2. Model

2.1. Model Ingredients

Three manifestations of food insecurity with increasing degrees of severity are: worry about food, inadequate diet quality, and insufficient calorie intake (Wilde, Reference Wilde2011). Economists have focused on the latter manifestations, which can be measured through objective indicators while the former is subjective as it depends on perceptions. When an individual is below the minimum calorie intake there is a severe disutility from the physical and psychological discomfort of hunger. The marginal utility of extra calories is therefore high for calorie-deprived individuals, leading them to acquire the cheapest available source of calories, typically cereals, roots and tubers. Once more favorable combinations of income and food prices lift individuals above subsistence calorie levels, the marginal utility of extra calories declines rapidly and they substitute away from staples, introducing better tasting and more expensive foods into their diets, like meat, fish and fruits.

Nonhomothetic preferences for food and non-food

For the third and most severe aspect of food insecurity we present a stylized partial equilibrium model of food insecurity. Households buy food and non-food to maximize utility from consumption. Those that fall below an exogenous food consumption level are classified food insecure. Since the model features only one type of food, we are abstracting from any diet composition channel that, in case of negative shocks, would slow down (but not reverse) households descent into food insecurity.Footnote ⁹ A mass one of households is indexed by $h$ on the continuum $h\in [0,1]$ . Each household has identical affine non-homothetic preferences, $U$ , over a food bundle, $C_F$ , and a non-food bundle of consumption, $C_{NF}$ , such that

\begin{gather*} U= \begin{cases} v(C_F) & \operatorname{for } 0\lt C_F\lt \underline {C_F}\\ u (C_F-\underline {C}_F,C_{NF}) & \operatorname{for } C_F\ge \underline {C_F} \end{cases} \end{gather*}

where $\underline {C}_F\gt 0$ is the food subsistence requirement, or basic need threshold, such that $\sup _{C_F} v(c_F)\lt \inf _{C_F,C_{NF}} u(C_F,C_{NF})$ , with $C_F$ and $C_{NF}$ defined in their respective support intervals. This utility function captures the notion that households have lexicographic preferences until food consumption reaches a basic need level.

We follow a number of recent papers in the literature on structural change, including Matsuyama (Reference Matsuyama2019), Comin et al. (Reference Comin, Lashkari and Mestieri2021), and Nath (Reference Nath2023), by assuming that subsistence utility $U$ is implicitly defined through the following nonhomothetic utility function:

(1) \begin{equation} \left ( \alpha ^{\frac {1}{\eta }} U^{\frac {\epsilon _{F} - \eta }{\eta }} (C_{F} - \underline {C_{F}})^{1-\frac {1}{\eta }} + (1- \alpha )^{\frac {1}{\eta }} U^{\frac {\varepsilon _{NF} - \eta }{\eta }} C_{NF}^{1-\frac {1}{\eta }} \right )^{\frac {\eta }{\eta - 1}} \equiv 1\quad \operatorname{ for } C_F\ge \underline {C_F} \end{equation}

where $\eta \in (0,1) \cup (1,\infty )$ and $\epsilon _F\gt 0$ ( $\epsilon _{NF}\gt 0$ ) are parameters representing the constant elasticity of substitution and the income elasticity of food (non-food) respectively, and where we have extended the isoelastic nonhomothetic CES function by including a food subsistence requirement. The latter adjustment has been made to ensure that under certain parameter restrictions (i.e. $\epsilon _{NF}=\epsilon _F=1$ ) equation (1) reduces to Stone-Geary CES, until recently the most popular choice for modeling nonhomothetic preferences.

Each household earns an income $y(h)$ , with households ranked on the continuum in ascending order such that $y(h^{\prime })\gt y(h^{\prime \prime }) \Leftrightarrow h^{\prime }\gt h^{\prime \prime }$ . Gross disposable income is $y^d(h) = y(h) + \tau (h)$ where $\tau :[0,1]\rightarrow \mathbb{R}$ is a redistributive tax scheme such that $\int _0^1 \tau (h)dh=0$ . Let $\mu _y$ denote average per capita income. Then social protection expenditures collected by (or taxes paid by) household $h$ are determined as follows $\tau (h)=-\kappa (y(h)- \mu _y)$ such that

(2) \begin{align} y^d(h)=(1-\kappa )y(h)+\kappa \mu _y. \end{align}

The variable $\kappa \in [0,1]$ controls the degree of redistribution and determines minimum income as a fraction of per capita income, $\kappa \mu _y$ . The size of the redistribution is thus proportional to $\kappa$ and to income dispersion. If $\kappa =0$ or $y(h)=\mu _y$ for all $h$ then $y^d=y$ .

