A Framework for Fiscal Policy Coordination and Economic Stability: Countercyclical Transfer for Infrastructure

Abstract

This chapter examines the theory behind the design of fiscal transfers; it identifies acyclicality of transfers as a drawback and proposes countercyclical design as an optimal fiscal framework to mitigate cyclical economic fluctuations. It further advocates countercyclical transfers for infrastructure investment, service maintenance, and business tax relief as incentives for states to save in boom years then build from recession into recovery. The framework is set to operate on time-consistent policy rules, as automatic stabilizers with triggers from key economic indicators. It advocates dynamic equity to compensate boom-year donor states ­during recession for more effective macroeconomic stabilization. This chapter focuses on fiscal policy; it does not discuss the political dynamics or details in implementation.

Appendix: A Model of Countercyclical Infrastructure Investment

1.1 Taylor’s Model for Multi-period Construction Projects

John Taylor (1982) devised a model for multiple-period construction projects (hereafter “Taylor’s model”) that well suits my topic. Here I use the Taylor’s model to illustrate why and how a countercyclical stabilization program of infrastructure investment may work in theory.

As discussed in Section 4.1, better infrastructure leads to higher productivity. We can take total existing infrastructure as capital and the building of new infrastructure as accumulating capital stock. Infrastructure takes time to build, with construction duration varying from 1 to 3 years (census data on construction projects show that this is a reasonable assumption). Thus, we can establish a capital accumulation equation:

$$ {K}_{\rm{t}+\rm{n}}={K}_{\rm{t}+\rm{n}-1}+{S}_{\rm{t}}-h{K}_{t+n-1}, $$

(11.1)

where Kt is capital stock at the beginning of year t, S _t is the level of construction project starts in year t, and h is the rate of deterioration (or depreciation) of existing infrastructure.

Depreciation reduces the value, or capital stock, of infrastructure; deterioration requires either advance resource reserves for replacement or current year outlay for maintenance of existing structure and facilities. Infrastructure is expensive to build, so governments try to minimize the expected cost of building them; thus, we have a cost equation:

$$ {\displaystyle {\sum }_{\rm{t}=1}^{\infty }{b}^{\rm{t}+\rm{n}}}\left[\rm{d}/2{(v{Y}_{\rm{t}+\rm{n}}-{K}_{\rm{t}+\rm{n}})}^{2}+{c}_{\rm{t}+\rm{n}}{I}_{\rm{t}+\rm{n}}\right],$$

(11.2)

where b is a discount factor, Y _t is exogenous demand for infrastructure, I _t is infrastructure investment in year t, v is the desired or optimal infrastructure-GDP ratio, and c _t is cost of investment goods in year t.

Demands for infrastructure (Y _t), capital stock (K _t), investment (I _t), project starts (S _t), and cost (c _t) are all cyclical deviations from their secular growth trend; so their long-run averages are zeroes. If a project is completed within 1 year and thus its value is accounted for in the stock in the same year, then I _t  =  S _t. Demand for infrastructure is exogenously driven and determined by demographic and other factors that are external to the government. The infrastructure-GDP ratio (v) is assumed to be constant since we consider the cyclical (short-term) changes in the relative cost (c _t) of investment into infrastructure. The quadratic term in the cost function implies a U-shaped cost of having too little or too much infrastructure relative to the demand.

In order to minimize the cost, a level of project starts is chosen, as indicated by the project starts equation:

$$ {S}_{\rm{t}}=v{Y}_{\rm{t}+\rm{n}}-(1-h){K}_{\rm{t}+\rm{n}-1}-({b}^{-\rm{n}}/\rm{d}){\displaystyle {\sum }_{\rm{i}=0}^{\rm{n}-1}{b}^{\rm{i}}{\rm{w}}_{\rm{i}+1}\left({\rm{C}}_{\rm{t}+1}-b(1-h){\rm{C}}_{\rm{t}+\rm{i}-1}\right)}\rm,$$

(11.3)

where K _{t + n −1} is predetermined at time t and w_i is the fraction of expenditure on projects during the ith year of construction.

