5 Galor and Özak (2016)
Galor and Özak (2016) provide empirical evidence that differences in time preference (how much people discount the future) may have agricultural origins. This is important for development because more patient, future-oriented individuals generally have a greater propensity to save. Although the paper’s major contribution is empirical, the authors develop an OLG model from which they derive testable implications.
The theoretical model shows how the composition of a population can be modelled using the OLG framework. The most influential model in that regard is Bisin and Verdier (2001). However, the model in Galor and Özak is relatively simple and illustrates well some population dynamics.
5.1 The model
We use an OLG framework and assume that the economy is agricultural and at a very early stage of development. In every period, the economy consists of individuals who live for three periods.
- During the first period of life, individuals are children and are economically passive; their parents provide their consumption.
- In the second and third periods of life, individuals work.
- All individuals can choose between two modes of production:
- Endowment mode: it provides an equal payoff during the second and third periods of life. For instance, individuals may be hunters.
- Investment mode: it pays little during the second period of life, but the payoff during the third period is much larger. This represents farmers, who must sow seeds and wait for crops to grow.
Lastly, a crucial assumption of the model is the lack of financial markets and long-term storage technology. This implies that individuals cannot transfer consumption between periods two and three. Hence, output produced in each working period must be consumed in that period.
5.1.1 Production modes
All adults must choose either the endowment mode or the investment mode. The endowment mode provides a constant level of output, \(R^0>1\), in each working period. If investment is instead chosen, it requires an investment during the first working period, which implies that fewer resources are available for consumption. In particular, we assume that it leaves individuals with \(1\) unit of consumption during their second period of life. However, the output it yields during the third period, \(R^1\), is higher than under endowment: \(\ln(R^1) > 2 \ln(R^0)\).
Finally, depending on the chosen production mode, the income of individual \(i\) is given by:
\[ (y_{i,t}, y_{i,t+1}) = \begin{cases} (R^0, R^0) & \mathrm{if\, endowment} \\ (1, R^1) & \mathrm{if\, investment} \end{cases} \]
5.2 Preferences
A key mechanism generating dynamics in the evolution of individual traits is the fertility decision. This approach is common: typically, fertility is linked to a trait through income. That is, individuals with more income will be able to have more children. If the trait is transmitted from parents to children, then the trait associated with greater income will become increasingly prevalent in the economy.1 Preferences are therefore important in this type of model because fertility decisions are derived from them.
In every period \(t\), a generation of size \(L_t\) becomes economically active; that is, it reaches the second period of life. Those individuals were born in period \(t-1\). At this stage, each individual will live for two more periods. Recall that financial markets do not exist and that it is impossible to transfer resources between periods through storage.
We assume that, during the second period of life, individuals only consume what they produce. During the third and final period, individuals consume and have children. In particular, utility is given by: \[ u^{i,t} = \ln c_{i,t} + \beta^i_t [\gamma \ln n_{i,t+1} + (1-\gamma) \ln c_{i,t+1}],\quad \gamma \in(0,1), \]
where \(c_{i,t}\) and \(c_{i,t+1}\) are consumption in the second and third periods of life and \(n_{i,t+1}\) is the number of children. The parameter \(\beta^i_t \in (0,1]\) is individual \(i\)’s discount factor: it measures how much the individual values the future relative to the present. The larger \(\beta^i_t\) is, the more the individual values the future and, hence, the more patient the individual is. Notice that \(\beta^i_t\) varies across individuals and can evolve over time.
During the second period, individuals do not really make any decision: since resources cannot be transferred, all production must be consumed. Hence, \(c_{i,t} = y_{i,t}\). However, during the final period, individuals can trade off utility from consumption against utility from children. The paper assumes that each child costs \(\tau\) units of consumption, which gives rise to the final-period budget constraint: \[ y_{i,t+1} = c_{i,t+1} + \tau n_{i,t+1}. \]
Given these preferences, utility maximisation implies: \[ c_{i,t+1} = (1-\gamma) y_{i,t+1}, \] \[ n_{i,t+1} = \frac{\gamma}{\tau} y_{i,t+1}. \]
Lastly, the indirect utility (\(v_{i,t}\)) of individual \(i\) is given by: \[ v_{i,t} = \ln y_{i,t} + \beta^i_t [\ln y_{i,t+1} + \xi],\quad \xi \equiv \gamma \ln\left(\frac{\gamma}{\tau} \right) + (1-\gamma)\ln(1-\gamma). \]
5.3 Hunters or farmers
Since individuals can decide on their mode of production, they are free to choose to become either hunters or farmers. That is, each individual chooses the mode of production (endowment or investment) that maximises lifetime utility. Hence,
\[ v_{i,t} = \begin{cases} \ln R^0 + \beta^i_t (\ln(R^0) + \xi) & \mathrm{if\, endowment} \\ \ln 1 + \beta^i_t (\ln(R^1) + \xi) & \mathrm{if\, investment} \end{cases}. \]
An individual is indifferent between modes of production if they obtain the same utility from both. That is, the individual with \(\beta^i_t = \hat{\beta}\) is indifferent between becoming a hunter and becoming a farmer if and only if \[ \ln R^0 + \hat{\beta} (\ln(R^0) + \xi) = \ln 1 + \hat{\beta} (\ln (R^1)+\xi). \]
Solving for \(\hat{\beta}\) allows us to identify this individual:2
\[ \hat{\beta} = \frac{\ln R^0}{\ln R^1 - \ln R^0} \in (0,1). \]
Thus, all individuals with \(\beta^i_t < \hat{\beta}\) optimally choose the endowment technology, while those with \(\beta^i_t > \hat{\beta}\) find the investment technology optimal. Note that, as the return to agriculture increases (\(R^1\) increases), the cutoff value \(\hat{\beta}\) decreases:
\[ \frac{\partial \hat{\beta}}{\partial R^1} = \frac{-\ln R^0}{R^1(\ln R^1 - \ln R^0)^2} < 0, \]
so, as agriculture becomes more profitable, more individuals find it optimal to become farmers.