Each household is a price taker in the food market. Let $p_F$ and $p_{NF}$ be the price of food and nonfood respectively. Then households maximize equation (1) subject to the budget constraint, $p_F C_F + p_{NF} C_{NF}=y^d(h)-p_F \underline {C_F}$ , resulting in the following demand for food and non-food respectively:

(3) \begin{gather} C_F(h)= \begin{cases} \dfrac {y^d(h)}{p_F} & \operatorname{for } C_F \lt \underline {C_F} \\ \underline {C_F} + \underbrace {\alpha \left ( \frac {y^d(h)-p_F \underline {C_F}}{P}\right )^{\epsilon _{F}-1} \left ( \frac {p_F}{P} \right )^{1-\eta }}_{\equiv m_F} \left (\frac {y^d(h)-p_F \underline {C_F}}{p_F}\right ) & \operatorname{for } C_F\ge \underline {C_F}\\ \end{cases} \end{gather}

(4) \begin{gather} C_{NF}(h)= \begin{cases} 0 & \operatorname{for } C_F \lt \underline {C_F}\\ \underbrace {(1-\alpha ) \left ( \frac {y^d(h)-p_F \underline {C_F}}{P}\right )^{\epsilon _{NF}-1} \left (\frac {p_{NF}}{P} \right )^{1-\eta }}_{\equiv m_{NF}} \left (\frac {y^d(h)-p_F \underline {C_F}}{p_{NF}}\right ) & \operatorname{for } C_F\ge \underline {C_F} \\ \end{cases} \end{gather}

where $m_F$ and $m_{NF}$ constitute the budget shares for food and non-food respectively, and where the price index $P=P(p_F,p_{NF},y^d(h))$ , satisfying $U=\frac {y^d(h)-p_F \underline {C_F}}{P}$ , is implicitly given by

(5) \begin{equation} \left [ m_F + m_{NF} \right ]^{\frac {1}{1-\eta }} \equiv 1 \end{equation}

Defining food insecurity

A household is undernourished if its calorie intake falls below a certain minimum $\underline {Q}$ , that is, $C_F \leq \underline {Q}$ , where $\underline {Q}$ is potentially higher than the subsistence or basic need threshold, i.e., $ \underline {C_F} = \lambda \underline {Q}$ with $\lambda \leq 1$ . Armed with this definition and making use of equations (3) and (2), we can then derive the following expression for the income level that coincides with the undernourishment threshold, that is,

(6) \begin{equation} y^{\operatorname{{PoU}}} = \frac {\tilde {\alpha } p_F \underline {Q} - \kappa \mu _y}{1-\kappa }, \end{equation}

where

(7) \begin{equation} \tilde {\alpha } \equiv \frac { \lambda \alpha ^{\frac {1}{\epsilon _F}} + \left ( (1-\lambda ) \underline {Q}^{1-\epsilon _F} \left ( \frac {p_F}{P} \right )^{\eta -\epsilon _F} \right )^{\frac {1}{\epsilon _F}} }{ \alpha ^{\frac {1}{\epsilon _F} } } \gt 0. \end{equation}

Note that $1 / \tilde {\alpha }$ is the share of nominal income that affords the household the minimum food basket $\underline {Q}$ . The right-hand side of equation (6) depends on $y^{\operatorname{{PoU}}}$ (via $P$ ), and thus represents an implicit solution to $y^{\operatorname{{PoU}}}$ . In the standard CES case ( $\epsilon _{NF}=\epsilon _F=1$ ), however, $P$ no longer depends on income and thus (6) represents an explicit solution.

