Risk-Sharing with Network Transaction Costs

Abstract

Transaction costs can impede transfers, causing consumption to co-vary with endowment. We set up a risk-sharing model that incorporates bilateral transaction costs and demonstrate both theoretically and empirically that these costs impede global risk-sharing. Using data on production, consumption, and trade networks for staple food commodities, we highlight the practical implications of our model. Our findings reveal that transaction costs can constrain risk-sharing in global food markets. Using counterfactual analysis, we also show how changes in transaction costs impact global risk-sharing and examine the consequences of a complete disruption of wheat trade between Ukraine, Russia, and major importers.

Appendices

Appendix

Model Details

1.1 Symmetric Trade Result

Symmetric trading is not optimal. This motivates excluding the smaller importer in such cases.

Lemma 1

Given a linear transaction cost of the form in Eq. (4) and with nonzero fractional bilateral trade parameter $\delta _{ij}\in (0,1]~\forall i\forall j \not =i$ (assuming both not equal to 1), any nonzero bilateral trade between i and j is one directional, meaning $y_{ij}y_{ji}=0, y_{ij}\ge 0, y_{ji}\ge 0$.

Proof

Suppose both i and j trade with each other, meaning $y_{ij}>0$ and $y_{ji}>0$. Then, based on the FOC system in Eq. (6), the two conditions would need to hold simultaneously:

$$\begin{aligned} \frac{\partial U}{\partial c_i}\delta _{ji}=\frac{\partial U}{\partial c_j} \cap \frac{\partial U}{\partial c_j}\delta _{ij}=\frac{\partial U}{\partial c_i} \end{aligned}$$

These can only hold if $\delta _{ji}=1/\delta _{ij}$. However that can only happen if both are equal to 1 or if one is greater than 1 (either case violating the assumptions). Thus one of these first order conditions does not hold with equality, meaning at least one export is zero. $\square$

1.2 Optimal Consumption Result

Proposition 1

Optimal $c^*_i$ is a linear combination of endowments.

Proof

Given a known set of exporters and for a given state, the system of equations that define the optimal export choices can be written as the following:

$$\begin{aligned} \Delta _{ij}\left( y_j-\sum ^N_{k\ne j}y^E_{jk} + \sum ^N_{k\ne j}y^E_{kj}\delta _{kj}\right) =\left( y_i-\sum ^N_{k\ne i}y^E_{ik} + \sum ^N_{k\ne i}y^E_{ki}\delta _{ki}\right) ~~\forall ~\text {exporter { i}} \end{aligned}$$

Implicit in this equation is that $y_{ij}>0$ and thus $y_{ji}=0$ (by Lemma 1 in Appendix A). This can be rearranged into an $\mathbf {Ax=b}$ matrix format, with $\textbf{x}=[y^E_{ij},...]$. $\textbf{b}=[y_i-\Delta _{ij}y_j,...]$. The formula for $\textbf{A}$ is convoluted and involves the products of transaction costs, relative Pareto weights, and the preference parameter.

Then the solution by inversion, due to being square (and assuming full rank), is $\mathbf {x=A}^{-1}\textbf{b}$. This is simply a linear combination of the vector $\textbf{b}$, which in this case is just a linear combination of endowments $y_i\forall i$. Thus the export choices are linear in endowments. Since consumption is also just a linear combination of exports and endowments, optimal consumption will be a linear combination of endowments. $\square$

Specifically, the endowments are $y_i$ and $y_j$ from the same trade network. Optimal consumption has the following pattern, where $\Omega \equiv [\Delta _{ij},\delta _{ij}~\forall i\forall j\not =i]$, and $f, g^{ik}, h^{ki}$ are all functions $c^*_i=(y_i\cdot f(\Omega ) - \sum ^N_{k\ne i}g^{ik}(y_k\forall k,\Omega ) + \sum ^N_{k\ne i}h^{ki}(y_k\forall k,\Omega )\delta _{ki} )/f(\Omega )$. Both agent pair specific functions ($g^{ik}$ and $h^{ki}$) are simply weighted sums of all endowments. The endowments of any agent connected via the trade network has non-zero weight. Any agent in a separate network has zero weight.

