{\left(1 + r_{t+1}^G\right)^2} \right ). \end{align}

(15) is analogous to the household Euler equation for consumption and debt but uses public goods and local government debt. It captures the utility trade-off of public goods in period $t$ for public goods in period $t+1$ , taking into account the microeconomic effect of local government debt on prices. When the yields on municipal bonds move strongly with interest rate shocks, these fiscal policy responses will tend to be greater, while the opposite is true when responses of yields to monetary shocks are weak.

The condition that the government take into account its effect on interest rates is not an inconsequential one. It affects the response of debt to transitory shocks, but it also has an effect on the steady state of the model. If the government’s debt level did not affect its borrowing costs, it would run up unrealistic levels of debt at normal interest rates. Failing to account for the effects of debt on borrowing costs unrealistically understates the market’s response to high municipal debt levels.

3.4 Financial sector

Household and government debt are traded on an external financial market, resulting in two interest rates $r_t^H$ and $r_t^G$ . These interest rates reflect the aggregate interest rate $r_t^*$ , debt stock/purchases $d_t$ , $d_{t+1}$ , $d_t^G$ , and $d_{t+1}^G$ , and monetary shocks. The interest rates can be thought of as being determined by the functions

(16) \begin{align} r_t^H = f^H\left(d_t,d_{t+1},r_t^*,m_t\right) \end{align}

and

(17) \begin{align} r_t^G = f^G\left(d_t^G,d_{t+1}^G,r_t^*,m_t\right). \end{align}

The function $f^G(m_t)$ is of particular interest in this paper, as it will be a key determinant of the passthrough of monetary policy to local governments. The response of municipal government borrowing costs could vary based on a multitude of factors, including trading costs and illiquidity in the muni market, as I will show in the empirical section.

These pricing functions could arise from a number of financial market specifications. For example, a common formulation in the international literature is the debt-elastic interest rates of Schmitt-Grohe and Uribe (Reference Schmitt-Grohe and Uribe2003). In Appendix B, I show one alternative to this formulation, which incorporates a standard OTC asset market model. That feature serves as a microfoundation for the functions $f^H(m_t)$ and $f^G(m_t)$ and provides intuition into the mechanism working behind the empirical investigations. Given that a number of factors may give rise to heterogeneity over these functions, it is helpful to keep them general.

3.5 New Keynesian production

3.5.1 Final goods production

The non-tradable good $y_t^N$ is produced by a final goods producer which buys intermediate goods $y_{it}^N$ from a continuum of intermediate goods producers in the local economy. Production of the final good from intermediates is determined by the aggregating equation

(18) \begin{align} y_t^N = \left ( \int _0^1 \left(y_{it}^N\right)^{1 - \frac{1}{\mu }} di\right )^\frac{1}{1 - \frac{1}{\mu }}, \end{align}

and final goods firm profits are given by $ P_t^Ny_t^N - \int _0^1P_{it}^Ny_{it}^Ndi .$ Profit maximization on the part of the final goods producer implies the demand equations for the monopolistically competitive intermediate goods producers $ y_{it}^N = y_t^N\bigg (\frac{P_{it}^N}{P_t^N} \bigg )^{-\mu }.$ Here, the domestically produced good is used both for consumption—by both types of agents—and government spending,

(19) \begin{align} y_t^N = (1-\kappa )c_t^N + \kappa c_t^{N,H} + g_t, \end{align}

so the relevant demand equations become

(20) \begin{align} y_{it}^N = ( (1-\kappa )c_t^N + \kappa c_t^{N,H} + g_t)\left (\frac{P_{it}^N}{P_t^N} \right )^{-\mu }. \end{align}

$P_t^N$ here is the price of final non-tradable goods, which is given by the aggregator $P_t^N = \Big ( \int _0^1 (P_{it}^N)^{1 - \mu } di \Big )^{\frac{1}{1-\mu }}$ .