Closing the model: inequality in urban and rural economies

Let income per capita $y$ be a continuous random variable with probability density function (PDF) $f(y;\; \boldsymbol{\theta })$ , cumulative distribution function (CDF) $F(y;\; \boldsymbol{\theta })$ , where $\boldsymbol{\theta }$ is a set of (typically two of three) parameters. The mean and standard deviation of the distribution are denoted by $\mu _y$ and $\sigma _y$ respectively. The CDF can be inverted to obtain the income level of a household with income rank $h$ , that is, $y(h)=F^{-1}\left (h;\; \boldsymbol{\theta } \right )$ . This function can be used to derive the Lorenz Curve, i.e., fraction of total income earned by the bottom $x$ % of the population,

(8) \begin{equation} L(x;\; \boldsymbol{\theta })\equiv \frac {\int _0^x F^{-1}\left (h;\; \boldsymbol{\theta }\right ) \,\operatorname{d}h}{\int _0^1 F^{-1}\left (h;\; \boldsymbol{\theta } \right ) \,\operatorname{d}h}, \end{equation}

which subsequently feeds into the calculation of the Gini inequality coefficient:

(9) \begin{equation} G(\boldsymbol{\theta })=1-2\left ( \int _0^{1} L(x;\; \boldsymbol{\theta }) dx\right ). \end{equation}

The share of the population that is undernourished is the fraction of households whose income is below the threshold income level $y^{\operatorname{{PoU}}}$ :

(10) \begin{equation} \operatorname{{PoU}} = F\left (y^{\operatorname{{PoU}}};\boldsymbol{\theta } \right ) \end{equation}

We consider CDF’s that satisfy the following conditions.

Assumption 1 (Growth is a rising tide that lifts all boats).There exists at least one parameter $\theta _i$ that maps into income per capita, $\mu _y=g(\theta _{i}, \boldsymbol{\theta _{-i}})$ with $\frac {\partial \mu _y}{\partial \theta _i}=\frac {\partial g }{ \partial \theta _i}$ and for which the following holds: $\frac {d F}{d \mu _y}=\frac {\partial F}{\partial \theta _i} / \frac {\partial \mu _y}{\partial \theta _i}\lt 0$ for all $y$ .

Assumption 2 (Growth preserves the income distribution).The CDF allows a closed form solution to the Gini coefficient, $G=G(\boldsymbol{\theta })$ , but is not affected by the growth parameter $\theta _i$ , i.e., $\frac {\partial G}{\partial \theta _i}=0$ .

Assumption1 says that the probability distribution is controlled by at least one parameter that allows us to increase GDP per capita and shift the entire CDF to the right such that the fraction of undernourished people unambiguously declines (“trickle-down growth”). This thought-experiment will look somewhat different for each distribution: for the normal distribution mean GDP per capita is identical to the location parameter $\mu$ , i.e., so that $\frac {\partial \mu _y}{\partial \mu }=1$ and $\frac {\partial F}{\partial \mu _y}=\frac {\partial F}{\partial \mu }\lt 0$ ; for the log-normal distribution the mean is also controlled by the location parameter $\mu$ (but not exclusively) with $\frac {\partial \mu _y}{\partial \mu }=\mu _y$ ; and for yet other distributions such as the Pareto Type 1 CDF both a scale and shape parameter satisfy Assumption1.

Assumption2 says that the parameter used to induce GDP per capita growth should not affect the degree of Gini inequality. This ensures that the results from comparative statics on the relationship between income per capita growth and the PoU are not conflated by changes in the income distribution. This assumption of “neutral income growth” also narrows down the number of parameters that can be used to induce GDP per capita growth, as for some distributions more than one parameter may satisfy Assumption1.Footnote ¹⁰

In addition to their effects on the affordability of the food minimums ( $\underline {Q}$ ), higher food prices may also affect food insecurity by boosting income per capita through windfall gains for farmers. To incorporate such an income effect, let us differentiate in a stylized manner between urban and rural economies. In the urban economy, as before, households have a claim on national income, as captured by the income distribution function. We assume that food prices do not affect income in the urban economy. In the rural economy, instead, the income distribution is an agricultural endowment distribution, i.e., it represents the heterogeneous claims that households hold to aggregate agricultural output or the total land endowment. Here higher food prices will boost household nominal income in proportion to each households’ claim on the country’s agricultural output. Let $\unicode {x1D7D9}{(Rural)}$ be an indicator function that’s equal to one when the condition between brackets, i.e., the economy is rural, is satisfied, and zero when the economy is urban.