1.3 Indirect Trade Weight Result

Proposition 2

The (geodesic path) indirect trade weight from index i to ending partner k is:

$$\begin{aligned} \Delta _{i\rightarrow k}=\left( \frac{\alpha _k}{\alpha _i}\prod _{(r,s)\in \mathcal {P}_{ik}}\frac{\delta _{rs}^{\mathcal {S}(r,s)}}{ \delta _{sr}^{\mathcal {S}'(r,s)}}\right) ^{-1/\gamma }\end{aligned}$$

(15)

Proof

Consider the undirected geodesic path from i to k, which exists if $i,k\in \mathcal {T}$, but may not be unique. There exists a path $i, k_{1},...,k_{n-1},k_{n}=k$ with n steps. The first FOC comparing i to $k_{1}$ is either $c_i=\Delta _{i,k_1}c_{k_1}$ or $c_{k_1}=\Delta _{k_1,i}c_i$ depending on which is the importer or exporter; either $\mathcal {S}'(i,k_1)=1$ or $\mathcal {S}(i,k_1)=1$, which due to Lemma 1 are mutually exclusive. The second is either $c_{k_1}=\Delta _{k_1,k_2}c_{k_2}$ or $c_{k_2}=\Delta _{k_2,k_1}c_{k_1}$. The combined chain is then:

$$\begin{aligned} c_i= \Delta _{i,k_1}^{\mathcal {S}(i,k_1)}\cdot \Delta _{k_1,i}^{-\mathcal {S}'(i,k_1)}\cdot \Delta _{k_1,k_2}^{\mathcal {S}(k_1,k_2)}\cdot \Delta _{k_2,k_1}^{-\mathcal {S}'(k_1,k_2)} c_{k_2} \end{aligned}$$

If $n=2$ then the expression above is the final answer. Now suppose it holds for $n=m$, meaning the path from i to k with m steps yields: $c_i=\prod _{(r,s)\in \mathcal {P}_{ik_m}}{\Delta _{rs}^{\mathcal {S}(r,s)}}{ \Delta _{sr}^{-\mathcal {S}'(r,s)}}c_{k_m}$.

Now suppose we actually need one more link to reach k from i, meaning $n=m+1$. If m exports to $m+1=k$, then $c_m=\Delta _{m,k}c_k$ and if k exports to m then $c_k=\Delta _{k,m}c_m$. Then we can compare $c_i$ to $c_k$ through $c_{k_m}$:

$$\begin{aligned} c_i=\Delta _{i\rightarrow k_m} c_{k_m}=\Delta _{i\rightarrow k_m}\cdot (\Delta _{m,k}^{\mathcal {S}(m,k)}\Delta _{k,m}^{-\mathcal {S}'(m,k)} c_k) \end{aligned}$$

Thus $\Delta _{i\rightarrow k}=\prod _{(r,s)\in \mathcal {P}_{ik}}{\Delta _{rs}^{\mathcal {S}(r,s)}}{ \Delta _{sr}^{-\mathcal {S}'(r,s)}}$. Substituting out $\Delta _{ik}\equiv \left( {\delta _{ik}\alpha _k}/{\alpha _i}\right) ^{-1/\gamma }$ immediately yields the result. $\square$

Note that the geodesic path is not in general the “lowest cost” path; geodesic paths only consider the number of links and there may be a “longer” path that has a lower indirect cost.³⁹ Also, in the case of heterogeneous $\gamma$, the ratios of $\gamma _i$ along the path change the formula in scaling each intermediary direct $\Delta$.