It is reasonable to assume I  =  1, 2, 3 because most projects are normally completed within 3 years. Project starts in year t are determined by the estimated demand n years later; estimates are done in the infrastructure improvement (capital ­budgeting) process and are given in the capital budget. This estimated number of project starts is also the discounted and expenditure-weighted sum of investment costs in the next n years. In other words, this equation discounts the expected gap in infrastructure n years into the future (which can be seen by multiplying the right-hand side by b ⁿ); the optimal level of starts is a weighted average of the infrastructure gap and the infrastructure outlay to close the gap.

In the process of federal infrastructure improvement planning or state capital budgeting, forecasts are (have to be) made on infrastructure demand nationwide or in each state. These are done with data from the past years; we can assume that it is done with an autoregressive model. The forecasting equation is

$$ \begin{array}{cc}{\widehat{Y}}_{\rm{t}+\rm{n}}={\alpha }^{\rm{n}}{Y}_{\rm{t}}, 0\le \rm\rm\alpha \rm\rm\le \rm\rm1\end{array}$$

(11.4)

Autoregressive forecasts are the best that federal and state governments can do about the unknown future, but such forecasts are not in line with shocks to demand from business cycle fluctuations. (Section 3.2, below, will dwell on this.) To better handle the drastic cyclical shocks, an infrastructure investment stabilization policy can be designed, with the following policy rule equation to calculate the expected future cost of infrastructure outlay:

$$ \begin{array}{cc}{\widehat{\rm{C}}}_{\rm{t}+\rm{i}}=\rm{g}{\alpha }^{\rm{i}}{\rm{Y}}_{\rm{t}}, i=1,2,\mathrm{....}\end{array}$$

(11.5)

where g is a positive policy parameter. As infrastructure demand fluctuates around the long-run trend, the effective price of investment goods moves in the same direction. Substituting Eqs. 11.4 and 11.5 into 11.3, the project starts equation, we obtain

$$ {S}_{\rm{t}}={A}_{\rm{n}}{Z}_{\rm{t}}+h{K}_{\rm{t}+\rm{n}-1},$$

(11.6)

where ${A}_{\rm{n}}=\upsilon {\alpha }^{\rm{n}}-({b}^{-\rm{n}}/\rm{d})g{\displaystyle {\sum }_{\rm{i}=0}^{\rm{n}-1}{\alpha }^{\rm{i}}{b}^{\rm{i}}{\rm{w}}_{\rm{i}+1}\left(1-\alpha b(1-h)\right)}$and ${z}_{\rm{t}}={Y}_{\rm{t}}-{Y}_{\rm{t}-1}$.

The first term (A _n z _t) is net investment, and the second is replacement of the portion of capital stock that depreciates between years. Net investment materializes through a linear accelerator mechanism. A is the accelerator coefficient; it is positively related to the desired infrastructure ratio, v, and negatively related to the depreciation rate, h. The faster the depreciation, the more expensive it is to maintain a given size of infrastructure stock.

The accelerator (A) and its response to the stabilization policy (g) both depend on n. Thus, the stabilization policy has different total and marginal effects for each value of the length of construction. Infrastructure in year t is now a distributed lag of starts over n years, with lag weight being weights for new value:

$$ \begin{array}{cc}{I}_{\rm{t}}={\rm{w}}_{1}{\rm{S}}_{\rm{t}}+{\rm{w}}_{2}{\rm{S}}_{\rm{t}-1}+\dots +{\rm{w}}_{\rm{n}}{\rm{S}}_{\rm{t}-\rm{n}+1}, {\rm{w}}_{1}+{\rm{w}}_{2}+\cdots +{\rm{w}}_{\rm{n}}=1,\end{array}$$

(11.7)