3.5.2 Intermediate goods producers

Intermediate goods firms exist on the continuum $[0,1]$ and produce differentiate inputs to the final non-tradable good using household labor:

(21) \begin{align} y_{it}^N = h_{it}^{\alpha }, \quad \alpha \in (0,1]. \end{align}

The choice variable for these firms is the price for good $i$ , $P_{it}^N$ , which determines demand for the intermediate good as in (20). Profit for an individual intermediate goods firm is given by $P_{it}^Ny_{it}^N - (1 - \frac{1}{\mu })W_th_{it}$ , where $W_t$ is the raw wage and $(1 - \frac{1}{\mu })$ is a labor subsidy meant to offset the distortions from monopolistic competition. Prices are sticky according to a Calvo mechanism, that is, intermediate goods firms may only change prices in each period with probability $(1-\theta )$ . In Appendix A, I show that maximization of expected profits on the part of intermediate goods firms, since all price-adjusting firms choose the same price, result in choosing the flexible relative price $\tilde{p_t}^N$ to equate present value marginal costs and marginal revenues, $mr = mc$ , where

(22) \begin{align} mr_t = \frac{\mu - 1}{\mu }y_t^Np_t\left(\tilde{p_t}^N\right)^{1-\mu } + \beta \theta \mathbb{E}_t\frac{\lambda _{t+1}}{\lambda _t} \left(\frac{\tilde{p_t}^N}{\tilde{p_{t+1}}^N}\frac{1}{\pi _{t+1}^N}\right )^{1-\mu }mr_{t+1} \end{align}

and

(23) \begin{align} mc_t = \frac{1 - \frac{1}{\mu }}{\alpha }\left(y_t^N\right)^{\frac{1}{\alpha }}w_t\left(\tilde{p_t}^N\right)^{-\frac{\mu }{\alpha }} + \beta \theta \mathbb{E}_t\frac{\lambda _{t+1}}{\lambda _t}\left (\frac{\tilde{p_t}^N}{\tilde{p_{t+1}}^N}\frac{1}{\pi _{t+1}^N}\right )^{-\frac{\mu }{\alpha }}mc_{t+1}. \end{align}

I also show in Appendix A that inflation dynamics are given by

(24) \begin{align} 1 = \theta \left(\pi _t^N\right)^{\mu -1} + (1 - \theta )\left(\tilde{p_t}^N\right)^{1 - \mu }, \end{align}

where $\pi _t^N = \frac{P_t^N}{P_{t-1}^N}$ , and aggregate production is given by $y_t^N = s_t^{-\alpha }h_t^{\alpha }$ , where $s_t = \int _0^1 \Big ( \frac{P_{it}^N}{P_t^N} \Big )^{-\frac{\mu }{\alpha }}$ represents the amount of price dispersion in the intermediate goods sector that has a dampening effect on aggregate output. Price dispersion evolves according to

(25) \begin{align} 1 = \theta s_{t-1}\left(\pi _t^N\right)^{\mu /\alpha } + (1 - \theta )\left(\tilde{p_t}^N\right)^{- \mu /\alpha }, \end{align}

3.6 Aggregation

Thus far I have focused on the problem of a single locality. In a full version of the model, there are a large number of localities subscripted by $s \in \{1,\ldots, S\}$ . These economies exchange tradable goods and debt with each other, as well as with the rest of the world.Footnote ²⁰ The nationwide interest rate $r_t^*$ is set by the economy’s monetary authority in response to aggregate output and inflation, which are made up of local values:

(26) \begin{align} r_t^* = R\left(\left\{y_{st}^T\right\}_{s},\left\{y_{st}^N\right\}_{s},\left\{\pi _{st}\right\}_{s}\right). \end{align}

3.7 Equilibrium

A competitive equilibrium is a set of quantities $y_t^N$ , $c_t^T$ , $c_t^N$ , $h_t$ , $g_t$ , $d_t$ , $d_t^G$ , $\lambda _T$ , $\pi _t^N$ , $s_t$ , $mr_t$ , and $mc_t$ and prices $p_t$ , $w_t$ , $\tilde{p_t}^N$ , $r_t^H$ , and $r_t^G$ for each locality $s$ satisfying:

1. (i) The optimizing household problem is solved by equations (6), (7), (9), and (8)

2. (ii) Hand-to-mouth quantities satisfy equations (13), (11), and (12)

3. (iii) The government’s problem is solved by equation (15)

4. (iv) Marginal revenue and marginal cost are given by equations (22) and (23), where $mr = mc$

5. (v) Aggregate production satisfies $y_t^N = s_t^{-\alpha }((1-\kappa )h_t + \kappa h_t^H)^{\alpha }$