1.4 Market Equilibrium

Consider each agent i solving the following individual maximization problem under uncertainty:

$$\begin{aligned} \max _{\{c_{it}(s_t),\, y_{ijt}^E(s_t)\ge 0\}} \mathbb {E}_0\left[ \sum _{t=0}^{T}\beta ^t u_i\left( c_{it}(s_t)\right) \right] , \end{aligned}$$

(16)

subject to the state-contingent budget constraint, where $p_{it}(s_t)$ is the market price of agent i’s endowment in state $s_t$:

$$\begin{aligned} p_{it}(s_t)c_{it}(s_t) = p_{it}(s_t)y_{it}(s_t) - \sum _{j\ne i} p_{jt}(s_t)\frac{y_{ijt}^E(s_t)}{\delta _{ijt}} + \sum _{j\ne i} p_{it}(s_t)y_{jit}^E(s_t), \end{aligned}$$

(17)

Market clearing conditions for each pair (i, j) ensure equilibrium:

$$\begin{aligned} y_{ijt}^E(s_t)\ge 0, \quad p_{it}(s_t)y_{ijt}^E(s_t) = p_{jt}(s_t)\delta _{ijt}y_{jit}^E(s_t). \end{aligned}$$

(18)

In equilibrium, state-contingent prices $p_{it}(s_t)$ adjust to reflect agents’ marginal valuations, explicitly capturing consumption risk. Specifically, equilibrium prices satisfy:

$$\begin{aligned} \frac{p_{it}(s_t)}{p_{it}(s_t')} = \frac{\partial u_i(c_{it}(s_t))/\partial c_{it}(s_t)}{\partial u_i(c_{it}(s_t'))/\partial c_{it}(s_t')} \quad \forall s_t,s_t'. \end{aligned}$$

(19)

1.5 Other Network Concepts

Home bias refers to the observed preference of agents towards domestic goods and services over foreign alternatives, due to factors like transaction costs or preferences. In our model, home bias is implicitly captured by $\delta _{ijt}$. Lower values of $\delta _{ijt}$ correspond to higher transaction costs and therefore a greater degree of home bias. The aggregate risk function $\mathcal {A}^*_{\mathcal {T}_t}$ incorporates this into equilibrium consumption allocations by aggregating these bilateral efficiencies across the entire network. Formally, higher transaction costs lead to lower aggregate risk sharing efficiency, increasing the reliance on endowments and increasing consumption volatility across the network.

Multilateral resistance, as introduced in gravity models (Anderson and Van Wincoop 2003), captures how bilateral trade flows are influenced by direct trade barriers between two agents and their relative trade costs with all other trading partners. In gravity frameworks, multilateral resistance terms summarize indirect effects through equilibrium price indices. In our risk-sharing network, the indirect trade weight $\Delta _{i\rightarrow k}$ generalizes this concept by explicitly accounting for the indirect impact of bilateral transaction costs through intermediary agents. Specifically, the indirect trade weight in Eq. 10 aggregates transaction costs across the shortest trading path linking two indirectly connected agents. Thus, it operationalizes multilateral resistance in the risk-sharing context by determining how transaction costs between any two agents propagate through the entire network, affecting equilibrium consumption allocations.

Estimation Details

1.1 Trading Participation Selection

The risk-sharing equation is derived from the FOC of the model; for agents who are not part of any network in a given year, their consumption is simply equal to their production: $\ln (c_{jt})=\ln (y_{jt})+\varepsilon _{jt}$, and so there is no “risk-sharing” parameter to test since they are unable to share risk in that period. Note however that the fact that the agent is not trading is endogenous to the model; the realization of shocks in a year combined with transaction costs could lead to an agent not trading. Thus it does raise the question of selection bias present in the sample if we only estimate the risk-sharing regression for trading agents.

If one simply wants to capture how being in a network reduces dependence on own production, then conditioning on the network participants is sufficient. However if one is interested in understanding the extent of global risk sharing induced by variation in endowment shocks, then one should include excluded countries as their autarkic status in a given year reveals the extent of risk sharing.⁴⁰ If the selection is based on observables, then an augmented risk-sharing equation is sufficient to capture membership as follows (expressed in terms of a single network):

$$\begin{aligned} \ln (c_{it})=\ln (c^*_{it})\cdot \mathbbm {1}[i_t\in \mathcal {T}_t]+\ln (y_{it})\cdot \mathbbm {1}[i_t\not \in \mathcal {T}_t]+\eta \ln (y_{it})+\varepsilon _{it} \end{aligned}$$

(20)