Therefore, the n-year infrastructure outlay equation is a distributed lag ­accelerator equation:

$$ \begin{array}{l}{I}_{\rm{t}}={A}_{\rm{n}}({w}_{1}{z}_{\rm{t}}+{w}_{2}{z}_{\rm{t}-1}+\dots +{w}_{\rm{n}}{z}_{\rm{t}-\rm{n}+1})+h({w}_{1}{K}_{\rm{t}+\rm{n}-1}+\dots +{w}_{\rm{n}}{K}_{\rm{t}})\\ \rm{ }\rm\rm={\rm{A}}_{\rm{n}}{\displaystyle {\sum }_{\rm{i}=0}^{\rm{n}-1}{\rm{w}}_{\rm{i}+1}{\rm{z}}_{\rm{t}-1}{\displaystyle {\sum }_{\rm{i}=1}^{\rm{n}}{\rm{w}}_{\rm{i}}{\rm{K}}_{\rm{t}+\rm{n}-\rm{i}}}}\end{array}$$

(11.8)

1.2 Two-State Regime Switching

Over 60 years ago the famous study by Burns and Mitchell (1946) told us that the evolution of the business cycle is nonlinear, with regime switching at turning points of the cycle. The Taylor model was devised in the early 1980s. Though the author saw the need for a two-state switching component, it was computationally “very complicated.” By the late 1980s advances in time series techniques provided the tools for the breakthrough. In this section I bring in a two-state regime switching component, developed by James Hamilton, to extend the Taylor model.

Incremental forecasts are the best that federal and state governments can do, but they are not in line with shocks to demand from business cycle fluctuations. In the autoregressive forecasting Eq., 11.4, infrastructure demand deviates around (above or below) the secular trend; the deviations are expected to return to the trend gradually over time. That is, assuming demand deviations are due from a simple first-order autoregressive process: Y _t  =  a ⁱ Y _{t - 1}  -  u _t, which is a gradual, continuous process, typical of normal years in a boom but not characteristic of the business cycle.

From the perspective of business cycles, infrastructure demands embed “turning points” that are transition from the expansionary phase (peak) into the contraction phase, and then from contraction (trough) back into expansion (Burns and Mitchell 1946). These turning points are regime switching from boom to bust and back to boom.⁸ Hamilton (1993) and Kim and Nelson (1999) provide the algorithm for such a model.

In his seminal paper, Hamilton (1989) proposed a tractable approach to modeling regime changes. He uses the parameters in an autoregression as the outcome of a discrete-state Markov process, which draws probabilistic inferences on the unobservable regime changes with a nonlinear iterative filter that he devised an algorithm for. Research prior to this paper assumed linear stationary process in the first difference of the log of key macroindicator series; Hamilton proposed that these series follow a nonlinear stationary process (Hamilton 1989, 357). The ­macroeconomy switches between a fast growth phase and a slow (even negative) growth phase of the cycle; the switch is governed by a Markov process (ibid, 362). The Kalman filter is a linear algorithm for continuous unobserved state vectors; the Markov filter that Hamilton devises provides nonlinear inferences which can well handle the phase switches of business cycles.

The Hamilton’s Markov model of trend is (different from Section 3.1, s here indicates state)

$$ \begin{array}{cc}{\rm{n}}_{\rm{t}}=\rm\rm{\alpha }_{1}{\rm{s}}_{\rm{t}}+\rm\rm{\alpha }_{0}+{\rm{n}}_{\rm{t}-1}, {\rm{S}}_{\rm{t}}=0\rm{or}1,\end{array}$$

(11.9)

The transition between states is governed by a stochastic first-order Markov process that is strictly stationary, with the following probability functions:

$$ Pr[{\rm{S}}_{\rm{t}}=1|{\rm{S}}_{\rm{t}-1}=1]=p,$$

$$ Pr[{\rm{S}}_{\rm{t}}=0|{\rm{S}}_{\rm{t}-1}=1]=1-p,$$

(11.10)

$$ Pr[{\rm{S}}_{\rm{t}}=0|{\rm{S}}_{\rm{t}-1}=0]=q,$$

$$ Pr[{\rm{S}}_{\rm{t}}=1|{\rm{S}}_{\rm{t}-1}=0]=1-p.$$

The stochastic stationary process follows an AR(1) representation:

$$ \begin{array}{cc}{\rm{S}}_{\rm{t}}=(1-\rm{q})+l{\rm{S}}_{\rm{t-}1}+{\rm{v}}_{\rm{t}}, l\equiv -1+\rm{p}+\rm{q,}\end{array} $$