6. (vi) Inflation and price dispersion evolve according to equations (24) and (25)

7. (vii) Inflation is defined by $\pi _t^N = \frac{p_t}{p_{t-1}}$

8. (viii Market clearing in tradable goods implies $(1- \kappa )c_t^T + \kappa c_t^{T,H} + d_t = y_t^T + \frac{d_{t+1}}{1 + r_t^H}$

9. (ix) Interest rates satisfy equations (16) and (17)

10. (x) The risk free rate is set by the monetary authority according to equation (26)

given the exogenous processes $y_t^T$ and initial conditions $s_{-1}$ , $d_0$ , and $d_0^G$ .

3.8 Elasticities of interest

A few key elasticities are important for understanding the passthrough of monetary policy through municipal public finance and its potential heterogeneity using this model. First, we need to know the effect of borrowing costs on government spending. Results from the literatureFootnote ²¹ suggest these effects could be quite sizeable. Additionally, it is important to know the local government spending multiplier: a helpful review in Chodorow-Reich (Reference Chodorow-Reich2019) suggests a point estimate of 1.8, suggesting a meaningful role of fiscal policy at the local level.

The empirical section detailed in depth the effect of monetary shocks on municipal borrowing costs, which is the main component of this transmission channel. In the baseline, local government borrowing costs move by 26 bp in response to a 100 bp monetary movement. This dampened response will, in the context of this model, reduce the size of transmission relative to assuming local governments have access to the risk-free rate. Heterogeneity of this elasticity will also result in heterogeneity across localities. The following quantitative section confirms and quantifies these conclusions.

4.1 Calibration and solution

Before moving on to the quantitative results, it is necessary to discuss specifics on the calibration of the model for proper interpretation, which in this case is a US municipality as a small open economy. I first discuss functional form choices, then parameters which are taken from the literature or estimated. I also calibrate a set of parameters to match some average statistics on state and local spending, revenues, consumption, and debt, followed by an investigation into the effects of local openness on the local fiscal multiplier. Finally, I briefly discuss the solution method.

4.1.1 Functional forms

I use the model outlined above, opting for an ad hoc version of financial markets, in the vein of models with external debt-elastic interest rates in the open economy literature. I use a simpler formulation for simplicity and ease of mapping the empirical response coefficients. Furthermore, because multiple sources of heterogeneity in borrowing costs have been explored, this formulation allows an abstraction from which of these courses is most important. To begin, assume the exogenous tradable endowment $y^T$ reflects shocks in the aggregate economy. Then let the aggregate risk-free $r_t^*$ be determined by the system

(27) \begin{align} z_t = Bz_{t-1} + \varepsilon _t^z, \end{align}

where $z_t = \Big [\log \frac{y_t^T}{{y^T}^*} \ \log \frac{1 + r_t^*}{1 + r^*} \Big ]^{\prime}$ , and $y^{T^*}$ and $r^*$ are parameters reflecting steady state values. $\varepsilon _t^z = [\varepsilon _t^y \ m_t]^{\prime}$ reflect the exogenous shocks, with $m_t$ being our shock of interest. Household interest rates are given by a standard debt-elastic interest rate formulation,

(28) \begin{align} r_t^H = r_t^* + \phi ^H\left(\exp (d_t^H - \bar{d^H}\right) - 1). \end{align}

The benefit of this formulation is the ability to set any arbitrary steady state debt level; in the baseline calibration, I make the representative local household a saver, that is, $\bar{d^H} \lt 0$ .Footnote ²² I set the government’s borrowing costs in a similar fashion, but with a friction on the adjustment to the treasury rate:

(29) \begin{align} r_t^G = r^* + \theta ^G\left(r_t^* - r^*\right) + \phi ^G \left(\exp (d_t^G - \bar{d^G}) - 1\right). \end{align}

First, note the imperfect response of actual borrowing costs to the risk-free rate, governed by $\theta ^G$ , which will be the key parameter of interest capturing the response of muni yields to aggregate interest rate shocks.Footnote ²³

Utility over consumption in this model is CRRA, with log utility over leisure and public goods consumption:

\begin{align*} U(c) = \frac{c^{1-\sigma } -1 }{1 - \sigma };\quad V(h) = \Phi \log \left(\bar{h} - h\right);\quad W(g) = \gamma \log (g). \end{align*}

Furthermore, the consumption aggregator is CES, $A(c^T,c^N) = \Big ( A{(c^T)}^{1 - \frac{1}{\xi }} + (1 - A)(c^N)^{1 - \frac{1}{\xi }} \Big )^{\frac{1}{1 - \frac{1}{\xi }}},$ where $\xi$ determines the substitution elasticity across tradable and non-tradable consumption, and $A$ represents the “openness” of the economy, which will be a key factor in the size of the local fiscal multiplier.