1.2 Asymptotic Derivations

The risk sharing test controlling for aggregate risk function and transaction costs is given as:

$$\begin{aligned} \text {ln}(c_{jt}) = \text {ln}(\mathcal {A}_{\mathcal {T}_t t}^*)+\eta \cdot \text {ln}(y_{jt})-\text {ln}(\Delta ^*_{jt})+\varepsilon _{jt}, \end{aligned}$$

(21)

for each $j\in \mathcal {N}_N=\{1,\ldots ,N\}$ and $t=1,2,\ldots , T$. Define the variables $\textbf{x}_{it} = (x_{it1}, x_{it2})$ and $\textbf{z}_{it} = \left( \{z_{it}^j\}_1^{|I_{it}|}, \{z_{it}^k\}_1^{|E_{it}|}\right)$ where $|I_{it}|$ and $|E_{it}|$ denote the cardinality of the importer and exporter sets, respectively of country i at time t. Let $\textbf{f}_{t} = \left( f1_{t}, \ldots , fT_{t}\right)$ denote the vector of binary time indicators. Let $\textbf{w}_{it} = \left( \textbf{d}_t, \textbf{x}_{it}, \textbf{z}_{it}\right)$ and $\varvec{\theta } = \left( \varvec{\alpha }, \varvec{\beta }\right) ^\prime$. Here $\varvec{\beta } = \left( \beta _1, \beta _2, \beta _3, \beta _4\right) ^\prime$ and $\varvec{\alpha } = \left( \alpha _1, \ldots , \alpha _T\right) ^\prime$. Then, one may rewrite the risk sharing regression as:

$$\begin{aligned} y_{it}&= \textbf{f}_{t}\varvec{\alpha }+\beta _1x_{it1}+\beta _2x_{it2}+\beta _2\left[ \sum _{j\in I_{it}} g(z_{it1}^j,z_{it2}^j; \beta _3, \beta _4)-\sum _{l\in E_{it}} g(z_{it1}^l,z_{it2}^l; \beta _3, \beta _4)\right] +\varepsilon _{it} \nonumber \\&\equiv h(\textbf{w}_{it}, \varvec{\theta })+\varepsilon _{it} \end{aligned}$$

(22)

where $g(\cdot ; \cdot ) = \text {log}\left( \frac{\text {exp}(\cdot )}{1+\text {exp}(\cdot )}\right)$. The summations over j and l are adding over countries that indirectly connect i with the index country $\bar{i}$ in the importer and exporter sets, respectively. Hence, $\{(y_{it}, \textbf{w}_{it}); \ i\in \mathcal {N}_N; t=1,2,\ldots , T\}$ represent the observed sample from the population. Then, stacking Eq. (22) across time periods, we can write the regression more compactly as⁴¹

$$\begin{aligned} \textbf{y}_i = \textbf{h}_i(\varvec{\theta })+\varvec{\varepsilon }_{i}, \ i\in \mathcal {N}_N \end{aligned}$$

(23)

where $\textbf{y}_i = \left( y_{i1}, \ldots , y_{iT}\right) ^\prime$, $\textbf{h}_i(\varvec{\theta }) = \left( h_{i1}(\varvec{\theta }), \ldots , h_{iT}(\varvec{\theta })\right) ^\prime$, and $\varvec{\varepsilon }_{i} = \left( \varepsilon _{i1}, \ldots , \varepsilon _{iT}\right) ^\prime$.

Assumption 1

$\mathbb {E}(\varepsilon _{it}|\textbf{w}_{i}) = \mathbb {E}(\varepsilon _{it}|\textbf{w}_{it})=0$

Assumption 1 implies that strict exogeneity holds with respect to risk sharing regression equation. Given the estimating Eq. in 23, we minimize the following nonlinear least squares objective function,

$$\begin{aligned} \underset{\varvec{\theta }\in \varvec{\Theta }}{\text {min}}\frac{1}{NT}\sum _{i\in \mathcal {N}_N}\varvec{\hat{\varepsilon }}_{i}^{2} \equiv \underset{\varvec{\theta }\in \varvec{\Theta }}{\text {min}}\frac{1}{NT}\sum _{i\in \mathcal {N}_N}(\textbf{y}_{i}-\textbf{h}_{i}(\varvec{\hat{\theta }}))^2 \end{aligned}$$

