# Climate-related Agricultural Productivity Losses through a Poverty Lens

Alkis Blanz  University of Leipzig, Faculty of Economics and Management
Sciences, Grimmaische Str. 12, 04109, Leipzig, Germany Mercator Research
Institute on Global Commons and Climate Change (MCC), Torgauer Str. 12-15,
10829, Berlin, Germany

In this paper, we analyze the long-term distributive impact of climate change
through rising food prices. We use a standard incomplete markets model and
account for non-linear Engel curves for food consumption. For the calibration
of our model, we rely on household data from 92 developing countries,
representing 4.5 billion people. The results indicate that the short-term and
long-term distributive impact of climate change differs. Including general
equilibrium effects change the welfare outcome especially for the poorest
quintile. In the presence of idiosyncratic risk, higher food prices increase
precautionary savings, which through general equilibrium affect labor income
of all agents. Furthermore, this paper studies the impact on inequality for
different allocations of productivity losses across sectors. When climate
impacts affects total factor productivity in both sectors of the economy,
climate impacts increase also wealth inequality.

I am very grateful to Matthias Kalkuhl, Francesca Diluiso, Ottmar Edenhofer,
Maik Heinemann and Ulrich Eydam for useful discussion and comments.
Furthermore, I want to thank participants of the SURED 2022, the 27th Annual
Conference of the European Association of Environmental and Resource
Economists, the CESifo Area Conference on Energy and Climate Economics 2022
and the Potsdam Research Seminar in Economics for their useful feedback and
comments.

##  1 Introduction

In recent decades, climate change has reduced global crop yields and slowed
agricultural producticity growth (Ortiz-Bobea et al., 2021). Relatedly,
climate impacts already affect all dimensions of food security by disrupting
production, quality, storage and retail (Shukla et al., 2019). Agricultural
productivity and food security are the most important resource for the
livelihood of households in the developing world. Increasing weather and
climatic events have exposed millions of people to acute food insecurity (Lee
et al., 2023). As developing countries will continue to be severely affected
by climate impacts over a long time horizon, it is important to improve our
understanding of the impact of agricultural productivity on long-term
inequality.

The distributive impact of climate change in the short and long run may differ
fundamentally. The direct, short-term impact treats income (and wealth) as
fixed, and thus focuses on demand dynamics. In contrast, the indirect long-
term impact considers changes in income and wealth through general equilibrium
effects. Poverty is multidimensional, and assets play a detrimental role for
the coping ability of households. In general, households with more assets tend
to be more resilient and are able to cope better with shocks. In the presence
of idiosyncratic risk, households accumulate assets in good times and deplete
them in bad times to smooth consumption (Angus, 1991; Rosenzweig and Wolpin,
1993). For instance, wealth inequality could not only be affected by the
direct impact on assets, but additionally may depend on how households will
adjust their saving behavior in reaction to climate impacts. However, we still
have limited knowledge about how climate change affects saving behavior across
the wealth distribution and what the key driving forces are.

In this paper, we address this research gap by analyzing the long-term impact
of food prices on inequality. By focusing both on the direct and indirect
effects of climate-related food price changes, we aim to align micro- and
macroeconomic analyses of the distributive impact of food prices in the
developing world. In addition to the direct impact on consumption, higher food
prices could affect the ability to accumulate assets over time. Food prices
could increase consumption and wealth inequality. Within a single framework,
we are thus able to compare the welfare impact of higher food prices in
partial and general equilibrium. To properly model these dynamics, we extend
the workhorse heterogeneous-agent macroeconomic model (Aiyagari, 1994; Bewley,
1977; Huggett, 1993) to account endogenously for nonlinear Engel curves for
food consumption. This allows to explicitly consider how both the consumption
and the saving decision of heterogeneous households are affected by changes in
food prices, while considering the impact of idiosyncratic risk. We calibrate
the model to capture the characteristics of the developing world with 4.5
billion people, based on the Global Consumption Database from the World Bank.

In the analysis, we proceed as follows. In a first step, we isolate the
distributive impact of past agricultural productivity losses by comparing the
developing world to a baseline, where without climate-related productivity
losses, food prices are lower. There, we compare the short-term, partial
equilibrium to long-term, general equilibrium outcome. While the direct impact
focuses on changes in consumption, the general equilibrium outcome also
considers changes in income. The poorest households suffer by far the largest
direct impact. This is due to the fact that these households spend a large
share of their expenditures on food. However, at the bottom of the
distribution, we find the largest difference between partial and general
equilibrium outcomes. Due to higher labor income, poor households can
partially alleviate the large direct effect of higher food prices. In a second
step, we compare the impact on inequality for different allocations of
productivity losses across sectors. When productivity declines in both
sectors, climate impacts also increase wealth inequality.

###  1.1 Related Literature

The key contribution is to bridge the gap between detailed micro-analyses of
distributional effects of food price changes and general equilibrium model
analyses of agricultural productivity losses. Thereby, this paper relates to
several branches of the literature. First, we contribute to the macroeconomic
development literature has studied the nexus between food and agricultural
productivity. In the developing world, a large part of the resources are used
to produce food for subsistence needs (Gollin et al., 2007). Since food is a
subsistence good, low agricultural productivity can limit growth, because high
food prices adversely affect the savings rate (Irz and Roe, 2005). We
contribute to this literature by studying how in the presence of idiosyncratic
risk agricultural productivity losses affect the savings decision across the
distribution.

By analyzing the impact of agricultural productivity losses through food
prices in a heterogeneous agent macroeconomic model, this paper aims to bridge
the gap to the microeconomic literature that analyzes the distributive impact
of food price increases. Microeconomic methods and macroeconomic models
generally find opposite effects of higher food prices on poverty (Headey,
2014). The general equilibrium effects are usually wage adjustments due to the
agricultural supply-side response (Ivanic and Martin, 2014; Jacoby, 2016).
Otherwise, households could adjust their labor supply to generate additional
income to cope with higher food prices. For example, households may increase
their off-farm work to smooth consumption (Kochar, 1995, 1999; Rasmus Heltberg
and Talukdar, 2015).

Furthermore, we relate to the literature that explicitly considers the fact
that households in developing countries face a severe difficulty in insulating
themselves from income volatility (Udry, 1994; Townsend, 1994). Existing
evidence suggests that households may sell livestock or use savings when faced
with health shocks (Islam and Maitra, 2012; Rosemary E. Isoto and Kraybill,
2017; DeLoach and Smith-Lin, 2018) or when faced with droughts (Fafchamps et
al., 1998; Kazianga and Udry, 2006). In developing countries, uninsurable
idiosyncratic risk is relevant for welfare of households through its effect on
risk sharing and migration (Lagakos et al., 2023; Morten, 2019) and for the
macroeconomy for agricultural productivity (Donovan, 2021) and the severe
limitations of risk sharing capabilities of privately owned
businesses(Angeletos, 2007). The contribution of this paper is to study the
role of uninsurable idiosyncratic risk in the context of climate-related
agricultural productivity losses. Thereby, we are able to study the impact on
both consumption and wealth inequality.

In the context of climate change, we build on the climate economic literature,
where climate impacts are modeled as reductions in sectoral productivity (e.g.
(Schlenker and Roberts, 2009)). This article studies agricultural productivity
losses in a macroeconomic model with heterogeneous agents. Models with
uninsurable idiosyncratic risk are increasingly used in the context of climate
change (Fried, 2021; Malafry and Brinca, 2022; Krusell and Smith, 2022). By
using a heterogeneous agent model, we are able to link this literature to the
microeconomic literature that studies the distributive impact of increases in
food prices related to climate change. Hallegatte and Rozenberg (2017) combine
household survey data from 92 countries with SSP scenarios and find that,
through food prices alone, climate change could push 100 million people into
extreme poverty by 2030. However, this widely used approach for studying the
distributional incidence of climate or price shocks captures only the short-
term, partial equilibrium effect and disregards how households accumulate
assets and how labor income changes. Our general-equilibrium framework with
heterogeneous agents allows households’ consumption decisions to interact with
macroeconomic dynamics. Thereby, we are able to to provide novel insights on
how climate impacts affect the willingness and the ability to accumulate
assets across the distribution.

The paper proceeds as follows. Section 2 presents the dynamic incomplete
markets model of food consumption. Section 3 discusses the calibration of the
model. Section 4 reports the results of the quantitative results on the
distributive impacts of rising food prices. Section 5 concludes.

##  2 A Dynamic Incomplete-Markets Model of Food Consumption

Our model is at its core a standard incomplete market general equilibrium
model with infinite horizon of the Aiyagari-Bewley-Hugget type (Aiyagari,
1994; Bewley, 1977; Huggett, 1993). The special feature of the model studied
here is that it accounts for Engel’s law. As an endogenous outcome of the
model, poorer households spend a larger share of their income on food. In the
present model type, individual labor productivity is affected by an
uninsurable idiosyncratic risk. This creates a precautionary saving motive
and, through this, an endogenous wealth distribution. Higher food prices are
an outcome of climate-related agricultural productivity losses. We abstract
from aggregate uncertainty. Without aggregate uncertainty, in the baseline
model agricultural productivity losses do not influence the idiosyncratic risk
of households.