4.1.2 Parameters

The parameters in the model are set using a combination of data and literature. Table 5 gives the basic parameters from the model which are standard in the literature. Nothing of extreme note is here, other than to note the decreasing returns to scale in intermediate production, which are consistent with the labor share of production in the USA.

Table 5. Fixed parameters

The exogeneous process for $y_t^T$ and $r_t^*$ is defined by the coefficient matrix $B$ , which amounts to a VAR process with a lag of one period. For the baseline results, I estimate the matrix $B$ as a bivariate VAR on the series $[\log(Y_t) \log(r_t)]^{\prime}$ , where $Y_t$ is real GDP and $r_t$ is a treasury rate, in this case, the USA 10-year. This estimation results in the (quarterly) coefficient matrix

\begin{align*} \hat{B} = \begin{bmatrix} 0.985 & -0.0004 \\ 0.012 & 0.96 \\ \end{bmatrix} . \end{align*}

The baseline elasticity of municipal borrowing costs to the exogenous component of the aggregate interest rate is taken directly from the main time series result for all GO bonds in the empirical section, $\theta ^G = 0.26$ . In the section studying the effects of heterogeneity in this elasticity, I examine the 10th, 50th, and 90th percentiles for $\theta ^G$ .

The remaining six parameters,Footnote ²⁴ ${y^T}^*$ , $A$ , $\tau ^G$ , $\bar{d^H}$ , $\bar{d^G}$ , $\phi ^G$ , and $\kappa$ , are calibrated to a set of moments that represent averages for state and local governments in the US economy, in addition to a selected point estimate from the literature on local spending multipliers. These targets and their values—approximated for simplicity—are given in Table 6.

At this point, a discussion is necessary on the exact interpretation of this small open economy. Is it a state government or a local government, or something else? The issue, of course, is that households in the USA are under the jurisdiction of multiple tiers of governments, each of which exerts its own sphere of responsibility. Why are state and local expenditures being used for calibration? A robust literature exists describing the determination of public policy in federalist systems, but it is not the goal of this paper to enter in to that exciting discussion. For now, it suffices to say that the “government” imagined in this model is some sufficiently small combination of government roles that can be thought to be representative of its constituents’ value functions. In the baseline calibration, I set a tradable consumption to total consumption ratio of 0.66, matching the proportion of shipments in the Commodity Flow Survey that travel further than 50 miles. This yields a value for $A$ of 0.2683.

Table 6. Calibration targets

The degree of openness, given in this paper by the parameter $A$ , can also be thought of as defining the “size” of the locality in question. As the locality’s area increases, a higher proportion of household consumption is produced within the locality; for example, a good produced in San Francisco but consumed in Oakland is considered an import if the locality is defined by the Oakland city limits, but as a domestic good if we define the locality as the state of California or the more nebulous “Bay Area.” The impact multiplier of fiscal policy depends crucially on the definition of a locality, or its openness. Alternative definitions of a locality, such as a state, in which tradable consumption is less important, will result in larger multipliers. As the economy becomes increasingly closed, the multiplier increases, as we approach the closed economy case.

The multiplier, in addition to depending on openness, also depends crucially on the proportion of non-Ricardian agents in the household, $\kappa$ . These two determinants stand in agreement with the papers of Chodorow-Reich (Reference Chodorow-Reich2019) and Farhi and Werning (Reference Farhi and Werning2017a), which analyze fiscal multipliers in monetary unions. A resulting implication is that if openness $A$ decreases, fewer non-Ricardian agents will be required to match the preferred multiplier. Finally, this impact multiplier does not take into account the dynamic effects of local fiscal policy; the next section presents these effects in fuller detail.