(24)

with the first order conditions for $\varvec{\hat{\theta }}$ given as

$$\begin{aligned} \frac{1}{NT}\sum _{i\in \mathcal {N}_N}\nabla _{\varvec{\theta }}\textbf{h}_{i}(\varvec{\hat{\theta }})^\prime \varvec{\hat{\varepsilon }}_{i} = \frac{1}{NT}\sum _{i\in \mathcal {N}_N}\textbf{h}_{i}^{(1)}(\varvec{\hat{\theta }})\varvec{\hat{\varepsilon }}_{i} =\textbf{0} \end{aligned}$$

(25)

where

$$\begin{aligned} \textbf{h}_{i}^{(1)}(\varvec{\hat{\theta }})^\prime = \nabla _{\varvec{\theta }}\textbf{h}_{i}(\varvec{\hat{\theta }})&= \begin{pmatrix} \frac{\partial h_{i1}(\varvec{\hat{\theta }})}{\partial \theta _1} & \cdots & \frac{\partial h_{i1}(\varvec{\hat{\theta }})}{\partial \theta _M} \\ \frac{\partial h_{i2}(\varvec{\hat{\theta }})}{\partial \theta _1} & \cdots & \frac{\partial h_{i2}(\varvec{\hat{\theta }})}{\partial \theta _M} \\ \vdots & \ddots & \vdots \\ \frac{\partial h_{iT}(\varvec{\hat{\theta }})}{\partial \theta _1} & \cdots & \frac{\partial h_{iT}(\varvec{\hat{\theta }})}{\partial \theta _M} \ \end{pmatrix}_{(T\times M)} \end{aligned}$$

Then, by first-order taylor expansion of Eq. (25) around the true $\varvec{\theta }$, we have the following influence function representation for $\hat{\varvec{\theta }}$

$$\begin{aligned} \sqrt{N}(\varvec{\hat{\theta }}-\varvec{\theta }) = -\textbf{Q}_T^{-1}\left( \frac{1}{\sqrt{N}}\sum _{i\in \mathcal {N}_N}\textbf{v}_i \right) +o_p(1) \end{aligned}$$

(26)

where, $\textbf{v}_i = \textbf{h}_{i}^{(1)}(\varvec{\theta })\varvec{\varepsilon }_{i} = \frac{1}{T}\sum _{t=1}^{T}\nabla _{\varvec{\theta }}h_{it}(\varvec{\theta })\varepsilon _{it}$ and $\textbf{Q}_T= \underset{N\rightarrow \infty }{\text {plim}}\frac{1}{NT}\sum _{i\in \mathcal {N}_N}\nabla _{\varvec{\theta }}\{\textbf{h}_{i}^{(1)}(\varvec{\tilde{\theta }})\varvec{\tilde{\varepsilon }}_{i}\} = -\frac{1}{T}\mathbb {E}[ \textbf{h}_{i}^{(1)}(\varvec{\theta })\textbf{h}_{i}^{(1)}(\varvec{\theta })^\prime ]$.

Now, in order to obtain the asymptotic distribution of $\varvec{\hat{\theta }}$, we need to apply the LLN and CLT for network dependent data to the sequence, $N^{-1/2}\sum _{i\in \mathcal {N}_N}\textbf{v}_i$, where $\mathbb {E}(\textbf{v}_i|\textbf{w}_i) = \textbf{0}$ due to assumption 1a).⁴² Provided that certain regularity conditions hold (mentioned below) which are informed by KMS, we will obtain the following asymptotic normality result.

$$\begin{aligned} \sqrt{N}(\varvec{\hat{\theta }}-\varvec{\theta }) \sim \text {N}\left( \textbf{0}, \textbf{Q}_T^{-1} \varvec{\Omega }_T\textbf{Q}_T^{-1}\right) \end{aligned}$$