###  2.1 Economic Environment

Households
The economy is populated by a unit mass of infinitely lived agents whose
preferences are represented by:

$\sum_{t=0}^{\infty}\beta_{t}U(c_{t},f_{t})$

where

$U(c_{t},f_{t})=\frac{(c_{t}^{\phi}(f_{t}-\underline{F})^{1-\phi})^{1-\eta}}{1-\eta}$
(1)

Each agent maximizes utility (1) subject to the budget constraint

$\theta_{t}w_{t}+r_{t}a_{t}=c_{t}+p_{t}f_{t}+a^{\prime}_{t}$

$f_{t}$ describes food consumption and $c_{t}$ non-food consumption of the
household. $\eta$ is the coefficient of relative risk aversion (CRRA) and
$\beta$ is the discount factor. Note that the price of food $p_{t}$ is
expressed in units of non-food consumption. Agents can buy real, one-period
bonds $a_{t}$ paying an interest rate $r_{t}$.

The utility function has two properties that can be described as follows. The
first part is a Cobb-Douglas structure between food and non-food consumption.
This functional form implies a unitary substitution possibility between both
goods. At the same time, this utility function is non-homothetic as agents
have to satisfy at all times a subsistence level of food consumption
$\underline{F}$. It can be thought of as the minimal amount of calories a
human has to consume to avoid starvation. Thereby, the utility function has
the well known Stone-Geary properties (Stone, 1954; Geary, 1950) and are thus
non-homothetic. Non-homothetic preference imply that the Engel’s law holds.
Households spend a lower share on food, when households become richer. This
property holds over time, when households get richer or poorer over time due
to economic development. However, with heterogeneous agents Engels law holds
also within each time period. Finally, both components may also interact. When
poor households are close to the subsistence level, then substituting food
consumption with non-food consumption becomes more difficult.

This type of utility function implies nonlinear Engel curves and is widely
used in the literature (for example (Kongsamut et al., 2001; Jacobs and van
der Ploeg, 2019)), but not in the context of studying the distributive impacts
of climate-related food price increases. There are multiple benefits from
using this kind of utility function. First, the share of total consumption
spend on food is endogenous in the model. Thereby, we can analyze how these
shares change across the income distribution over time. Second, this kind of
utility function is still analytically tractable. Finally, we can use micro
data to estimate the parameters $\phi$ and $\underline{F}$ to replicate
consumption pattern that are consistent with their empirical counterpart.

As in the standard incomplete markets model, agents face idiosyncratic risk to
their labor productivity $\theta$ and experience transitory shocks to their
labor endowment. As households supply one unit of labor inelastically, the
time-varying labor productivity is the only source of difference in labor
income $w_{t}\theta_{t}$ across households. As markets are assumed to be
incomplete, this idiosyncratic labor risk is uninsurable. The income process
itself is exogenous to the model. Together with the possibility of being
borrowing constrained, this opens up a precautionary saving motive for
households (Aiyagari, 1994). The different history of labor productivity
realizations yields an endogenous wealth distribution. When agents experienced
a lot of low labor productivity realizations, their labor income is relatively
low and thereby it is difficult for these households to accumulate assets.
Note that this difficulty increase when the cost of living increase for
households.

Apart from the idiosyncratic labor endowment risk, the possibility of being
borrowing constrained is crucial for the precautionary savings motive. As in
Aiyagari (1994), we include a fixed borrowing limit in the form of
$a^{\prime}\geq\underline{a}$. We assume that the natural borrowing limit
$\underline{a}^{n}$ holds. Even if an agent has the lowest labor productivity
level $\theta_{min}$ at all periods (assuming that there is a positive
probability that an agent with any current $\theta$ will keep receiving
$\theta_{1}$), she can maintain a positive level of consumption. However, in
the presence of the subsistence level of food consumption $\underline{F}$, we
need to modify the natural borrowing limit. In our setup, it is not sufficient
that non-food consumption is non-negative, but food consumption must satisfy
the subsistence level. The budget constraint thus requires that:

$\underline{a}r+\theta_{1}w=a^{\prime}+\tilde{c}+p\underline{F}$

Once we solve for $\tilde{c}$ and assume it is slightly larger than 0. we get
the adjusted natural borrowing limit:

$\underline{a}^{n}=\frac{w\theta_{1}-p\underline{F}}{1-r}$

Production

The production side of the economy is kept simple and consists of two
symmetric sectors. The agricultural sector produces food and the other sector
produces the non-food consumption good. As both sectors are symmetric
production is described in both sectors by the following Cobb-Douglas
technology:

$Y^{j}=A_{j}K_{j}^{\alpha_{j}}L_{j}^{1-\alpha_{j}}$

where $j\in(c,f)$.

We assume that $\alpha_{c}=\alpha_{f}$ implies that the only difference
between the sectors will be the level of total factor productivity. With
levels of technology being the only difference, capital and labor are
perfectly mobile across sectors. Thereby, we can focus on the demand side
effects of rising food prices and isolate them from employment dynamics and
structural change. For the same reason, we exclude the role of land ownership.

Prices

Intuitively, we assume that the non-food sector is relevant for investment.
Considering the usual law of motion for capital, the setup implies that prices
$w,r$ are given by:

$r_{t}=1+A_{t}^{c}f^{\prime}(k_{t})-\delta$ (2)

$w_{t}=A_{t}^{c}[f(k_{t})-f^{\prime}(k_{t})k_{t}]$ (3)

The useful feature of this setup is that food prices $p_{t}$ are entirely
described by the Agricultural Productivity Gap $g_{apg}$, namely the
differences in total factor productivity level between the two sectors:

$p_{t}=\frac{A_{t}^{c}}{\tilde{A_{t}^{f}}}$ (4)

where $\tilde{A^{f}}=(1-\xi^{f})A^{f}$ is the agricultural productivity net of
the climatic shock. In the baseline, climate change affects only agricultural
productivity and $\xi^{f}$ describes the cumulative percentage loss due to
athropogenic climate change. By lowering agricultural productivity $A^{f}$ ,
climate impacts widen the technological gap between both sectors and increase
food prices. However, the parameter could be adjusted to account for trade
responses or other adaptation measures. The idea is that $p$ are net food
prices that are relevant for the consumer. In an extension, we consider
additionally impacts on non-agricultural productivity $A^{C}$.

###  2.2 Optimization and Steady State Equilibrium

The main decision of households is to choose total consumption expenditures
over time. How households split up consumption between food and non-food
consumption is static, meaning that it does not change over time. Instead of
having two control variables $(c_{f},f_{t})$, we can simplify the optimization
problem. It will be convenient to express the problem of households in
indirect utility, as it simplifies the optimization problem. There the only
control variable will be to choose income or total consumption expenditures,
$y^{exp}=c+pF$. It will be useful for the analysis later on to distinguish
between the decisions of households across and within periods. Using the
definition of total expenditures, we derive the demand functions for non-food
consumption and for food consumption:

$c_{t}=\phi(y^{exp}_{t}-p_{t}\underline{F})$ (5)

$f_{t}=(1-\phi)\frac{y^{exp}_{t}}{p_{t}}+\phi\underline{F}$ (6)

These demand functions describe how total consumption expenditures $y^{exp}$
are split up between food and non-food consumption within each period. Using
these demand function the corresponding indirect utility function to 1 is
described by:

$I(p,y)\equiv\frac{1}{1-\eta}\bigg{(}\Phi
p^{\phi-1}(y^{exp}-p\underline{F})\bigg{)}^{1-\eta}$ (7)

with $\Phi=\phi^{\phi}(1-\phi)^{1-\phi}$.

The optimization over the indirect utility function implies that the
intertemporal decision is how to choose the total consumption expenditures
$y^{exp}_{t}$, where as the intratemporal decision on how to split this
consumption up into food and non-food consumption is governed by the demand
functions derived above.

Given these prices the Bellman equation of households (in indirect utility)
is:

$V(a,\theta)=\max_{a^{\prime}\geq 0}\frac{1}{1-\eta}\bigg{(}\Phi
p^{\phi-1}\bigg{)}^{1-\eta}(ra+\theta
w-a^{\prime}-p\underline{F})^{1-\eta}+\beta
E_{\theta^{\prime}}V(a^{\prime},\theta^{\prime})$ (8)

subject to

$a^{\prime}\geq\underline{a}$ $r$ and $w$ taken as given.

The Bellman equation balances the current reward of choosing a specific level
of total consumption expenditures to the expected future reward. The future
reward is affected by idiosyncratic risk. Depending on the future expected
labor productivity $\theta^{\prime}$, agents will choose incomes over time.
The value function $V(a,\theta)$ considers the value a certain level of
consumption will yield given that decisions are taken optimally. Solving the
Bellman equations delivers policy functions. A policy functions describe
optimal decisions rather than optimal sequences of consumption. In the present
setting, policy function $y(a,\theta)$ describes the optimal level of savings
given current realization of the labor productivity and current level of
wealth.