Finally, for matters of convenience to simplify the comparative statics for urban and rural economies we assume any CDF to be “homothetic,” a quality usually preserved for utility and production functions. We can formally summarize these assumptions:

Assumption 3 (Higher global food prices increase income only in rural economies).In rural economies, but not in urban economies, household nominal income constitutes a claim on the agricultural endowment, i.e., $y(h)=p_F r_F(h)$ and $\mu _y = \int _{0}^{1} y(h) dh= p_F R_F$ where $R_F \equiv \int _0^1 r_F(h)dh$ , such that $\frac {d \mu _y}{d p_F} = \unicode {x1D7D9}{(Rural)} R_F$ .

Assumption 4 (Eligible CDFs are homogeneous of degree zero in in $y$ and $\mu _y$ .).CDFs must satisfy $F\left ( \lambda y;\; \lambda \mu _y \right ) = F( y;\; \mu _y )$ .For rural economies this implies:

(11) $\begin{align} \nonumber\\[-18pt]F\left (\frac {y}{p_F};\; R_F \right ) = F(y;\; p_F R_F ) = F(y;\; \mu _y) \\[-10pt]\nonumber\end{align} $

Two examples of CDFs satisfying Assumptions1, 2 and 4 are:

Example 1. The income (or agricultural endowment) CDF is log-normal:

$F(y;\mu ,\sigma )=\Phi \left (\frac {\ln (y)-\mu }{\sigma }\right )=\Phi \left (\frac {\ln (\frac {y}{\mu _y})+ \sigma ^2 /2}{\sigma }\right )$ , where $\Phi (x)$ is the standard normal distribution, and Gini is given by $G=2\Phi \left (\frac {\sigma }{\sqrt {2}} \right )-1$ .

Example 2. The income (or endowment) CDF is Pareto-like: $F(y;\mu ,\rho )=\left ( \frac {1}{1+\rho } \frac {y}{\mu } \right ) ^{1/\rho } = \left ( \frac {1}{1+\rho } \frac {y}{\mu _y} \right ) ^{1/\rho }$ .Gini is given by $G=\frac {\rho }{\rho +2}$ .

2.2. Comparative statics

Under Assumptions (1)–(4), we can then summarize the effects of changes in food prices and GDP per capita on the PoU in the following proposition.

Proposition 1 (Income, food prices, and the PoU. General case).Differentiating $\operatorname{{PoU}}$ with respect to $p_F$ , $\mu _y$ , and $\kappa$ , we can then state the following the results using semi-elasticities: Footnote ¹¹

(12) \begin{align*}{\quad\,\,\, \frac {\operatorname{d} \operatorname{{PoU}}}{\operatorname{d} p_F / p_F} =\left ( f\left (y^{\operatorname{PoU}}\right ) \underset { (\pm )}{\frac {\operatorname{d} y^{\operatorname{PoU}}}{\operatorname{d} p_F}} + \underset {(0/-)}{\frac {\partial F}{\partial \mu _y} \frac {\operatorname{d} \mu _y}{\operatorname{d} p_F}} \right ) p_F }\end{align*}

(13) \begin{align*}{\quad\,\, \frac {\operatorname{d} \operatorname{PoU}}{\operatorname{d} \mu _y / \mu _y} = \left ( \frac {\partial F}{\partial \mu _y} - f\left (y^{\operatorname{PoU}}\right ) \frac {\kappa }{1- \kappa } \right ) \mu _y \lt 0 }\end{align*}

(14) \begin{align*}{ \frac {\operatorname{d} \operatorname{PoU}}{\operatorname{d} \kappa / \kappa } = f\left (y^{\operatorname{PoU}}\right ) \left ( \frac { y^{\operatorname{PoU}} - \mu _y }{1-\kappa } \right ) \kappa \gtrless 0\quad \operatorname{ for } y^{\operatorname{PoU}} \gtrless \mu _y }\end{align*}

1. (i) Higher food prices change the nominal income level needed to buy the minimum calorie intake. This changes the PoU in proportion to the share of people $f\left (y^{\operatorname{PoU}}\right )$ that stand at the income threshold. In rural economies, the CDF also shifts to the right because higher food prices lead to windfall gains, subsequently raising GDP per capita. The overall effect of higher food prices is unclear.