(27)

with the network-HAC variance is given by

$$\begin{aligned} \varvec{\Omega }_T = Avar (N^{-1/2}\sum _{i\in \mathcal {N}_N}\textbf{v}_i)&= \underset{N\rightarrow \infty }{lim }\frac{1}{NT^2}\sum _{d\ge 0}\sum _{i\in \mathcal {N}_N}\sum _{j\in \mathcal {N}_N^\partial (i,d)}\mathbb {E}(\textbf{v}_i\textbf{v}_{j}^\prime |\textbf{w}_i, \textbf{w}_j) \end{aligned}$$

(28)

where $\mathcal {N}_N^\partial (i,d)= \{j\in \mathcal {N}_N: D_N(i,j) = d\}$ is the set of neighbors of node i who are exactly d-links away. In our context, these cross-correlations result from network dependence among the errors $\varvec{\varepsilon }_i$’s. This is because

$$\begin{aligned} \begin{aligned} \mathbb {E}(\textbf{v}_i\textbf{v}_{j}^\prime |\textbf{w}_i, \textbf{w}_j)&= \textbf{h}_{i}^{(1)}(\varvec{\theta })\mathbb {E}(\varvec{\varepsilon }_{i}\varvec{\varepsilon }_{j}^\prime |\textbf{w}_i, \textbf{w}_j)\textbf{h}_{j}^{(1)}(\varvec{\theta })^\prime \end{aligned} \end{aligned}$$

In particular, we assume weak dependence which means that these covariances diminish as the network-distance between any two pairs of nodes, i and j, grows large. To limit the strength of dependence, we impose regularity conditions that assume the network data to be conditionally $\psi$-dependent given the $\sigma$-algebra, $\mathcal {C}_N$.

Note that we have assumed conditional $\psi$-dependence with respect to $\varvec{\varepsilon }_i$. This is easily translated into conditional $\psi$-dependence of the linear transformation, $\textbf{v}_i= \textbf{h}_{i}^{(1)}(\varvec{\theta })\varvec{\varepsilon }_{i}$ given $\mathcal {C}_N$ with dependence coefficients, $\lambda _N$, from using Lemma 2.1 of KMS. The other conditions required for the LLN and CLT include moments conditions for $\textbf{v}_{i}$, as follows

Assumption 2

(Moment conditions) i) For some $\epsilon >0$, $sup _{N\ge 1} max _{i\in \mathcal {N}_N}||\textbf{v}_{i}||_{\mathcal {C}_N, 1+\epsilon }<\infty$ a.s; ii) For some $p>4$, $sup _{N\ge 1} max _{i\in \mathcal {N}_N}||\textbf{v}_{i}||_{\mathcal {C}_N, p}<\infty$ a.s. where $||\textbf{v}_{ i}||_{\mathcal {C}_N, p} = \left( \mathbb {E}[|\textbf{v}_{i}|^p|\mathcal {C}_N]^{1/p}\right)$.

Assumption 3

(Denseness and strength of network dependence) i) $\frac{1}{N}\sum _{d\ge 0}\zeta _N^{\partial }(d)\lambda _{N,d}\rightarrow _{a.s.} 0$ where $\zeta ^{\partial }_N(d) = N^{-1}\sum _{i\in \mathcal {N}_N}|\mathcal {N}_N^{\partial }(i,d)|$; ii) There exists a positive sequence, $m_N\rightarrow \infty$ such that for $k=1,2$

$$\begin{aligned} \begin{aligned}&N^{1-k} \cdot \varvec{\Omega }_N^{-k}\sum _{d\ge 0} a_N(d, m_N; k)\lambda _{N,d}^{1-\frac{2+k}{p}} \rightarrow _{a.s.} 0 \\&N^{3/2}\cdot \varvec{\Omega }_N^{-1/2} \lambda _{N, m_N}^{1-1/p}\rightarrow _{a.s.} 0 \end{aligned} \end{aligned}$$

as $N\rightarrow \infty$, where $p>4$ and $a_N(\cdot , \cdot ;)$ is the measure of network denseness used in KMS.

where $\zeta ^{\partial }_N(d)$ gives us the average neighborhood size and $a_N(\cdot , \cdot ; )$ is a combination of average neighborhood size and average neighborhood shell size. These are being used as measures of network denseness. As KMS put it, this condition appears in the form of a tradeoff between network denseness and strength of dependence needed to ensure that covariance decays as a function of network-distance, d.