Use the first order conditions and the Envelope condition to get:111A more
detailed derivation can be found in appendix A

$(y(a,\theta)-p\underline{F})^{-\eta}=\beta(y(a^{\prime},\theta^{\prime})-p\underline{F})^{-\eta}r$

where

$y(a,\theta)=ra+w\theta-a^{\prime}$

and

$y(a^{\prime},\theta^{\prime})=y(ra+w\theta-y(a,\theta),\theta^{\prime})$

We start by studying the stationary equilibrium, where the wealth distribution
is constant over time. However, even if the wealth distribution, and
aggregates are constant, there is still movement on the individual level,
where agents are hit by idiosyncratic shocks and adjust their savings behavior
accordingly. For the numerical solution of the model, we rely on the
endogenous grid method by Carroll (2006), which is widely used for solving
models with heterogeneous agents.

Next, we turn to the description of the stationary equilibrium. A recursive
competitive equilibrium consists of prices $r$,$p$ and $w$, the value function
$V(a,\theta)$, the optimal decision rule $y(a,\theta)$, the wealth
distribution $x(a,\theta)$, the aggregate capital stock $K$, and the aggregate
labor supply $L$, such that:

  1. 1.

Households: Given prices $r$,$p$ and $w$, the value function $V(a,\theta)$ is
a solution to the optimization problem of the consumer. $y(a,\theta)$ is the
optimal decision rule.

  2. 2.

Firms: Prices $r$, $p$ and w satisfy the following conditions:
$r=1+A^{c}f^{\prime}(k)-\delta$
$w=A^{c}[(f(k)-f^{\prime}(k))k]$
$p=A^{c}/\tilde{A^{f}}$

  3. 3.

Consistency: $x(a,\theta)$ is a stationary distribution consistent with the
optimal decision rule $y(a,\theta)$ and the associated Markov chain.

  4. 4.

Aggregation: The aggregate capital stock is consistent with the stationary
distribution $x(a,\theta)$ or:
$K=\sum_{i=1}^{n_{s}}\int_{A}a\hskip 7.11317ptdx(a,\theta)$

##  3 Calibration

The model described in the previous section is calibrated to a large sample of
developing countries. A strength of the model class we use is that many
parameter values can be estimated based on micro data. However, due to data
availability and quality issues in developing countries, we could not take all
parameter values from the data. For the calibration of the consumption
pattern, we relied on extensive household data. For the calibration of the
main climate impact channel, namely through agricultural productivity, we rely
on existing empirical estimates. The rest of the calibration follows the
standard approaches in the macroeconomic literature.

The calibration is summarized in table 1.

Table 1: Parameter Values Parameter |  Value
---|---
Production |
Capital Share: $\alpha$ |  0.36
Depreciation: $\delta$ |  0.08
Agricultural Productivity Gap: $g_{apg}$ |  2.49
Preferences & Consumption |
Discount factor: $\beta$ |  0.975
CRRA coefficient: $\eta$ |  2
Persistence of Income shocks $\rho$ |  0.23
Subsistence Food level $\underline{F}$ |  0.0564
Food preference parameter: $\phi$ |  0.8196
Climate Impacts |
Baseline Productivity Loss |  0.25
Low Productivity Loss |  0.11
High Productivity Loss |  0.4

###  3.1 Food Consumption and Preferences

The first step is to use micro data to match the consumption pattern of
households in the developing world. We rely on the Global Consumption Database
(The World Bank, 2023), which is the most comprehensive data on household
consumption expenditures covering 92 developing countries. These consumption
pattern are reported based on four consumption segments222Four consumption
segments are based on the following thresholds: Lowest$<\$2.97$ , $\$2.97<$
Low $<\$8.44$, $\$8.44<$ Middle $<\$23.03$, Higher $>\$23.03$ (per capita a
day). The task at hand is to estimate the subsistence level of food
consumption $\underline{F}$ and the preference parameter $\phi$. For the
estimation of $\underline{F}$ and $\phi$, one can rearrange the demand
function for non-food (5) and food consumption (6) to get an expression for
the food expenditure share:

$\frac{p*f}{y^{exp}}=\underbrace{(1-\phi)}_{\beta_{0}}\hskip
7.11317pt+\underbrace{p*\phi*\underline{F}}_{\beta_{1}}*\frac{1}{y^{exp}}$ (9)

Next, the model equation can be turned into a regression equation.

Using the expenditure data from the Global Consumption Database, we regress
the inverse of total consumption expenditures on the share spend on food in
the data. As indicated in equation 9, we then can calculate from the
regression coefficients the desired parameter values. In detail, from the
intercept we can calculate the preference $\phi$. Conditional on this value,
one can then calculate the subsistence level of food consumption
$\underline{F}$. Notably, as the price of non-food consumption in the model is
normalized to 1, a conversion factor is used to translate the subsistence
level into model units. For this, we use the value of mean total consumption
expenditures in the data. From the regression, we obtain $\phi=0.8196$ and
$\underline{F}=0.0564$.

###  3.2 Wealth and Income Inequality

The parameters of the income distribution are important for the endogenously
determined stationary wealth distribution. As standard in the literature, we
discretize the shock process and describe it as an N-state Markov chain. The
stationary distribution of this Markov chain determines the income inequality
in the steady state. Absent of aggregate uncertainty, the idiosyncratic risk
to individual labor productivity is exogenous to the model and not affected by
climate change.

The idiosyncratic labor productivity risk can be estimated from the data.
Where possible, both the standard deviation between labor productivity levels
and the probabilities to change the productivity level can be obtained using
household panel data. However, the panel data must have high quality and
coverage over a long time horizon on the income sources of households. Such
data are even rare in developed countries (Peralta-Alva et al., 2019). In the
context of developing countries, the problem goes beyond the quality of
available panel date. First, many households do not engage in employments with
formal wage income. For example, subsistence farmers do not receive a wage
each month that can be measured. Second, these arrangements are often in the
informal sector. Thus, it is difficult to obtain the income dynamics of
households in developing countries. Consequently, we abstract from estimating
the idiosyncratic shock process directly from household data.

In principle, we follow the calibration approach taken by Malafry and Brinca
(2022). Inequality in the stationary equilibrium depends on a initial wealth
endowment and the idiosyncratic labor productivity risk. Due to scarcity of
high quality data on inequality for all countries of the sample, the
calibration target is consumption inequality from the expenditure data used
for the calibration of the preference parameter. In detail, inequality
parameters are set to match the 80-20 ratio of total consumption expenditures
from the Global Consumption Database. While in the data the 80-20 ratio of
total consumption expenditures is $21$, the baseline steady state of the model
produces 80-20 ratio of $20.94$. Thus, the model is able to match consumption
inequality from the data well. The best model fit is achieved by using large
differences in the spread of income shocks and low persistence. Robustness
checks show that using more persistent income shocks with a lower spread does
not affect qualitatively the main results of the analysis.

###  3.3 Production and Agricultural Productivity

For the calibration, we are interested in the difference between the
productivity levels in both sectors. The difference in technology levels
determines the level of food prices in the baseline steady state. Following
Gollin et al. (2014), we refer to this difference as the agricultural
productivity gap. We use their database for the countries in our sample and
calculate the productivity gap for our sample and optain a value
$g_{apg}=2.49$. This value corresponds to the agricultural productivity gap
that accounts differences in hours worked and human capital.

The remaining parameters are chosen as standard in the literature. The capital
share $\alpha$ is taken from Peralta-Alva et al. (2019) and is set to $0.37$.
For the depreciation rate $\delta$, a range of values is applied within
macroeconomic models that are calibrated to developing countries. We take an
average of the values found in Peralta-Alva et al. (2019) and Gollin et al.
(2007) set the depreciation rate to $0.08$. As the economy there is calibrated
to a developing country, the parameter values chosen there seem to be
suitable. However, the insights of our analysis are not sensitive on the
calibration of these parameters.

###  3.4 Climate Change

Anthropogenic climate change has changed weather patterns globally in the past
50 years. These patterns are important for agricultural production. Most of
the empirical literature focus on estimating the effect of climate change on
yields of specific crops. However, due to our interest in sectoral dynamics,
we are interested in the cumulative impact on the agricultural productivity.
For the estimates of the cumulative impact of climate change on agricultural
productivity in our sample, we rely on past estimated effects of climate
impacts (Ortiz-Bobea et al., 2021), which are reported in first columns of
table 3.

The developing countries in our sample experience heterogeneous impact on
agricultural productivity. Hotter countries, for example from Sub-Saharan
Africa, tend to face large losses. In contrast, the losses in Central Europe
or Asia could be significantly lower. In the baseline scenario, we focus on
the aggregate productivity loss through climate change. First, we have to
consider how many households in our sample live in which region of the world.
This corresponds to the population-weighted average of the regional
productivity losses. Thereby, we control for the regional differences and
consider the average loss.