4.2 Results of an interest rate shock

The response of this calibrated economy to a 25 bp expansionary monetary shock is shown in Figs. 3 and 4.Footnote ²⁷ The figure shows the percent deviations from the steady state in response to the monetary shock. Notice the logic of the channel shown in the second figure: borrowing costs decrease, increasing government debt and spending, resulting in output stimulus. In the baseline calibration, a 25 bp decrease in the risk-free rate results in about a quarter percent increase in output on impact.

Figure 3. IRFs, 25 bp expansionary shock.

Note: A time period corresponds to one quarter. IRFs plot the percentage response of the specified variable to an unforeseen 25 bp downward shock in the national risk-free interest rate (annualized).

Figure 4. IRFs, 25 bp expansionary shock.

Note: A time period corresponds to one quarter. IRFs plot the percentage response of the specified variable to an unforeseen 25 bp downward shock in the national risk-free interest rate (annualized).

Note, however, there is a long-run effect of the government debt buildup. Because the only margins of fiscal adjustment for the government are debt and spending—the tax rate is exogenous—the government has to reverse its debt accumulation through costly decreases in government spending later on. One immediate consequence of this result is the importance of thinking about the long-term consequences of stimulating debt-financed spending. Expansionary monetary policy allows local governments to shift spending from far in the future to the present, stimulating output in the short run but depressing it in the long run.

In the empirical section, I noted that the responses of municipal yields to monetary shocks were muted relative to what one might expect. When conditioned on trading activity, responses are higher but remain weaker than one-for-one. As a result, the stimulative or depressive effects of monetary policy through local public finance are lower in a model which takes these features of muni markets into account, relative to a model which assumes local governments can borrow at the risk-free rate. To see the magnitude of the difference between the calibrated model, a model with an alternate level of passthrough based on Table 3, and a model with borrowing costs at the risk-free rate, see Fig. 5.

Figure 5. IRFs compared to standard case.

Note: A time period corresponds to one quarter. IRFs plot the percentage response of the specified variable to an unforeseen 25 bp downward shock in the national risk-free interest rate (annualized). The alternate model sets the passthrough parameter equal to 0.68.

In the model that does not take the muted response of local borrowing costs to monetary shocks into account, the stimulative effects of monetary policy are more than double the baseline model and 40% higher than in the alternate data-based model. When local government borrowing costs are tied more closely to the risk-free rate, local government debt increases by almost five percent relative to the steady state, and the reaction of spending, output, and labor are much higher than before. This stark difference suggests that models which assume municipal governments have access to borrowing at the aggregate risk-free rate will significantly overstate the stimulative effects of monetary policy on local economies; such models will also understate the possibility for heterogeneity of stimulus across localities.Footnote ²⁸ I take up the extent of the heterogeneity in the next section.

In Appendix Figure 11, I plot an estimated response of output to a monetary shock according to the Christiano et al. (Reference Christiano, Eichenbaum and Evans1996) methodology mentioned earlier. The peak response of output to a one standard deviation monetary shock matches very well the magnitude of the peak response in the calibrated model, suggesting that the magnitudes of response in this model are reflective of the real world. Across the board, this type of “canonical” open economy DSGE model fails to generate the hump-shaped responses of economic variables to monetary shocks observed in VARs. One could imagine a number of features to supplement the model which might better match these hump shapes, but these features are not the focus of this paper.

4.3 Heterogeneity over monetary responses

Model results. A model that assumes a one-for-one relationship between municipal borrowing costs and national interest rates eliminates the possibility that borrowing costs might respond differently to monetary shocks in different localities. As a result, such a model will eliminate an important source of heterogeneity in the passthrough of monetary shocks across regions and localities in the USA. Above, I focus on one locality for ease of exposition; here, I consider a full model with a large number of localities.

In the data, I identified the municipal yield responses to monetary shocks of 41 US states. In this specification, then, I set $S = 41$ and assign each state a $\theta ^G$ corresponding to one of the empirical estimates, while all other parameters remain the same. Aggregate output is defined as total output, and aggregate inflation as average inflation of non-tradables across localities. The risk-free rate is set according to a Taylor rule which mimics the interest rate process above, replacing $y^T$ with total output and adding in a response of inflation, where the risk-free rate responds to a basis point of lagged inflation with a 1.5 basis point rate increase.