2. (ii) Higher GDP per capita lowers the PoU by shifting the CDF to the right, and by increasing social benefits for households below average income.

3. (iii) an increase in the degree of income redistribution lowers the PoU provided income of the marginally food insecure household is below the mean.

Proof. Follows from Assumption (1)-(4) and inspection of the derivatives. See appendix.

Proposition1 examines how changes in food prices and GDP per capita affect food insecurity.Footnote ¹² Our focus here is on the extensive margin of food insecurity, which considers the overall share of affected households (i.e., those below the threshold), rather than the intensive margin, which looks at thedegree of insecurity within households as measured by their food consumption level. Consequently, the change in the PoU in response to income and food price fluctuations is shaped not only by parameters of the demand function (as is true for the intensive margin of food insecurity), but also by attributes of the income distribution. For example, a marginal increase in food prices affects food consumption for all households to some degree, but how many households will actually slide below the threshold as a result depends on the fraction of people $f(y^{\operatorname{PoU}})$ that stand at said income floor.

Under quite general conditions, Proposition1 tells us that while economic growth and larger social transfers unambiguously serve to lower the PoU, the effects of higher food prices are unclear. In order to obtain sharper predictions (even regarding the sign of the price elasticity), one must assume specific functional forms and put further restrictions on model parameters.Footnote ¹³ Furthermore, particularly sharp qualitative predictions can be obtained when preferences are further restricted to CES ( $\varepsilon _F=\varepsilon _{NF}=1$ ).Footnote ¹⁴ For ease of exposition we also mute the redistribution channel ( $\kappa =0$ ):

Proposition 2 (Income, food prices, and the PoU. CES case.).Consider Example 2 with CES preferences. The closed-form solutions to the PoU in urban and rural economies are:

(15) \begin{align*}{ \operatorname{PoU}_U\ = \left ( \frac {1}{1+\rho } \frac {\tilde {\alpha } (p_F/P) \underline {Q} }{R} \right ) ^{1/\rho } = f_U( \underset {+}{p_F},\underset {-}{R}) }\end{align*}

(16) \begin{align*}{\operatorname{PoU}_R\ = \left ( \frac {1}{1+\rho } \frac { \tilde {\alpha } (p_F/P) \underline {Q}}{ (p_F/P) R_F} \right ) ^{1/\rho } = f_R( \underset {-/+}{p_F} ,\underset {-}{R_F)} }\end{align*}

1. (i) In the urban economy, the positive price elasticity of the PoU is equal to or strictly smaller than the negative income elasticity in terms of magnitude iff $\eta \leq 1$ (i.e., $\frac {{PoU}}{\operatorname{d}p_F / p_F} \leq - \frac {\operatorname{d}{PoU}}{\operatorname{d}R/R} = \frac {1}{\rho } {PoU} \lt 0$ ).

2. (ii) In contrast, in the rural economy the food price elasticity of the PoU is positive only for $\eta \gt 1$ , negative for $\eta \lt 1$ , and zero for $\eta =0$ .The (agricultural) income elasticity of the PoU is identical to the income elasticity in the urban economy (i.e., $\frac {\operatorname{d}{PoU}}{\operatorname{d}R_F/R_F} = - \frac {1}{\rho } {PoU} \lt 0$ ).

Proof. Follows from inspecting the derivatives of equations (15)–(16). See also the appendix.

The food price elasticity of the PoU is governed by at most three sub-effects: a positive price effect and an expenditure share effect ( $\gtrless 0$ for $\eta \gtrless 1$ ) driving the demand for food, and a negative income effect for farmers (which is only present in rural economies). In the urban economy the food price elasticity is strictly positive as the price effect always dominates the expenditure share effect (which is negative when food and non-food are strict complements). This explains part (i) of Proposition2. In the rural economy, however, the income effect and the price effect perfectly offset each other, such that the expenditure share effect fully determines the sign of the food price elasticity.Footnote ¹⁵ If food and non-food goods are complements (substitutes), the food price elasticity of the PoU is negative (positive). This explains part (ii). In the real world, which is probably a blend of the stylized urban and rural models, we can then postulate that the food price elasticity can be negative, positive, or close to zero.