First, we consider their mean estimate as the baseline damage in our analysis.
Furthermore, we consider as a robustness check the ($10\%$,$90\%$)-percentile
as the optimistic and pessimistic estimate for climate-induced productivity
losses. Thereby, we consider the uncertainty regarding the true loss in three
different scenarios.With the baseline estimation agricultural total factor
productivity is expected to drop by $25\%$. In contrast, the optimistic
estimate yields a decline of only $11\%$, whereas the pessimistic estimate is
$40\%$. The estimated loss in agricultural productivity is large. However, the
final impact on productivity may additionally depend on regional substitution
possibilities such as the location or the specific crop. The estimate is
however broadly in line with future estimates net of adaptation. Hultgren et
al. (2022) find that climate change, accounting for adaptation, will in
regions lower agricultural productivity between 10 and 30 percent until 2100.
Thereby, the present analysis may provide insights for future damages as well.

Region |  Baseline |  Optimistic |  Pessimistic |  Population Share
---|---|---|---|---
Africa |  $-32.9$ |  $-19.8$ |  $-45.3$ |  $28.2$
Latin America & Caribbean |  $-30$ |  $-19.4$ |  $-44.1$ |  $12.8$
Near East & North Africa |  $-24.6$ |  $-14.7$ |  $-40.2$ |  $4.7$
Asia |  $-21.4$ |  $-12,4$ |  $-37.4$ |  $29.4$
Europe & Central Asia |  $-16$ |  $5.2$ |  $-33.4$ |  $25.4$
weigthed Averages |  $-11$ |  $-25$ |  $-40$ |
Table 2: Cumulative Impacts of Climate Change on Regional Agricultural Total
Factor Productivity (in percent).

##  4 Results

The aim of this quantitative exercise is to analyze the nexus between climate-
related sectoral productivity losses and long-run inequality. The starting
point of the analysis is to focus on past agricultural productivity losses due
to anthropogenic climate change. In the present analysis, we compare the
actually observed inequality to a counterfactual world, where climate change
did not affect agricultural productivity. In the counterfactual world, a
higher agricultural productivity results in lower food prices. The first part
of the quantitative exercise focuses on the distributive impact of changes in
food prices in general equilibrium. The focus lies on decomposing consumption
and income changes across the distribution. In a second step, we compare the
results to the partial equilibrium outcome, where total consumption
expenditures are held constant. The benefit of this exercise is that one can
use closed-form solution from the model to compare the welfare impact with
different levels of complexity. Thereby, one can isolate the impact of general
equilibrium effects on welfare of poor and rich households.

The final subsection focuses on implications for future climate change.
Climate change does not only affect agricultural productivity, but lowers
simultaneously non-agricultural productivity. However, it is unclear how total
productivity losses are distributed across sectors. The aim of this exercise
is to analyze the distributional impact for different sectoral allocations of
damages. An immediate consequence is that depending on the composition of
damages different damage channels are active. These channels are directly
linked to the agricultural productivity gap and differ in their impact on
inequality.

###  4.1 Distributive Impact in General Equilibrium

This section aims to analyze the distributive impact of past agricultural
productivity losses in general equilibrium. The present analysis relies on a
comparison between two steady states. The difference between both steady
states is the impact of anthropogenic climate change on agricultural
productivity. We rely on estimates from Ortiz-Bobea et al. (2021), who
calculate the cumulative impact of climate change on agricultural productivity
from 1960 to 2016. The benchmark includes realized agricultural productivity
losses. We compare this baseline to a counterfactual world, where climate
change did not lower agricultural productivity in the developing world.

In the world without climate damages, agricultural total factor productivity
$A_{f}$ is higher is higher. As the productivity in the non-agricultural
sector is unaffected, climate impacts affect the agricultural productivity
gap. In detail, climate damages increase the agricultural productivity gap and
thus food prices are higher (see equation (4)). Table 3 summarizes the
baseline results of key aggregate variables to the decline in agricultural
productivity due to climate change.333A comparison for the other low and high
estimate can be found in appendix B. When we compare the world with climate
change to a counterfactual world, where climate impacts did not affect
agricultural productivity, we can observe the following. From a macroeconomic
perspective, as a result of the agricultural productivity loss, output in both
the agricultural and the non-agricultural sector decline. However, the decline
in the agricultural sector is significantly larger. At the same time,
aggregate capital is higher in the world with climate impacts.

The increase in aggregate capital is related to uninsurable idiosyncratic risk
to labor productivity. The underlying savings mechanism in the standard
incomplete market model are precautionary savings due to idiosyncratic labor
productivity risk (Aiyagari, 1994). In the presence of idiosyncratic risk,
higher food prices affect additionally the precautionary savings motive. Apart
from affecting the current ability to save, higher food prices increase the
cost of living in periods, where households draw a low realization of the
labor productivity shock. Relatively to the low income realization, the
increase in cost of living is especially severe. Consequently, households want
to increase the buffer savings for periods with low labor income. In settings
with complete markets this channel is absent, because the individual labor
productivity risk is insurable. With complete markets, higher food prices lead
to lower aggregate capital accumulation. Changes in agricultural productivity
only affect capital accumulation dynamics when preferences are non-
homothetic.444See Irz and Roe (2005) for a comprehensive analysis of the
interaction between agricultural productivity and growth with complete
markets. In contrast, in the present analysis the steady state with climate
damages features a higher capital stock due to the increase in precautionary
savings.

Before we turn to a more detailed analysis of the distributive impact, it is
useful to analyze the underlying general equilibrium dynamics in more detail.
The general-equilibrium dynamics do not depend including heterogeneous agents.
The flow of reactions and adjustments are similar with a single representative
agent (see for example Irz and Roe (2005); Wichmann (2012)). Households react
to the changes in food prices by adjusting both their food 6 and their non-
food consumption 5.555This direct impact of climate-related food price
increases is studied in section 4.2 in more detail. However, apart from the
partial equilibrium response, the changes in demand for food and non-food
consumption affect production sectors. As demand for both goods change, also
demand in the non-food sector change, which is used for investment. As labor
is mobile across sector, this implies that the market-clearing wage rate and
the interest rate change. In a second step, households adjust their saving
decision in reaction these changes of factor prices. Consequently, steady
states differ in their capital stock and output, because climate change
permanently changed the productivity of agriculture. Note that these general
equilibrium effects through changes in wages and interest rates are entirely
driven by the initial change in demand and not by climate change affecting
wages and interest rates.666In a later exercise, we study the impact when
climate change additionally affects non-agricultural total factor
productivity.

Scenario |  $f^{8020}$ |  $\mu^{f}$ |  $y^{8020}$ |  $gini_{W}$ |  $\Delta Y^{F}$ |  $\Delta Y^{C}$ |  $\Delta K$
---|---|---|---|---|---|---|---
Baseline |  $-1.11$ |  $0.0311$ |  $-3.13$ |  $0.00008$ |  $-0.1303$ |  $-0.015$ |  $0.028$
Table 3: Baseline results for key aggregate variables: 80-20 ratio of food
expenditures, mean share of food expenditures, 80-20 ratio of total
consumption expenditures, Gini index wealth, agricultural output (percentage
change), non-agriculture output (percentage change), aggregate capital
(percentage change)

Next, we turn to the results that are related to the distributional impact. At
first glance, the distributive impact depends on the indicator. In a world
with climate impacts, the difference in consumption expenditures is smaller
between the richest and the poorest quintile, both for food expenditures and
for total consumption expenditures. However, the decline is smaller for food
expenditures, which is in line with the fact that food is a subsistence good.
Intuitively, households spend on average a higher share of their expenditures
on food,when food prices are higher. While inequality measured through total
consumption expenditures decreases, wealth inequality, measured by the Gini
Index, does not increase in response to agricultural productivity losses.

As we are interested to understand how households across the distribution
adjust their consumption and savings decisions, our setting includes
heterogeneous households. With heterogeneous households the aggregate demand
dynamics differ compared to the representative agent case. Figure 2 summarizes
the consumption pattern across the distribution for both steady states, thus
including general equilibrium effects. In detail, the graph describes how both
food and non-food consumption expenditure shares differ across the
distribution. Total consumption expenditures are normalized by the mean level
of total consumption expenditures. Thus, in the baseline steady state (solid
lines), a household with total consumption expenditures equal to half of the
mean level of total consumption expenditures spends 40 percent on food and 60
percent on non-food consumption. In contrast, in the world without climate
impacts, the same household will spend only 36 percent on food due to lower
food prices.777For the normalization of the new steady state, we use the mean
of the baseline steady state. Thereby, the consumption pattern can be compared
across steady states. However, not all households adjust their consumption in
a similar magnitude. In detail, households at the top and the very bottom of
the distribution adjust their consumption much less. Thus, intuitively,
households spend a larger share of their expenditures on food, when climate
impacts affect the agricultural sector.

