Firms

Firms produce tradable and nontradable goods (y_x and y_n, respectively) according to a Cobb-Douglas production technology combining public infrastructure—a constant elasticity of substitution (CES) function of standard infrastructure zⁱ and adaptation capital z^a with productivity v_a—with private capital k_j and labor l_j. Firms operate in one single sector and benefit from a sector-specific total factor productivity (TFP) A_j. The natural disaster affects both sectors, but causes disparate damages D_j ∈ [0, 1] to output by permanently destroying public and private capital and resulting in a transitory loss of productivity. The impact of the natural disaster can however be counteracted by public investments in adaptation activities that reduce the extent of the damages (with π_j and v_D being scaling factors).

In each sector j, a firm maximizes profits according to

where w is the economy-wide wage and r_j is the sectoral return on capital in sector j. Output is defined as

and public capital as

where ρ_z is a weight parameter and ξ is the intratemporal elasticity of substitution between public capital inputs (standard and adaptation). This functional form is particularly useful in discussing adaptation because it allows us to view it either as a substitute or a complement to standard infrastructure (e.g. seawalls vs. climate-proofing roads; see discussion in the calibration section). Note that damages can be decomposed into damages to public and private capital as well as losses of productivity according to $(1-D_{z_j})={(1-\frac{D_j}{{(1+{\pi }_j{z}^a)}^{\nu }D})}^{{\varpi }_z},(1-D_{k_j})={(1-\frac{D_j}{{(1+{\pi }_j{z}^a)}^{\nu }D})}^{{\varpi }_k}$, and $(1-D_{A_j})={(1-\frac{D_j}{{(1+{\pi }_j{z}^a)}^{{\nu }_D}})}^{1-{\varpi }_z{\psi }_j-{\varpi }_{k^{\alpha }j}}$, respectively. he severity of the damages to public and private infrastructures is controlled by ϖ_z and ϖ_k, respectively, such that the larger ϖ_z is, the greater public infrastructure was destroyed relative to private capital and TFP (with 1 —ϖ_zψ_j —ϖ_kα_j ≥ 0). The literature on natural disasters distinguishes damages from economic losses. We thus target the size of D_Aj as representing economic losses and D_zj. and D_kj as damages to physical infrastructure. The disruption in the functioning of public and private infrastructure has a persistent effect on the TFP because infrastructure takes some time to rebuild and workers may have spend to their time cleaning roads or rebuilding their houses, so aggregate productivity is assumed to recover slowly.¹

Profit-maximizing firms demand capital and labor according to

and²

Output elasticities with respect to total public capital can also be derived as

where $R_t^z$ is the total return to public infrastructure, which can be expressed in terms of the returns to standard and adaptation infrastructure as $R_t^{zi}{\left(\frac{z_{t-1}^i}{{\rho }_z{z}_{t-1}}\right)}^{1/\xi }$ and $R_t^{za}{\left(\frac{{\nu }_a^{1-\xi }{z}_{t-1}^a}{(1-{\rho }_z)z_{t-1}}\right)}^{1/\xi }$, respectively. Note that firms do not take into account the added benefit of adaptation (namely the protection against natural disasters) in their optimal decisions and only factor in the benefits of having additional infrastructure as an input in their production processes. Infrastructure prices are a function of the price of the imported machinery needed to produce capital, P_mm, and of the price of the a_j (j = k, zi, za) nontradable inputs needed to produce capital P_n. The price of private capital is given by P_k,t = P_mm,t+a_kP_n,t. We allow public standard and adaptation infrastructures to have different imported and nontradable input contents to reflect the idea that adaptation infrastructure requires more technical inputs provided externally and sold at a premium pr or that they demand more or less construction materials than the standard infrastructure. The prices of standard and adaptation infrastructure are thus given by P_zi,t = P_mm,t + a_ziP_n,t and P_za,t = (1 + pr) P_mm,t + a_zaP_n,t, respectively.

Government

Government expenditures are directed to investments in standard and adaptation infrastructure i_zi and i_za, transfers to households T, and servicing public domestic and external debt. The government also collects revenues from VAT on households’ consumption and from fees on Ricardian households’ usage of standard public infrastructure (as a fraction f of recurrent costs δ_ziP_zi0 involved in the maintenance of the infrastructure (i.e. μ =fδ_ziP_zi0)).³

As Dablas-Norris et al. (2012) shows, infrastructure stocks vary considerably in low-income countries despite high levels of investment. Therefore, the economy only benefits from effectively produced capital, which depends on public investment efficiency s ∈ [0,1]. The public investment process is equally bureaucratic and inefficient across the two capital inputs. To account for the increased inefficiencies that arise post-disaster, s is time-varying and affected by D_s at the time of disaster. In particular, the ability to transform each dollar spent into capital takes some time to recover, with expenses being lost in non-productive activities. Public capital therefore evolves according to

Note that these two types of infrastructure depreciate at different rates in order to account for their different resilience to natural disasters. Adaptation infrastructure is able to better withstand the effects of climate change and natural disasters than standard infrastructure, and therefore δ_zi > δ_za. Adam and Bevan (2014), in contrast, consider the case of operations and maintenance expenditures affecting the level of depreciation of public capital to underline the need for accounting these in the government’s budget constraint.