In addition to the key testable implications laid out in Propositions2, the model makes predictions on how the semi elasticities of food prices and income per capita change with the level of per capita income and inequality:

Proposition 3 (Structural characteristics and the semi-elasticity of PoU to income.).Consider again Example 2 with CES preferences. The responsiveness of the PoU to income:

1. (i) can become greater or weaker at higher levels of income, i.e., $\frac {\operatorname{d}}{\operatorname{d}\mu _y} \left ( \frac {\operatorname{d}{PoU}}{\operatorname{d} \mu _y/ \mu _y} \right ) = \frac {\operatorname{d}{PoU}}{\operatorname{d}\mu _y} \left ( \frac {\operatorname{d}{PoU} /{PoU}}{\operatorname{d}\mu _y/\mu _y} + \frac {\kappa }{1-\kappa } \frac {\mu _y}{y^{{PoU}}} \right ) \gtrless 0 \Leftrightarrow \rho \kappa \lessgtr \frac {\tilde {\alpha } p_F \underline {Q}}{\mu _y}$ .

2. (ii) can become greater or weaker in countries with a higher degree of income inequality, i.e., $ \frac {\operatorname{d}}{\operatorname{d} \rho } \left ( \frac {\operatorname{d}{PoU}} {\operatorname{d}\mu _y/ \mu _y} \right ) \gtrless 0 \Leftrightarrow {PoU} \gtrless e^{- \frac {2+\rho }{1+\rho }}$ .

Proof. See appendix.

Consider how the income elasticity of the PoU changes when income increases. There are two effects. On the one hand, there will be fewer households to push out of undernourishment at higher levels of income and so the income semi-elasticity weakens. On the other hand, the absolute amount of income redistribution increases at higher income levels. Proposition3 (i) says that in theory the overall effect could go in either direction. Using parameter values representative of a low-income country, our model predicts that in practice the reaction of PoU to income growth tends to decline at higher levels of income (see Figure 1), which we empirically confirm in Section 6. Figure 1 also shows that the relation between the semi-elasticity and key parameters is highly non-linear.

The effect of inequality on the semi-elasticity of PoU to income is also ambiguous. There are two opposing effects. First, there is a direct effect that increases the impact of GDP growth on PoU because PoU increases with inequality ( $\frac {d\operatorname{PoU}}{d\rho }\gt 0$ ) and the income semi-elasticity increases in magnitude with PoU. The second effect works in the opposite direction: when inequality is high an increase in GDP growth is less “inclusive,” that is, in absolute terms it is less sizeable for poor households, thereby moving a smaller mass of households around the food security threshold. A simple, low-income country calibration exercise (see also the next section) predicts that the semi-elasticity of PoU to income is larger in magnitude in more unequal societies (see Figure 1), which implies that the first effect dominates.

3. Model calibration

We calibrate our model to match key macroeconomic data moments and targets. This calibration yields alternative estimates and bounds for income and price semi-elasticities of food insecurity, complementing our regression results. The calibration incorporates additional data not used in the regressions, including Gini coefficients, social protection spending, and external demand parameters.

4. Data and descriptive statistics

From a measurement perspective, we focus on objective manifestations of food insecurity related to qualitative adequacy and quantitative sufficiency of diets (Wilde, Reference Wilde2011). To capture aspects of diet quality, we use the share of dietary energy from cereals, tubers and roots and the average supply of animal protein measured in grams per capita per day. For the most severe manifestation of food insecurity related to insufficient caloric intake we use the Prevalence of Undernourished (PoU). The PoU is defined as the share of a country’s population whose habitual food intake is insufficient to conduct an active and healthy life.