Figure 1: Decomposition of income changes across the distribution for capital
and labor income

This partial equilibrium effect differs due to non-linear Engel curves in food
consumption. Households respond heterogeneously on the changes in food prices
depending on their level of total consumption expenditures. Poorer households
spend a larger share of their income on food and are thus more affected by
price changes even if they face identical prices as richer households.
Additionally, saving decisions will be affected heterogeneously across the
distribution, as households differ in their initial wealth level, as well as
in their dependency on capital and labor income. For understanding the long-
term impact of food price changes it is not sufficient to study consumption
expenditures.

Essentially, the long-term distributive impact of higher food prices is driven
by heterogeneous effects on incomes across the distribution. Depending on the
direction of these changes in income, this could amplify or alleviate the
short-term effect. For instance, if income in the steady state with climate
damages is higher, households may increase their total consumption
expenditures to cope with higher food prices. In general, there are two forces
that affect savings behavior of households. First, when food prices are high
the ability to save may be impacted due to non-homothetic preferences. Non-
homothetic preferences imply that changes in food prices affect the cost of
living of households directly. With higher costs of living it becomes more
difficult for households to accumulate assets. Additionally, higher food
prices affect the willingness to save in our setting. Thus, it is useful to
analyze the income side of households in more detail.

Figure 2: Consumption pattern across the distribution for steady state with
damages (solid) and for steady state without damages (dashed), expenditures
are normalized by mean level of total consumption expenditures.

In the steady state with damages, wages are slightly higher and interest rates
are lower due to a higher capital stock. These changes in factor prices, will
affect both labor and capital income. Figure 1 decomposes the income changes
of households across the distribution relative to the world without climate
change. For all households, total incomes are higher in the world with climate
change. However, households across the distribution differ in their income
sources. Intuitively, while richer households receive additional income from
their asset holdings, poor households depend on labor income. Labor income
$y_{inc,i}^{w}=w\theta^{i}$ is heterogeneous across the distribution. Even if
the wage rate is the same for all households, as labor is assumed to be mobile
across sectors, they differ in their individual labor productivity
$\theta^{i}$ due to uninsurable idiosyncratic productivity risk. However, as
we abstract from aggregate risk, the shock process for idiosyncratic risk is
unaffected by climate change. Thus, we compare the same realizations of the
labor productivity $\theta$ across steady states. Consequently, all households
will benefit from the wage gain, but to a different degree based on the level
of individual labor productivity.888Even in the steady state, individual labor
productivity is fluctuating constantly. The present comparison is conditional
on a specific realization of the shock.

Capital income $y_{inc,i}^{c}=ra^{i}$ is also heterogeneous across the
distribution, but on more dimensions. Even if the interest rate is common to
all households, the capital holdings $a^{i}$ are not. In contrast to the
realizations of the idiosyncratic labor productivity, households can adjust
their capital holdings. We decompose the changes in capital income according
to:

$\Delta y_{inc,i}^{c}=a_{NoDam}^{i}\Delta r+r_{Dam}\Delta a^{i}$

Both terms are specific to each household and together with the changes in
labor income explain the differences across the distribution. The first
component $a_{NoDam}^{i}\Delta r$ describes the income change due to the
changes in the return of assets. As the interest rate is lower in the new
steady state, this term tracks the income loss for agents conditional on fixed
initial capital holdings. This term increases with asset holdings. Thus, for
poorer households the absolute income loss from the change in the return is
small to zero, as they have very little or no asset holdings at all. The
second component $r_{Dam}^{*}\Delta a_{t}^{i}$ describes the change in income
that is associated with adjusting the asset holdings at the new return. As
previously discussed, in the presence of idiosyncratic risk all households
want to increase their savings. This term captures the the actual ability of
households to increase their savings. The results indicate that in the
baseline analysis most of the households are able to increase their asset
holdings, albeit at different scale. In detail, for the richest quintile the
change in asset holdings is the largest, while the poorest quintile still has
zero asset holdings. In the middle of the distribution, households are able to
increase their asset holdings as well.

When we study the demand-induced general equilibrium effects of climate-
related food price changes, most households are able to increase their
savings. The underlying driver are increases in labor income. Thus, poorer
households benefit from higher wage income, because in the presence of
idiosyncratic risk capital accumulation becomes more attractive. The aim of
the present exercise is not to imply that climate impacts increase wages of
the poorest households, but to isolate the impact of demand changes in the
presence of incomplete markets and idiosyncratic risk. Depending on the
strength of the precautionary savings motive and the size of the price
increase, wage and capital income may change differently depending on the
position in the distribution. However, higher incomes do not relate to higher
consumption and thus welfare. Higher consumption expenditures in a world with
climate impacts and thus higher food prices seems intuitive. To understand how
welfare is affected across the distribution, it is useful to study consumption
dynamics in more detail. Therefore, we turn next to the welfare analysis
including a comparison to the partial equilibrium outcome.

###  4.2 Comparison to Partial Equilibrium Outcome

The aim of this section is to compare the previously discussed to the short-
term, direct impact of higher food prices. The benefits are twofold. First, we
can isolate the strength of the general equilibrium channel. Furthermore, one
can study welfare changes across the distribution for different metrics. For
the evaluation of the direct impact of higher food prices on can use the can
use the steady state without damages to evaluate the direct effect of higher
food prices.

For the partial equilibrium outcome the total consumption expenditures of each
household are kept constant. Given the new prices, households can re-evaluate
how they will split it up between food and non-food consumption. As we keep
the total consumption expenditures for each household constant, the underlying
model allows for deriving these dynamics analytically. Recall that the demand
functions for food $f$ and non-food consumption $c$ are given by:

$c_{t}=\phi(y^{exp}_{t}-p_{t}\underline{F})$
$f_{t}=(1-\phi)\frac{y^{exp}_{t}}{p_{t}}+\phi\underline{F}$

Taking the partial derivative of the demand functions for food and non-food
consumption with respect to food prices yields:

$\frac{\partial c(p)}{\partial p}=-\phi\underline{F}$ $\frac{\partial
f(p)}{\partial p}=-(1-\phi)\frac{y}{p^{2}}$

For increasing food prices both derivatives are strictly negative. Thus, the
immediate response of households to higher food prices is to consume less of
both food and non-food goods. While the decrease of non-food consumption is
the same across the expenditure deciles, the decline in food consumption is
not. The decline increases with the level of total consumption expenditures.
Richer households reduce their food consumption more strongly than poor
households. Consequently, households end up adjusting differently their share
of total consumption expenditures spent on food:

$\frac{\partial f(p)p/y}{\partial p}=\frac{\underline{F}\phi}{y}$

This emphasizes the role of non-homothetic preferences. Poorer households
spend a larger share of their total consumption expenditures on food and are
closer to the subsistence level. The direct impact on consumption of food and
non-food consumption is larger than the observed changes including general
equilibrium effects. Notably, the difference between the partial and general
equilibrium analysis is largest for the poorest households. While this is an
indication that changes in total consumption expenditures matter for these
households, the consumption pattern cannot be translated directly to welfare.
Thus, we next compare changes in welfare across the distribution for the
partial and the general equilibrium outcome.

Figure 3: Change in consumption equivalent variation for the partial and
general equilibrium outcome and for an approximation based on the initial food
expenditure share.

Figure 3 shows the welfare outcome in partial equilibrium in terms of
consumption equivalent variation.999The formulas are included in appendix A.
In partial equilibrium, poor households suffer the most from lower food
prices, as they are closer to the subsistence level of food consumption and
initially spend a larger share on food. Households at the bottom of the
distribution are sensible to changes in wages due to their dependence on labor
income. These households do not own any wealth, they are insensitive to
changes in the interest rate. For the poorest households, the large real
income decrease from the direct effect is dampened by an increase in labor
income. The higher labor income is due to a higher wage rate in the new steady
state. Even if the increase in the wage rate is small, it has an impact on the
poorest households given their very low total income level. While poor
households can consume less for a given level of consumption expenditures
(partial equilibrium effect), they are partly able to cope with this by
increasing their total consumption expenditures due to higher wage income.

Similar to the poor, the welfare impact for households in the middle of the
distribution differs in partial and the general equilibrium. The difference is
driven by changes in total consumption expenditures. Households in the middle
of the distribution increase their consumption expenditures to cope with
higher food prices. While these households benefit too from higher wage
income, these households additionally receive capital income. In contrast to
the poorest households, households in the middle of the distribution are able
to increase their capital holdings. Thus, the difference in welfare between
partial and general equilibrium for these households is lower, as they
increase not only their total consumption expenditures, but also their capital
holdings due to a higher precautionary savings motive.

For the richest households, the difference in welfare between partial and
general equilibrium is negligible. First, the direct impact of higher food
prices on these households is small. In detail, as rich households spend a
smaller share on food, they are less affected by higher food prices.
Furthermore, the income of the richest households increase compared to the
baseline, but relatively less then for poorer households. Due to the stronger
precautionary savings motive and the smaller direct impact, these households
prefer to increase their capital holdings rather than their consumption
expenditures. Consequently, total consumption expenditures and thus the
welfare impact for these households does not differ significantly between the
partial and general equilibrium outcome.