Another channel through which natural disasters can affect fiscal policy is through market interest rates. As S&P (2015) points out natural disasters can, in addition to the physical destruction of infrastructure, damage a sovereign’s creditworthiness. While the real interest rate on concessional debt is held constant (r_d,t = r_d), the natural disaster shock weakens the sovereign’s rating by D_rt. And as common in the literature, large accumulations of external debt (in deviations from its steady state level) aggravate the country risk premium, in line with Schmitt-Grohe and Uribe (2003). Given the risk-free world interest rate r^f and nominal GDP (y_t = P_n,ty_n,t + P_x,ty_x,t), the real interest rate on external commercial debt follows

In addition to building resilience with adaptation infrastructure, the government can decide to set up a fund to build up its savings against a natural disaster. Let S_t be government savings in a bank abroad, paying real interest rate r^f, the fund can only be drawn down in case of a major natural disaster to finance the fiscal deficit that arises with reconstruction activities. In addition to this buffer, the government can finance its fiscal deficit with external grants G and a combination of tax adjustments and borrowing: external concessional debt ∆d_t = d_t —d_t-1 as well as domestic ∆b_t = b_t —b_t-1 and external commercial ∆d_c,t = d_c,t —d_c,t-1 debt according to the rule (1 —v) ∆b_t = v∆d_c,t. The government budget constraint is thus given by

We can rewrite the budget constraint above in terms of the fiscal gap (Gap) as

where Gap_t = Erp_t —Rev_t, i.e. given by the difference between total expenditures and revenues when the consumption tax is kept at its initial value $\left({\tau }_o^c\right)$ and concessional debt and grants are exogenous, is covered by domestic and external commercial borrowing, VAT adjustments, and/or savings withdrawals, such that

and

The fiscal rule for the consumption tax rate allows τ^c to respond to both tax and debt deviations from their target, with λ₁ controlling the response of the tax adjustment to the tax rate that would prevail if all the other financing instruments were not available to close the fiscal gap and λ₂ controlling the response to deviations from the debt target

where the tax target ${\left({\tau }_t^c\right)}^{target}={\tau }_o^c+\frac{\mathfrak{G}\mathfrak{a}{\mathfrak{p}}_t}{P_t{c}_t}$, ensuring debt sustainability in the long run⁴. or the scenario analysis that we conduct, ${\tau }_t^c={\tau }_o^c$ for the case in which the government finances its reconstruction entirely with borrowing and ${\tau }_t^c={\left({\tau }_t^c\right)}^{target}$ when the reconstruction program is entirely financed with taxes.

Households

The economy features two types of households: Ricardian and liquidity-constrained (denoted by r and c, respectively), who consume tradable goods produced domestically c_x,t and abroad c_m,t as well as domestic nontradable goods c_n,t. The consumption basket is defined as the CES function

where ρ_m and ρ_x are weights and ε is the intratemporal elasticity of substitution between consumption goods. The associated consumer price index is given by⁵

Ricardian households face the intertemporal optimization problem

where β is the discount factor and ς_c is the intertemporal elasticity of substitution of consumption. In addition to consuming goods, this representative household invests in private tradable and nontradable capital (i_x and i_n, respectively), which encompasses some adjustment costs $A{C}_{j,t}^r\equiv \frac{v}{2}{(\frac{i_{j,t}^r}{k_{j,t-1}^S}-\delta )}^2{k}_{j,t-1}^r$; pays levies on standard public infrastructure μzⁱ; saves through domestic bonds b; and buys foreign debt b*, which carries a real interest rate r* as well as portfolio adjustment costs ${\Theta }_t^{r*}\equiv \frac{\eta }{2}{(b_t^{r*}-\overline{b^{s*}})}^2$, where $\overline{b^{r*}}$ is the steady-state value of external private debt. The government levies a value-added tax (VAT) τ^c on consumption, which affects both Ricardian and liquidity-constrained households.