The PoU indicator as derived by FAO uses a log-normal probability density function for yearly dietary energy intake in the population, which requires the estimation of two parameters in order to fully characterize this distribution. The share of undernourished people in a country is then estimated as a cumulative probability of being below the minimum dietary energy requirements (MDER), i.e., the undernourishment threshold. More formally,

(17) \begin{equation} \operatorname{PoU}=\int _{x\lt \operatorname{MDER}} f(x|\Theta ) \,\operatorname{d}x \end{equation}

where $f(x|\Theta )$ is the log-normal distribution of caloric intake and $\Theta$ includes a location and a scale parameter. This function is intended to capture two dimensions of food security, namely food availability through the location (i.e., mean) parameter and differential food access through the higher moments (variance, skewness, and kurtosis). The mean is estimated every year from aggregate data on national food utilization accounts or from household survey data, when these are available. The other parameter and the MDER are estimated from micro data less frequently.

The literature has highlighted some limitations of the PoU, due to its top-down and aggregate approach. These limitations include the quality of the national food accounts data, the choice of a single undernourishment threshold for the whole country, and the lack of information on the severity of food insecurity. We refer the interested reader to FAO’s technical documentation for a detailed discussion (Cafiero, Reference Cafiero2014).

An alternative measure of food insecurity is the Food Insecurity Experience Scale (FIES), which overcomes the feasibility constraints faced by the PoU, as it does not require a costly dietary intake survey and the assumption of an undernourishment threshold. The main criticism against the FIES questions the comparability of perceived hunger experience across different individuals, countries and cultural contexts, while other authors highlight the possibility that it may overestimate food insecurity compared to insufficient-intake measures like the PoU (Barrett, Reference Barrett2010). We opt for the PoU as an objective and aggregate measure of national food insecurity, which has been selected by the United Nations to track progress towards the achievement of zero hunger by 2030. Further, the PoU has a much longer time coverage, spanning almost two decades, while the FIES is available only for three waves. This makes the PoU more suitable for our cross-country analysis on how food insecurity varies with business cycle fluctuations.

Data on GDP per capita and social protection expenditure come from the World Bank. We use real GDP per capita measured in constant 2015 dollars. Social protection expenditure in constant 2015 dollars includes current in-kind or cash transfers to households intended to provide for the needs that arise from social risks. They fall mostly under two broad categories: social security (e.g., unemployment benefits, retirement pension) and social assistance. Food inflation is defined as the year-on-year change in the food component of a country’s Consumer Price Index and is taken from the global inflation database constructed by Ha et al. (Reference Ha, Kose and Ohnsorge2023).

Our estimation sample comprises 142 countries across all continents and income groups and spans a period between 2001 and 2021. Countries with missing observations in a given year for the dependent or independent variables were dropped from the sample. We also drop observations if in a given year the country has experienced a high food inflation episode of above 50%, or a strong deflationary episode (bottom 10% of all observations). Our final estimation sample is an unbalanced panel.

The left-hand side graph of Figure 2 shows the evolution of the PoU in the sample period by income group. Low-income countries show the most significant rates of undernourishment, with a 30% incidence compared to the global average of 10%. This gap with the rest of the world has widened in the recent decade.

The right-hand side graph of Figure 2 gives a descriptive view of the relationship between undernourishment and real GDP per capita. Undernourishment decreases with GDP per capita, but it also becomes less responsive to income growth as countries become richer.

Table 3 describes the independent variables used in this study. GDP growth in the developing world has outpaced that in high-income countries. The level of food inflation in developing countries at the beginning of the observation period was almost double the one observed in the rich world (column 5), providing suggestive evidence of a positive association between food prices and undernourishment. The next column shows a secular reduction towards lower food inflation in upper-middle-income and high-income countries. Inflation is also least volatile in the group of high-income countries.

5. Empirical strategy

We are interested in estimating the relationship between aggregate food insecurity and its macroeconomic determinants. We can cast this relationship through a linear model:

(18) \begin{equation} y_{it}=\alpha +(\beta _0+\gamma _i)t+ \beta _1X_{it}+u_i+\epsilon _{it} {\\}, \end{equation}

where $y_{it}$ represents our measures of food insecurity for country $i$ at time $t$ , namely, the prevalence of undernourished, the share of dietary energy from cereals, roots and tubers, and animal protein supply. On the right-hand side of equation (18) the matrix $X_{it}$ includes our two main macro drivers, i.e., log real GDP per capita and food inflation (annual percentage change of the food component of the Consumer Price Index), as well as global and country time effects to control for phenomena that impact all countries in a similar fashion ( $t$ ). The composite error term includes country-specific time-invariant unobserved heterogeneity ( $u_i$ ) to capture geographical, historical or slow-changing institutional factors and a mean zero idiosyncratic shock ( $\epsilon _{it}$ ).