We compare it to an approximation based on the initial consumption expenditure
share spend on food. The idea is to compare the partial equilibrium effect to
a reduced form approximation. The approximation is based on the initial food
expenditure share and the percentage change in food prices. Thus, the the
approximation relies simply on Engels law and does not use the explicit demand
functions. The results indicate that Engels law is a suitable approximation
for the short-term welfare impact of climate change on inequality through
changes in food prices. However, the initial share spent on food is not a
suitable predictor for how households will adjust their total consumption
expenditures when food prices increase. As the adjustment of total consumption
expenditures relies on heterogeneous income changes across the distribution,
the approximation is unable to capture the general equilibrium welfare impact.

Expenditure Decile |  Low Estimate |  Baseline |  High Estimate
---|---|---|---
1 |  $-0.071$ |  $-0.136$ |  $-0.189$
2 |  $-0.067$ |  $-0.128$ |  $-0.177$
3 |  $-0.039$ |  $-0.072$ |  $-0.098$
4 |  $-0.029$ |  $-0.051$ |  $-0.069$
5 |  $-0.0287$ |  $-0.05$ |  $-0.068$
6 |  $-0.027$ |  $-0.048$ |  $-0.063$
7 |  $-0.027$ |  $-0.46$ |  $-0.061$
8 |  $-0.019$ |  $-0.033$ |  $-0.043$
9 |  $-0.008$ |  $-0.007$ |  $-0.007$
10 |  $-0.008$ |  $-0.007$ |  $-0.006$
Table 4: Comparison of change in welfare between partial and general
equilibrium outcome for baseline productivity loss ($\xi^{f}=0.25$) for low
($\xi^{f}=0.11$) and high estimates ($\xi^{f}=0.4$). Change in welfare is
calculated as the difference in consumption equivalent variation between
partial and general equilibrium outcome.

Next, we perform a sensitivity analysis to the main result. The main focus of
the sensitivity analysis is the impact of climate change on agricultural
productivity. The estimation of the agricultural productivity losses include
some uncertainty. The hotter a region is the more uncertainty surrounds the
estimate of the productivity loss (Ortiz-Bobea et al., 2021). As more
developing countries in our sample are in hotter regions, which are associated
with higher uncertainty in the estimate, we consider three different
agricultural productivity losses. The baseline loss in the main analysis is
the mean estimate, where as here we consider the ($10\%$,$90\%$)-percentile as
the optimistic and pessimistic estimate for climate-induced productivity
losses. We consider this to be the main point for sensitivity analysis for
several reasons. First, it is the channel through which climate change enters
the analysis and determines fully the change in food prices. Furthermore, the
cumulative total factor productivity losses are significant in the baseline
analysis. In this sensitivity analysis, we want to ensure that the results are
not driven by the size of the decline. Finally, the main result builds on the
general equilibrium effects and the shock size is a main determinant of their
strength. As one can keep everything else constant between the steady states,
we abstract from changes to structural parameters as then also the baseline
steady state would change and we would be unable to do a comparison to the
main results.

Table 4 reports the results of the sensitivity analysis.101010A graphical
comparison of the difference in welfare per decile between partial and general
equilibrium outcomes for low and high estimate can be found in appendix C. The
main focus of the sensitivity analysis is the change in welfare. In detail, we
compare the difference in consumption equivalent variation between the patrial
and general equilibrium outcome for each expenditure decile. For example, if
the difference is negative, it indicates that the partial equilibrium welfare
loss is larger than the general equilibrium welfare loss. The results show
that the partial equilibrium welfare loss is larger in all scenarios. Thus, in
all scenarios, through a stronger precautionary savings motive, incomes of all
households increase. Intuitively, general equilibrium effects and thus the
impact on the incomes depend on the size of the productivity loss. A higher
productivity loss implies a stronger precautionary savings motive and through
higher savings o a higher wage income for all households. However,
quantitatively, we find differences across the distribution.

Across all sizes of productivity losses we can observe the strongest
differences between partial and general equilibrium effects at the bottom of
the distribution. Thus, the potential to overestimate the welfare loss of the
poor based on the direct impact of food prices increases with the size of the
productivity loss. Through higher incomes the poorest households, who are not
able to accumulate assets, the poor can adjust their total consumption
expenditures to cope with higher food prices. This adjustment in expenditures
drives the difference between partial and general equilibrium welfare for the
poorest. In contrast, households at the middle of the distribution will partly
also increase their asset holdings due to the stronger precautionary savings
motive. As these households still spend a significant share of their income on
food, they will increase their total consumption expenditures to cope with
higher food prices. However, as these households are able to increase their
savings, they will use part of their higher income to increase their asset
holdings due to a stronger precautionary savings motive. Thereby, households
at the middle of the distribution increase their buffer savings for periods
with low labor income, where due to climate change the cost of living (through
food prices) will be higher.

For the households at the top of the distribution, we do not observe
differences across the shock sizes. For these households the difference
between partial and general equilibrium analysis is similar across all
estimates. This indicates that these households are always able to cope with
the negative impact from higher food prices without adjusting their total
consumption expenditures. The richest households simply increase their asset
holdings due to a higher precautionary savings motive. Thus, for the richest
households the partial and general equilibrium welfare outcome is identical.

###  4.3 Climate Impacts and the Agricultural Productivity Gap

In the previous section, we compared the distributive impact of climate-
related food price changes in partial and general equilibrium. While the
previous sections relied on estimates of the past impact of anthropogenic
climate change on agricultural productivity losses, the underlying dynamics
are also relevant for future climate change. In the following section, we want
to explore how a potential bias towards agricultural productivity losses
affect the impact channels of climate change. As the sectoral composition of
damages is uncertain at best, we have to think what the implications of
asymmetric impacts are.

Figure 4: Changes Agricultural Productivity Gap (APG) depending on the
distribution of damages across sectors

In principle, climate change may lower the level of total factor productivity
in the agricultural sector $\tilde{A^{f}}=(1-\xi^{f})A^{f}$ and in the non-
agricultural sector $\tilde{A^{f}}=(1-\xi^{f})A^{f}$. The parameters $\xi^{f}$
and $\xi^{c}$ capture the respective loss in total factor productivity due to
climate change. However, it is uncertain if both sectors are affected equally
or if climate impacts affect one sector more strongly. In our setting, the
potential asymmetry in impacts is related directly to the agricultural
productivity gap. The agricultural productivity gap describes the difference
in productivity levels between agriculture and non-agricultural sectors.
Figure 4 illustrates how different distributions of damages across sectors
affect the agricultural productivity gap. Asymmetric climate impacts can widen
or lower the agricultural productivity gap. As we will see this will imply
different impact channels that can affect households heterogeneously.

The reference point is the case of symmetric productivity damages. With
symmetric damages agricultural and non-agricultural productivity are affected
equally, i.e. $\xi^{c}=\xi^{f}$, where $\xi^{f},\xi^{c}>0$. Consequently, the
agricultural productivity gap stays constant. In this case, the damage channel
is identical to a single sector model. There is no direct impact on households
through changes in food prices. Households are affected through changes in
factor prices. The strength of this impact depends on the size of the change
in non-agricultural productivity due to climate change ($\xi^{c}$). As
households depend heterogeneously on labor and capital income, the indirect
impacts can affect inequality. Even if damages do not hit households directly
inequality can increase in the long run.

Next, we turn to the case, where the agricultural sector bears a higher share
of the productivity losses, i.e. $\xi^{f}>\xi^{c}$. In this case, there are
two separate impact channels present. The first impact channel is the
previously described impact through changes in factor prices. The second
impact channel is related to the change in the agricultural productivity gap.
As agricultural productivity falls more than non-agricultural productivity,
the agricultural productivity gap widens. A higer agricultural productivity
gap translates to higher food prices. Thus, apart from the indirect impacts
through changes in factor prices, households are additionally affected by an
increase in consumption prices. The impact through consumption prices will
additionally affect factor prices. As seen in the previous section, where this
channel was studied in isolation ($\xi^{c}=0$), demand changes and
heterogeneous savings decision trigger a general equilibrium response.
However, when non-agricultural productivity is lower due to climate change
(but still less affected than agricultural total factor productivity), then
the demand-driven general equilibrium effects interact with the climate-
related impact on factor prices.

Figure 5: Changes in poverty indicators for different allocations of sectoral
productivity losses.

In the last case, the non-agricultural total factor productivity is affected
more by climate change, i.e. $\xi^{c}>\xi^{f}$. While in principle the same
impact channels are active, some interesting dynamics emerge. A higher impact
on non-agricultural productivity implies a lower agricultural productivity
gap. A lower agricultural productivity gap translates in our setup to lower
food prices. Thus, even if there is a strong impact on the income side of
households through changes in factor prices, lower consumption prices may
alleviate some of the impacts on the expenditure side.