Figure 6. IRFs under muni heterogeneity.

Note: A time period corresponds to one quarter. IRFs plot the percentage response of the specified variable to an unforeseen 25 bp downward shock in the national risk-free interest rate (annualized). Lines correspond to the 10th, 50th, and 90th percentiles of responsiveness to an interest rate shock in the full multi-region economy.

In Fig. 6, I plot the same impulse response functions to a 25 bp risk-free rate shock as in the previous section, for the 10th, 50th, and 90th percentiles of $\theta ^G$ in the large economy.Footnote ²⁹ The municipal bond market is the only difference between these states, yet monetary transmission is markedly different between them. A government whose borrowing costs fall more after an expansionary shock borrows more, spends more, and sees greater output and labor increases on impact; the impact effects on output and labor are almost 25% greater in the 90th percentile than the 10th percentile state. In the long run, of course, these effects will be flipped: the high-response governments have more debt to pay down, and thus a bigger future recession. The magnitude of response of municipal bond prices to monetary shocks is a key parameter, then, in determining both the size and path of local economic outcomes.

Empirical results. Realistic differences in the response of local government borrowing costs to monetary shocks have quantitatively important implications for heterogeneity in monetary transmission. For example, Francis et al. (Reference Francis, Owyang and Sekhposyan2012) estimate the responses of several US cities to monetary shocks, grouping them according to region, and finding that the difference between the smallest regional peak employment response and the largest is about tenfold. While the differences in this experiment are smaller, a 25% difference in responses is an not inconsequential proportion of this estimate.

A key testable implication of this model is the positive effect of municipal bond elasticities (to monetary shocks) on monetary policy transmission. Specifically, the model predicts that an increased elasticity of municipal bond yields with respect to monetary shocks implies increased effects of monetary policy on employment. To investigate this in the data, I use the peak employment responses at the MSA level estimated by Francis et al. (Reference Francis, Owyang and Sekhposyan2012).Footnote ³⁰ I take the Census of Governments subsample mentioned in Section 2.3.2 and Appendix C.1, assign governments to MSAs where possible, and merge to the CUSIP-6 muni issuer codes where possible. After estimating (2) to obtain coefficients $\theta$ on monetary shocks separately for each issuer code, I then estimate the following equation for each issuer $i$ in MSA $j$ :

(30) \begin{align} \log Peak_{ij} = 1 + \beta _1\theta _i + \beta _2{\% Gov_j} + \beta _3{\% Gov_j}*\theta _i + \beta _4\log Density_j + \beta _5\log Income_j + \varepsilon _{ij} \end{align}

Here, $Peak_{ij}$ is the peak employment response to an expansionary monetary shock and $\theta _i$ is the estimated CUSIP6-level response of municipal bonds to monetary shocks. $\% Gov_j$ , $Density_j$ , and $Income_j$ are MSA-level explanatory variables from Francis et al. (Reference Francis, Owyang and Sekhposyan2012) corresponding to percent employed in the government sector, population density, and median income; $\% Gov_j$ and $Density_j$ especially emerge in their paper as significant determinants of local monetary policy transmission.

Encouragingly for the model, higher muni yield responses are associated with more monetary transmission to employment: the estimated effect $\hat{\beta _1} = 3.40$ with p-value $p = 0.08$ indicate a positive effect of muni responses on monetary transmission. Additionally, the size of the government sector may be associated with stronger monetary transmission (though decreasingly so for higher  $\theta$ , and not significantly); I show in Appendix E.4 that this is also a prediction of the model. While the data is limited, these simple results align with predictions of the quantitative model.

Why is this heterogeneity important? First, such differences in passthrough may affect the desirability of a given central bank policy in a monetary union. The results here suggest that, for example, the Federal Reserve should take into account heterogeneous effects on municipal governments across the USA when it is considering a policy.Footnote ³¹ To the extent that a given policy can strengthen the relationship between treasuries and munis, that policy can increase the short-term output effects of monetary policy on the economy as a whole. Additionally, since we have a calibrated model of a local government in a monetary union, in which the muni market plays an important role, we can explore how financial markets have affected local governments in past crises.Footnote ³²