We adopt a first difference instrumental variables (FD-IV) estimation approach, to address concerns over omitted variables and reverse causality–which could lead to biased estimates from endogenous co-variation between the composite error term and the controls. In our setting, time-invariant unobservables ( $u_i$ ) may be correlated not only with food insecurity but also with the GDP and other right-hand side variables, potentially confounding the relationship. Taking the first difference of equation (18) eliminates $u_i$ , excluding the possibility of bias from correlation of $X_{it}$ with time-invariant confounders. This also allows to cast the relationship in terms of rates of change, since we are mostly interested in the short-term variations of food insecurity along the phases of the business cycle. It removes possible unit roots in the variables, making sure we are using stationary variables on both sides of the equation to avoid spurious findings. Finally, we utilize changes in food inflation rather than levels to net out the influence from monetary factors:Footnote ¹⁷

(19) \begin{equation} \Delta y_{it}=\beta _0+\theta y_{i0}+\beta _1 \Delta X_{it}+\Delta \epsilon _{it} , \end{equation}

where we assumed that $\gamma _i=\theta y_{i0}$ to capture a time-invariant regressor that controls for initial conditions allowing us to investigate whether there is convergence among countries towards some “equilibrium level” of food insecurity.Footnote ¹⁸ Countries that start out from higher levels of food insecurity should exhibit a faster pace of catching up.

Endogeneity may also stem from correlation of the controls with the idiosyncratic shock even after having netted out $u_i$ . This situation is often pervasive in a macro setting and potentially more problematic. For example, a given year’s shock to food security may feed back into next year’s fiscal budget as governments react quickly to distress situations by scaling up social protection spending, which in turn affects growth and prices (reverse causation). As a result, we may end up capturing a smaller (i.e. closer to zero) reaction of food insecurity to GDP growth than we would have in absence of this feedback effect. A similar result would be obtained for the coefficient of food inflation if governments reacted by introducing price controls for food items. All these scenarios invalidate the strict exogeneity assumption, which requires the error term to be uncorrelated with past, current and future values of the covariates ( $E[\Delta \epsilon _{it}|X_{is}]=0$ for $s=1,2..t..T$ ). The resulting least squares estimates of equation (19) would be biased towards zero.Footnote ¹⁹

We use external instruments for each endogenous variable to isolate plausibly exogenous variation. We instrument income growth with the average growth rate of a country’s trading partners and the change in the commodities terms of trade (Acemoglu et al. Reference Acemoglu, Johnson, Robinson and Yared2008; Burke, Reference Burke2012). In the average growth rate of trading partners the year-specific weights are given by the share in the previous three years of the country’s export to a given trading partner on total exports during the same period. For the second instrument of growth we use IMF’s commodity terms-of-trade (CTOT) index, which is the average of a range of commodity prices, each of which is weighed by the country’s share of net exports of a given commodity on its GDP over the previous three years (Gruss and Kebhaj, Reference Gruss and Kebhaj2021). (Details on the instruments are provided in the appendix).

Finally, food inflation is instrumented with a regional and a global (or “rest-of-world”) harvest shock. We use a Hodrick-Prescott filter to extract regional and global deviations from per capita trend production of (the calorie-weighted sum of) four grains - wheat, corn, soybeans, rice - which jointly provide more than two thirds of calories for human consumption, either directly or indirectly as livestock feedstuff. The regional supply shock for country $i$ represents the percentage deviation from per capita trend production across all countries within the region, except country $i$ . The rest-of-world shock for country $i$ represents the percentage deviation from total per capita trend production in all countries outside the country’s own region. To aggregate by region we follow the World Bank classification, which distinguishes between 7 regions: East Asia and Pacific, Europe and Central Asia, Latin America and Caribbean, Middle East and North Africa, North America, South Asia, and Sub-Saharan Africa (SSA). In the next section, we provide evidence of instrument relevance ( $E[X_{it}|Z_{it}] \neq 0$ ) and validity ( $E[\Delta \epsilon _{it}|Z_{is}]=0$ for $s=1,2..t..T$ ).