In the following, we analyze the impact of each of these channels in
isolation. Thereby, we aim to provide a basic understanding of their potential
distributional impact. The reference point is the case of symmetric
productivity losses. We compare this baseline to the extreme cases. First, we
consider the case of the previous section, where damages only affect
agricultural productivity ($\xi^{c}=0$). In this case, we can study in
isolation the general equilibrium impact of demand side dynamics through
changes in consumption prices. Second, we consider the case where only non-
agricultural productivity is affected ($\xi^{f}=0$). In this case, households
benefit on the expenditure side from lower food prices, but face instead lower
incomes through climate impacts. Finally, in the case of symmetric damages,
the consumption expenditure damage channel is muted and households only are
affected through direct impacts on incomes.

Figure 5 summarizes the impact on poverty indicators for each of the
scenarios. In detail, the comparison includes cumulative productivity losses
from 10 to 25 percent for each of the scenarios. Productivity losses in the
non-agricultural sector are weighted by the initial output share of non-food
output. Thereby, the losses in agricultural and non-agricultural productivity
are comparable. In the present analysis, we rely on the wealth Gini index, an
80-20 ratio for total consumption expenditures, as well as the share of
households that have no wealth. The change in inequality in total consumption
expenditures captures the changes in consumption inequality, which is relevant
for welfare. In contrast, through the combination of the share of wealthless
households and the wealth Gini index allows for a more elaborate focus on
wealth inequality.

When climate damages hit only the agricultural productivity, the dynamics are
similar to the previous exercise. Higher food prices increase the
precautionary savings motive. However, not all households have the ability to
increase their savings. The poorest households, who initially do not hold any
assets, are not able to increase their asset holdings. Instead, these
households increase their consumption expenditures to cope with higher food
prices. As richer households are not affected that strongly by higher food
prices, richer households increase their asset holdings rather than their
consumption expenditures. Consequently, inequality measured by the 80-20 ratio
in total consumption expenditures declines, when climate impacts hit only the
agricultural sector. This effect increases with the size of the cumulative
productivity loss.

The dynamics differ when we focus on the asset-based inequality measures.
First, there is only a small increase in the share of wealthless households.
The modest increase in wealthless households can be traced back to households
that initially held very little wealth, but spend a large share of their
consumption expenditures on food. The increase in cost of living due to higher
food prices force these households to run down their buffer stock and to
increase their consumption expenditures instead to cope with higher food
prices. However, only very few households are trapped in these dynamics. This
holds for all sizes of productivity losses. Similar dynamics can be observed
for the Gini index. Even if rich households increase their asset holdings,
households in the middle of the distribution also increase their asset
holdings due to higher labor income. As the share of households at the bottom
of the distribution does not increase, the total impact on the Gini index is
limited. For all loss sizes, the Gini index for wealth remains unchanged, when
climate damages solely affect food prices.

In contrast, when climate-related productivity losses are equally distributed
across sectors, the agricultural productivity gap, and thus food prices,
remain unchanged. In this case, climate impacts lower directly both capital
and labor income. Without the hike in food prices inequality decreases
measured by the 80-20 total consumption ratio. However, compared to the
previous case, the decrease in expenditure-based inequality indicator is
driven by changes at the top of the distribution. When climate damages affect
incomes directly, rich households decrease their consumption expenditures more
drastically than the poorest households. This difference increases in the size
of the cumulative productivity loss. Intuitively, the poorest households are
close to the subsistence level and thus are less able to decrease their
consumption.

With symmetric climate-related productivity losses, the impact on wealth
inequality depends on the indicator. For all sizes of loss, the asset-based
metrics have opposite signs. While the Gini index for wealth decreases and
thus indicates lower wealth inequality, the share of households without any
wealth increases in all cases. The share of the population without any assets
increases up to 10 percentage points for the highest cumulative loss. When
climate damages affect incomes, the asset holdings of all households decline.
The decrease in asset holdings is not equally distributed across the
distribution. Richer households decrease in absolute terms more their asset
holdings. However, as the households with little wealth are affected as well,
the decline in the Gini index is modest. Thus, while the Gini index decreases,
the impact on households with little wealth is significant, when damages
affect only incomes.

Finally, we analyze the case, where climate damages fall entirely on non-
agricultural productivity. While the direct impact on incomes is higher than
in the symmetric case, households benefit from lower food prices, because
climate damages lower the agricultural productivity gap. Similar to the case
with symmetric damages, inequality measured by the 80-20 ratio of consumption
expenditures decreases. The decrease is driven by the reduction in consumption
expenditures of the richest households. Thus, the impact of climate change on
incomes dominates the benefits of lower food prices. As the impact on incomes
is stronger, when damages fall entirely on non-agricultural productivity, the
decrease increases with the size of the cumulative productivity loss. The
reduction in consumption inequality for the largest productivity loss is
larger than for the other two damage allocations. This supports how income-
related damages affect richest households stronger, because the poorest
households are close to the subsistence level.

The impact on wealth inequality differs compared to the expenditure-based
inequality metric. When damages affect only non-agricultural productivity, the
Gini index for wealth decreases for productivity losses up to 15 percent, but
increases for the case with the highest productivity loss. In contrast, the
share of households without any wealth increases in the size of the
productivity loss. For the lower and medium sized productivity loss, the
impact on asset holdings of richer households are more pronounced than for
households with little wealth. Consequently, the Gini index decreases in this
case. In contrast, for the high loss scenario, the impact on the households
with little wealth dominates and the Gini index increases. This is reflected
by the increase in the share of households without any wealth. For the highest
cumulative loss, the share of households without wealth increases by 15
percentage points. This stresses the importance of income-related damages for
wealth inequality.

##  5 Conclusion

Climate change poses a significant threat to economic growth and human
development. Agricultural productivity losses are not only important for
structural change, but may increase poverty through their impact on food
prices. However, the long-term distributive impact of climate impacts, is not
well understood in this context. In this paper, we addressed this research gap
and analyzed the long-run distributive impact of past climate-related
agricultural productivity losses. Climate impacts reduced agricultural
productivity and thereby increased food prices in the developing world.
Depending on the initial food expenditure share, higher food prices have a
direct distributive impact through the partial equilibrium effect. However,
food price increases can additionally have an indirect impact on incomes
through general equilibrium effects. General equilibrium effects matter for
the distributive impacts of climate-related agricultural productivity losses.
The results show that the direct impact of higher food prices effect
overestimates the impact for the poorest households. The size of the potential
bias in the welfare measurement increases in the size of the impact on
agricultural productivity loss. This highlights the importance of considering
general equilibrium effects, when analyzing the welfare impact of food price
changes. The differences can be traced back to heterogeneous changes in
incomes and thus in the level of total consumption expenditures of households.
Higher food prices increase the cost of living and thus the precautionary
savings motive. For most households the stronger precautionary savings motive
dominates the need to increase their total consumption expenditures. Only for
the poorest households, who are unable to accumulate assets, the need to
increase consumption expenditures dominates in order to cope with the higher
cost of basic food consumption.

Our analysis has important implications for future climate impacts. Depending
on the allocation of productivity losses across sectors, different damage
channels affect households. Climate impacts may affect households through the
impact on food prices, by lowering incomes directly or through both channels
simultaneously. The analysis shows how poverty is dynamic and
multidimensional. The distributive impact of climate change can differ
depending on the inequality indicator used. The results highlight the need for
using more than one indicator to capture the dynamic effects of climate
damages across the distribution. When climate damages solely affect food
prices, the impact on wealth inequality is limited. However, as soon as
climate impacts affect incomes, the share of households without any wealth
increases in all cases.

The main limitation of the present analysis is related to data availability.
Most applications of heterogeneous agent models of the Aiyagari-Bewley-Hugget
type focus on advanced economies for specific reasons. Without high quality
household panel data that contains detailed information on income sources and
wealth holdings, calibrating the model is subject to caveats. However, as the
present analysis shows idiosyncratic risk can be an important factor to
consider in the context of the developing world. Poverty is a multidimensional
phenomenon and heterogeneous agent macroeconomic models can be an important
tool to increase the methodological scope. We leave several important aspects
for future research. First, climate impacts will affect also other savings
motives apart from precautionary saving. Existing literature, indicates that
intergenerational aspects such as bequests important in explaining saving
behavior of households. Furthermore, the permanent decrease in productivity in
Agriculture has also implications for structural change. As capital and labor
are perfectly mobile across sectors in our model, agricultural capital and
labor shares decrease significantly in our model. However, mobility
restrictions could tie labor and capital to the relative unproductive
agricultural sector and decrease directly the income of the poorest households
and strengthen the adverse impact of the poorest households. Analyzing the
implications for structural change in a model with occupational choice and
labor market frictions could be promising.

Our analysis indicates how macroeconomic dynamics interact with household-
level distributive impact. Bridging the gap between research fields and
approaches is essential for improving our understanding of the long-term
distributive effects in the developing world. This would be an important step
towards achieving the sustainable development goals and improve the livelihood
of millions of poor households.