A consistent estimator for the network HAC variance can then be obtained from constructing a sample analogue of the expression in (27) where the middle term is taken to be a weighted average of the sample covariances of $\textbf{h}_{i}^{(1)}(\varvec{\hat{\theta }})\varvec{\hat{\varepsilon }}_{i}$ and $\textbf{h}_{j}^{(1)}(\varvec{\hat{\theta }})\varvec{\hat{\varepsilon }}_{j}$ as follows:

$$\begin{aligned} \varvec{\hat{\Omega }}_T = \frac{1}{NT^2}\sum _{d\ge 0}\mathcal {K}_N(d)\left[ \sum _{i\in \mathcal {N}_N}\sum _{j\in \mathcal {N}_N^\partial (i,d)} \textbf{h}_{i}^{(1)}(\varvec{\hat{\theta }})\varvec{\hat{\varepsilon }}_{i}\varvec{\hat{\varepsilon }}_{j}^\prime \textbf{h}_{j}^{(1)}(\varvec{\hat{\theta }})^\prime \right] \end{aligned}$$

with $\mathcal {K}_N(d)=\mathcal {K}(d/b_N)$ as the weight given to sample covariances as a function of the distance.

Consistency of $\varvec{\hat{\Omega }}_T$ can be established using the results in KMS if we assume the following conditions hold along with Assumption 2) ii).

Assumption 4

(Consistency of the HAC estimator) There exists a $p>4$ such that

i) lim$_{N\rightarrow \infty }\sum _{d\ge 1}|\mathcal {K}_N(d)-1|\zeta _N^\partial (d)\lambda _{N,d}^{1-2/p} = 0$ a.s. and

ii) lim$_{N\rightarrow \infty }N^{-1}\sum _{d\ge 0}a_N(d,b_N;2)\lambda _{N,d}^{1-4/p} = 0$ a.s.

Results Appendix

1.1 Heterogeneous Preferences

As demonstrated in Schulhofer-Wohl (2011), agents may have specific preferences $\gamma _i$. In this case, the FOC still has an exponential proportional form $c_i^{\gamma _i}=c_j^{\gamma _{j}}(\alpha _j\delta _{ij}/\alpha _i)$. Just as before, we can, without loss, define $\mathcal {A}^*_{\mathcal {T}}$ to be agent $\bar{i}$’s consumption. Then by the FOC, $c_j=\tilde{\mathcal {A}}^{\gamma _{\bar{i}}/\gamma _j}/ (\tilde{\Delta }_{\bar{i}\rightarrow j})^{-1/\gamma _j}$, where $\tilde{\Delta }\equiv {\Delta }^{-\gamma }$. Then the risk-sharing regression has scaled model terms per country:

$$\begin{aligned} \ln (c_{jt})=\frac{\ln (\tilde{\Delta }_{jt})}{\gamma _j}+\frac{\gamma _{\bar{i}_t}\ln (\mathcal {A}_{\mathcal {T}_tt})}{\gamma _j}+ \eta \cdot \ln (y_{jt})+\varepsilon _{jt} \end{aligned}$$

(29)

Note that $\gamma _{\bar{i}_t}$ can change every year depending on the network structure; a previous member chosen for the index could be absent in the network the following year, and thus the identity of $\bar{i}_t=\bar{i}(\mathcal {N}_t)$ changes. Note the presence of the index country’s $\gamma$ term as a coefficient on $\mathcal {A}_{\mathcal {T}_tt}$. This is a network-specific term and thus even if one omits trading costs from the model, the correct model specification with heterogeneous preferences uses trade network information. As the heterogeneous specification has country specific $\gamma _j$, it can soak up time-invariant country specific factors that could affect the relationship between production and consumption. Fully addressing all possible alternative explanations for nonzero $\eta$ beyond transaction costs is beyond the scope of this analysis, but this specification aids in this.