(11.11)

where conditional on S _{t - 1}  =  1, V _t  =  (1  −  p), with probability p and V _t  =  −  p, with probability 1  −  p; where conditional on S _{t  −  1}  =  0, V _t  =  −  (1  −  q), with probability q, V _t  =  q, with probability 1  −  q. Equation 11.9 is a special case of a standard ARIMA model with normally distributed error terms (taken as “innovations”); V _t in Eq. 11.11 is uncorrelated with lagged values of S _t, such that

$$ \begin{array}{cc}E\left[{V}_{t}|{S}_{t-j}=1\right]=E\left[{V}_{t}|{S}_{t-j}=0\right]=0,\rm \rm{for}j=1,\rm2,\rm\dots \end{array}$$

(11.12)

Hamilton highlights two differences between Eq. 11.9 and a standard ARIMA. Under the Hamilton’s Markov model of trend, the first difference growth rate (n _t – n _t−1) changes as response to regime switches, but does not necessarily do so in every period. When this model is applied to a linear normal process, it generates a nonlinear process (Hamilton 1989, 362).

1.3 Model Integration

To link the two-state Markov regime switching with the identification of actual business cycles, Diebold and Rudebusch (1996) cleverly rewrite Eq. 11.10 as a transition probability matrix:

$$ \rm{M}=\left[\begin{array}{cc}{\rm{p}}_{00} 1-{\rm{p}}_{000}\\ 1-{\rm{p}}_{11} {\rm{p}}_{11}\end{array}\right],$$

(11.13)

and treat regime switching as different probabilistic objects. We can take ${\left\{{\rm{y}}_{\rm{t}}\right\}}_{\rm{t}=1}^{\rm{T}}$as the path of the observed time series. In our case, it is the macroeconomic indicator, GDP, or infrastructure investment. ${\left\{{\rm{y}}_{\rm{t}}\right\}}_{\rm{t}=1}^{\rm{T}}$depends on the state, ${\left\{{S}_{\rm{t}}\right\}}_{\rm{t}=1}^{\rm{T}}$. Thus, the density of y _t is conditional on s _t, as in

$$ f({\rm{y}}_{\rm{t}}{|}_{S}{}_{\rm{t}};\theta )=\frac{1}{\sqrt{2\pi \sigma }}\mathrm{exp}\left[\frac{-{({\rm{y}}_{\rm{t}}-\rm\rm\mu {\rm{s}}_{\rm{t}})}^{2}}{2{\sigma }^{2}}\right].$$

(11.14)

Based on this, they proposed a multivariate dynamic factor model with Markov regime switching:

$$ P({f}_{\rm{t}}|{h}_{\rm{t}};\theta )=\frac{1}{\sqrt{2\pi \rm{ }\sigma }}\mathrm{exp}\left[\frac{-{\left[({f}_{\rm{t}}-\rm\rm{\mu }_{{\rm{s}}_{\rm{t}}})-{\displaystyle {\sum }_{\rm{i}=1}^{\rm{p}}{\varphi }_{\rm{i}}({f}_{\rm{t-i}}-{\mu }_{{s}_{\rm{t-i}}})}\right]}^{2}}{2\rm{}{\sigma }^{2}}\right],$$

(11.15)

where ${\left\{{f}_{\rm{t}}\right\}}_{\rm{t}=1}^{\rm{T}}$is the sample path of our key factor (infrastructure or GDP). The conditional density summarizes the probabilistic dependence of f _t on h _t. The key factor moves around two means: μ ₀ is recession and μ₁ is boom. In a regression representation, it is

$$ \Delta {x}_{\rm{t}}=b+l{f}_{\rm{t}}+{\mu }_{\rm{t}},D(L){\mu }_{\rm{t}}={\epsilon }_{\rm{t}}. $$

(11.16)

The dimension of all elements is (N  ×  1) except $D(L)$which is (N  ×  N) and f which is (1  ×  1). Thus, econometricians have shown that a time-consistent stabilization policy can work in theory on infrastructure investment. My next step is to integrate infrastructure into a federal program and into state capital budgets; see Section 4.4 of the paper “Putting Infrastructure into Federal/State Programs.”