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##  Appendix A Analytical Derivations

###  A.1 Household Optimization

We start by deriving the demand functions for both goods (C,F). Use the
definition of income to get an expression for food:

$F=\frac{y}{p}-\frac{C}{p}$

Plug it into the utility function and take the derivative with respect to
consumption:

$U=\frac{(C^{\phi}(\frac{y}{p}-\frac{C}{p}-\underline{F}))^{1-\phi}}{1-\eta}$
$\frac{\partial U}{\partial
C}=(C^{\phi}(\frac{y}{p}-\frac{C}{p}-\underline{F}^{1-\phi}))^{-\phi}[\phi
C^{\phi-1}(\frac{y}{p}-\frac{C}{p}-\underline{F})^{1-\phi}+C^{\phi}(1-\phi)(\frac{y}{p}-\frac{C}{p}-\underline{F})^{-\phi}(-\frac{1}{p})]=0$

Now, rearrange and solve for C:

$\phi
C^{\phi-1}(\frac{y}{p}-\frac{C}{p}-\underline{F})^{1-\phi}=\frac{C^{\phi}}{p}(1-\phi)(\frac{y}{p}-\frac{C}{p}-\underline{F})^{-\phi}$

$\frac{\phi}{C}(\frac{y}{p}-\frac{C}{p}-\underline{F})=\frac{1-\phi}{p}$
$C(\frac{1}{1-\phi})=\frac{\phi}{1-\phi}(y-p\underline{F})$
$C=\phi(y-p\underline{F})$

Combining the demand function for consumption and the definition of income we
get the demand function for food:

$F=\frac{y}{p}-\frac{\phi(y-p\underline{F})}{p}=\frac{(1-\phi)y}{p}+\phi\underline{F}$

Next, we can plug both demand functions into the utility function and thus
express it in terms of the price and income. The indirect utility function
will become useful in the solution of the model and can be derived as follows:

$U(p,y)=\frac{((\phi(y-p\underline{F}))^{\phi}(\frac{(1-\phi)y}{p}+\phi\underline{F}-\underline{F})^{1-\phi}))^{1-\eta}}{1-\eta}$

$U(p,y)=\frac{1}{1-\eta}\bigg{(}(\phi(y-p\underline{F}))^{\phi}(\frac{(1-\phi)y}{p}-(1-\phi)\underline{F})^{1-\phi})\bigg{)}^{1-\eta}$

$U(p,y)=\frac{1}{1-\eta}\bigg{(}(\phi(y-p\underline{F}))^{\phi}(\frac{(1-\phi)}{p}(y-p\underline{F}))^{1-\phi}\bigg{)}^{1-\eta}$

$U(p,y)=\frac{1}{1-\eta}\bigg{(}(\phi^{\phi}(y-p\underline{F})^{\phi}(1-\phi)^{1-\phi}p^{\phi-1}(y-p\underline{F})^{1-\phi})\bigg{)}^{1-\eta}$

Define $\Phi=\phi^{\phi}(1-\phi)^{1-\phi}$ and simplify to get:

$U(p,y)=\frac{1}{1-\eta}\bigg{(}\Phi
p^{\phi-1}(y-p\underline{F})\bigg{)}^{1-\eta}$

We can write the optimization problem of households recursively with the
following Bellman equation:

$V(a,\theta)=\max_{a^{\prime}\geq 0}\frac{1}{1-\eta}\bigg{(}\Phi
p^{\phi-1}\bigg{)}^{1-\eta}(ra+\theta
w-a^{\prime}-p\underline{F})^{1-\eta}+\beta
E_{\theta^{\prime}}V(a^{\prime},\theta^{\prime})$

The first order condition is given by:

$\frac{\partial V}{\partial a^{\prime}}=\bigg{(}\Phi
p^{\phi-1}\bigg{)}^{1-\eta}(ra+\theta
w-a^{\prime}-p\underline{F})^{-\eta}(-1)+\beta\frac{\partial
V(a^{\prime},\theta^{\prime})}{\partial a^{\prime}}=0$

$\bigg{(}\Phi p^{\phi-1}\bigg{)}^{1-\eta}(ra+\theta
w-a^{\prime}-p\underline{F})^{-\eta}=\beta\frac{\partial
V(a^{\prime},\theta^{\prime})}{\partial a^{\prime}}$

We want to substitute the term $\frac{\partial
V(a^{\prime},\theta^{\prime})}{\partial a^{\prime}}$, because it is a
derivative for a yet unknown function. For that we can derive the Envelope
Condition.

$\frac{\partial V(a,\theta)}{\partial a}=\bigg{(}\Phi
p^{\phi-1}\bigg{)}^{1-\eta}(ra+\theta
w-a^{\prime}-p\underline{F})^{-\eta}R=\bigg{(}\Phi
p^{\phi-1}\bigg{)}^{1-\eta}(y(a,\theta)-p\underline{F})^{-\eta}R$

Update one period to get:

$\frac{\partial V(a^{\prime},\theta^{\prime})}{\partial
a^{\prime}}=\bigg{(}\Phi
p^{\phi-1}\bigg{)}^{1-\eta}(y(a^{\prime},\theta^{\prime})-p\underline{F})^{-\eta}R$

Plug it into the FOC to get:

$\bigg{(}\Phi p^{\phi-1}\bigg{)}^{1-\eta}(ra+\theta
w-a^{\prime}-p\underline{F})^{-\eta}=\beta\bigg{(}\Phi
p^{\phi-1}\bigg{)}^{1-\eta}(y(a^{\prime},\theta^{\prime})-p\underline{F})^{-\eta}R$

Plug it into the FOC to get:

$(y(a,\theta)-p\underline{F})^{-\eta}=\beta(y(a^{\prime},\theta^{\prime})-p\underline{F})^{-\eta}R$

where

$y(a,\theta)=ra+w\theta-a^{\prime}$

and

$y(a^{\prime},\theta^{\prime})=y(ra+w\theta-y(a,\theta),\theta^{\prime})$

###  A.2 Consumption Equivalent Variation

The equivalent satisfies the following condition:

Indirect Utility:

$I(p_{0},y_{0})=I(p,y+EV)$ $\frac{1}{1-\eta}\bigg{(}\Phi
p_{0}^{\phi-1}(y_{0}-p_{0}\underline{F})\bigg{)}^{1-\eta}=\frac{1}{1-\eta}\bigg{(}\Phi
p^{\phi-1}(y+EV-p\underline{F})\bigg{)}^{1-\eta}$
$p_{0}^{\phi-1}(y_{0}-p_{0}\underline{F})=p^{\phi-1}(y+EV-p\underline{F})$
$EV=(\frac{p_{0}}{p})^{\phi-1}(y_{0}-p_{0}\underline{F})-(y-p\underline{F})$

In Partial Equilibrium $y=y_{0}$, thus:

$EV_{PE}=(\frac{p_{0}}{p})^{\phi-1}(y_{0}-p_{0}\underline{F})-(y_{0}-p\underline{F})$

$EV_{PE}=(\frac{p_{0}}{p})^{\phi-1}y_{0}-\frac{1}{p})^{\phi-1}p_{0}^{\phi}\underline{F}-y_{0}-p\underline{F}$
$EV_{PE}=((\frac{p_{0}}{p})^{\phi-1}-1)y_{0}+(p-{p}^{1-\phi}p_{0}^{\phi})\underline{F}$

##  Appendix B Comparison Productivity loss

Scenario |  $f^{8020}$ |  $\mu^{f}$ |  $y^{8020}$ |  $gini_{W}$ |  $\Delta Y^{f}$ |  $\Delta Y^{c}$ |  $\Delta K$
---|---|---|---|---|---|---|---
Baseline |  $-1.11$ |  $0.0311$ |  $-3.14$ |  $0.00007$ |  $-0.1303$ |  $-0.015$ |  $0.028$
Low Loss |  $-0.47$ |  $0.0142$ |  $-1.4$ |  $0.0003$ |  $-0.06$ |  $0.001$ |  $0.0265$
High Loss |  $-1.77$ |  $0.0466$ |  $-4.83$ |  $0.0009$ |  $-0.1955$ |  $-0.003$ |  $0.0273$
Table 5: Comparison of baseline results ($\xi^{f}=0.25$) to low
($\xi^{f}=0.11$) and high estimates ($\xi^{f}=0.4$) for key aggregate
variables: 80-20 ratio of food expenditures, mean share of food expenditures,
80-20 ratio of total consumption expenditures, Gini index wealth, agricultural
output (percentage change), non-agriculture output (percentage change),
aggregate capital (percentage change)

##  Appendix C Additional Figures

Figure 6: Changes in consumption pattern across the distribution for the
partial equilibrium outcome Figure 7: Difference in welfare outcome between
partial and general equilibrium outcome for the low estimate ($\xi^{f}=0.11$).
A negative value indicates that the welfare loss in partial equilibrium is
higher than in general equilibrium Figure 8: Difference in welfare outcome
between partial and general equilibrium outcome for the high estimate
($\xi^{f}=0.4$). A negative value indicates that the welfare loss in partial
equilibrium is higher than in general equilibrium