Table 6 reports estimates from the heterogeneous risk preference model in Eq. (29) for baseline and transaction cost specifications for all three commodities.⁴³ These regressions include country fixed effects in all specifications. We compare the baseline to the trade-cost-constant specification as the full set of cost controls were not well estimated separately from the country-level slopes. Overall the results are noisier for production, consistent with Schulhofer-Wohl (2011).

The baseline maize model has a production coefficient $\eta$ of 0.255 (0.137), with the standard error (SE) in parentheses. The full model with transaction cost specification yields $\eta$ equal to 0.1711 (0.08845). This is a 33.1% decrease in $\eta$ and a 35.3% decrease in its SE. For wheat, the baseline $\eta$ is 0.11481 (0.038932), and the model yields 0.11695 (0.066531). This is a 1.86% increase in $\eta$ and a 71.9% increase in its SE. Thus, while the coefficient increases a trivial amount, the predictive power of production decreases significantly. Finally, for rice, baseline $\eta$ is 0.21078 (0.21541), and the model yields 0.18163 (0.20765). The coefficient and SE change by −13.8% and −3.60%. The trade cost constant model under heterogeneous preferences results in larger changes to risk-sharing than in the homogeneous case. This is intuitive as the effects of the network shape are likely dependent on a country’s location in the network, which the latter averages out.

Table 6 Heterogeneous Effects

1.2 Additional Results

To see how centrality of the network affects risk sharing, we consider specifications in which we only include countries that are within a certain distance (network-distance) from the index. When only including agents within a 1-link radius from the index, the risk sharing parameter should be smaller than the whole sample. As one increases the number of links included (meaning going away from the center of the network center), risk-sharing would naturally go down; the less connected countries are not as integrated into the trading network. The effects of controlling for transaction costs should also matter at all levels of the radius. For maize, at $\le 3$-links, the coefficient decreases by 9.9% from baseline, at $\le 2$-links it decreases by 11%, and at 1-link it decreases by 23%. In addition, at 1-link distance the statistical significance decreases. This indicates that near the center of the network, there is stronger evidence of risk-sharing in Maize after controlling for transaction costs. For the distance from center results, for wheat, $\le 3$-links yields a -5.7% change, $\le 2$-links has -6.6%, and 1-link has -10.4%. For rice, $\le 3$-links yields a -9.6% change, $\le 2$-links has a similar decrease, and 1-link has a -5.6% change.

As an alternative sample, we consider a 1 million population cutoff and keep all nonzero trade levels. Results are weaker (smaller or no difference between baseline and full model) and there is less aggregate risk sharing; this is intuitive as many more countries which are less connected (and trade less) are included in this sample. For functional form robustness, first we consider a probit instead of logit for the costs, $\delta$. Second, we consider a more substantial functional form change that still captures the indirect trade effects. We specify the log indirect trade weight as a single logit of the linear difference in covariates across export/import pairs. Across all commodities, we find similar risk-sharing estimates and some small changes to statistical significance of the controls.

Finally, we re-estimate the risk-sharing parameter for maize, wheat, and rice under two alternative Pareto weight specifications, namely constant weights and log yearly trade volume Pareto weights. Across commodities and weighting schemes, the baseline specification consistently produces larger and more precisely estimated coefficients than the trade-cost specification. Under constant Pareto weights, the baseline coefficient is 0.399 (0.156) for maize, 0.248 (0.047) for wheat, and 0.310 (0.062) for rice, compared to 0.232 (0.135), 0.216 (0.055), and 0.234 (0.048), respectively, when trade costs are incorporated. Similarly, with volume-weighted Pareto weights: the baseline coefficients are 0.380 (0.144) for maize, 0.214 (0.046) for wheat, and 0.296 (0.056) for rice, while the corresponding trade-cost-adjusted estimates are uniformly smaller at 0.213 (0.133), 0.191 (0.053), and 0.231 (0.047). In addition to being larger in magnitude, the baseline coefficients are generally more statistically significant, particularly for maize, indicating that ignoring transaction costs tend to overstate the sensitivity of consumption to domestic production.

Additional Figures

See Figure 9.

Fig. 9

Adding Trade Partners. This figure plots the network shapes of a 5 country example under different trading scenarios