The income side of the household’s budget constraint

is composed of an employment wage w on labor l^r supplied to firms; interest r_x and r_n received from tradable and nontradable capital, respectively; interest r received on domestic bonds; a fraction of remittances R and government transfers T corresponding to the share of Ricardian households in the labor market;⁶ as well as profits Φ^r from owning domestic firms. The accumulation of private capital, which depreciates annually at rate δ_k, is given by

The solution to the household’s constrained maximization problem yields the consumption Euler equation

and the non-arbitrage conditions defining that the returns on tradable and nontradable capital equate the return on domestic bonds

where ${\mathrm{\Upsilon }}_{j,t}^r=(\frac{i_{j,t}^r}{k_{j,t-1}^r}-\delta )$ for j = n, x, and that the return on domestic bonds equates the return on external private debt

where the real interest rate on external private debt $r_t^{*}=r_{dc,t}+u$ is a premium u over the sovereign’s real interest rate on external commercial debt r_dc and η governs the level of integration the private sector has in international capital markets.⁷

Liquidity-constrained households have the same preferences as Ricardian households but are prevented from saving and borrowing in the domestic and external financial market. These households have therefore to consume as much as their income from wages, remittances, transfers allow, and pay the same consumption taxes as the Ricardian households, i.e.

Aggregation and Market-Clearing Conditions

Aggregating across both types of households and firms, we have $x_t=\underset{\underset{j=n,x}{i=r,c}}{\text{Σ}}{x}_t^{i,j}\text{for}{x}_t^{i,j}=c_{j,t}^i,l_{j,t}^i,i_t^i,k_{j,t}^i,A{C}_{j,t}^i,b_t^i,b_t^{i,*},y_{j,t},{\Phi }_{j,t}^i$. Labor markets clear with labor demanded in the tradable and nontradable sectors supplied by both types of households, i.e. $l_{x,t}+l_{n,t}=l_t^r+l_t^c=l_t$. Nontradable output produced domestically must match the demand for nontradable goods, public and private investment used in the process of building nontradable public and private capital domestically (with a_k and a_z defining their nontradable content)

Finally, the balance of payment condition must hold

where the right-hand-side of (21) represents the current account deficit and the left-hand-side the capital account, i.e. the current account balance $c{a}_t=-\mathrm{\Delta }{b}_t^{*}-\mathrm{\Delta }{d}_{c,t}-\mathrm{\Delta }{d}_t-{\mathcal{G}}_t-{ℛ}_t+\mathrm{\Delta }{s}_t$.⁸

Natural Disaster

We analyze the response to a known disaster—2015 Cyclone Pam—for which damages and losses have already been accounted for. We therefore calibrate Dj to match Vanuatu’s known damages to infrastructure and economic losses following the cyclone. The authorities’ Post-Disaster Needs Assessment (2015) details damages and losses amounting to 60.8% of GDP (36.7% in damages and 24.2% in losses). Damages to the tradable sector, in particular damages and losses in agriculture, industry, and tourism, represented 23.7% of GDP (45% of which in damages and the remainder in losses); and 24.7% to the nontradable sector, which aggregates damages and losses in education and health, communication, energy, environment, transport, and water (57% of which in damages and the remainder in losses). The remaining 12.4% in damages and losses were in private housing.

The distribution of damages and losses among public and private capital destruction and productivity losses reflects our field assessment in Vanuatu. The distribution depends on output elasticities with respect to public infrastructure (ψ_n and ψ_x), private capital shares in value added (α_n and α_x) as well as the severity of damages to public and private capital (ϖ_z and ϖ_k, respectively). The former are pinned down by the initial return on public infrastructure with ψ_n = ψ_x. Estimates of the rates of return on public infrastructure abound and have been estimated to be between 15% and 30% for low-income countries (e.g., Foster and Briceño-Garmendia (2010)), but could also be greater than 50% depending on the type of infrastructure (Canning and Bennathan (1999)).⁹ For the baseline scenario (separable public capital inputs), the initial return on total public infrastructure $R_o^z=30\%$, which by definition equals the gross return on standard public investment $R_o^{zi}=30\%$ given our assumption on ξ as discussed below. On the other hand, estimates on the gross return on public adaptation investment $R_o^{za}$ are much less abundant. Our initial value is high (50%) but commensurate with the scarcity of adaptation infrastructure in developing economies or the additional benefit resilient infrastructure provides to the standard infrastructure.¹⁰ Therefore the implied value of the output elasticities with respect to public infrastructure is 0.0312. These low elasticities reflect the high inefficiencies in the public investment process. We calibrate the initial efficiency of public investment to be s₀ = 60%, which is in line with the relative distance of small developing states from one of the most efficient countries in the IMF’s PIMA and PIE-X assessments as done in Ghazanchyan et al. (2017).