Hence, we can rewrite an individual’s income as a function of \(\beta^i_t\):
\[ (y_{i,t}, y_{i,t+1}) = \begin{cases} (R^0, R^0) & \mathrm{if\, \beta^i_t \leq \hat{\beta}} \\ (1, R^1) & \mathrm{if\, \beta^i_t > \hat{\beta}} \end{cases}. \]
Because final-period income differs, hunters and farmers have different numbers of children. In particular, using the optimal number of children derived above:
\[ n_{i,t+1} = \frac{\gamma}{\tau} y_{i,t+1} = \begin{cases} \frac{\gamma}{\tau}R^0\equiv n^E & \mathrm{if\, \beta^i_t \leq \hat{\beta}} \\ \frac{\gamma}{\tau}R^1\equiv n^I & \mathrm{if\, \beta^i_t > \hat{\beta}} \end{cases}. \]
Because \(R^1 > R^0\), farmers have more children than hunters.
5.4 The evolution of preferences
Finally, we can trace how preferences change over time due to the differential fertility of farmers and hunters. If time preferences \(\beta^i_t\) are transmitted from parents to children, farmers’ higher fertility causes their share of the population to increase over time. The paper makes essentially this assumption but modifies the transmission of preferences for individuals engaged in farming. In particular: * \(\beta^i_t\) is perfectly transmitted if an individual is a hunter. * Farmers transmit a weakly larger value of \(\beta^i_t\) to their children, with equality at the fixed point, reflecting an acquired tolerance for waiting and delaying reward. Specifically,
\[ \beta^i_{t+1} = \begin{cases} \beta^i_t & \mathrm{if\, \beta^i_t \leq \hat{\beta}} \\ \phi(\beta^i_t, R^1) & \mathrm{if\, \beta^i_t > \hat{\beta}} \end{cases}, \]
where \(\phi\) has the following properties:
- \(\phi(\beta,R^1)<1\) for \(\beta\in[\hat{\beta},1]\), so the transmitted discount factor remains below one.
- Along the transition to the steady state, \(\beta^i_t \leq \phi(\beta^i_t,R^1)\): the child’s discount factor is weakly greater than the parent’s. Equality means that the dynasty has reached a fixed point.
- \(\phi(\hat{\beta},R^1) > \hat{\beta}\), so transmission strictly raises the discount factor at the farming cutoff.
- \(\phi_\beta(\beta,R^1)>0\): the transmitted discount factor is increasing in the parent’s discount factor.
- \(\phi_{\beta\beta}(\beta,R^1)<0\): the marginal effect of the parent’s discount factor decreases as that factor rises.
- \(\phi_R(\beta,R^1)>0\): a higher agricultural return raises the discount factor transmitted to the child.
Suppose an individual at the beginning of time has \(\beta^i_0 < \hat{\beta}\). This individual optimally chooses to be a hunter, and the transmission process implies that the individual’s children inherit \(\beta^i_1 = \beta^i_0\). Because \(\hat{\beta}\) is constant over time, all descendants also choose to be hunters and transmit the same time preference. Hence,
\[ \beta^i_0 \leq \hat{\beta} \implies \lim_{t\rightarrow \infty} \beta^i_t = \beta^i_0. \]
Suppose instead that \(\beta^i_0 > \hat{\beta}\) and lies below the farmer-dynasty fixed point defined below. The individual becomes a farmer and transmits \(\phi(\beta^i_0,R^1) > \beta^i_0\). All descendants are therefore also farmers, and their discount factors rise weakly until they converge to the fixed point:
\[ \lim_{t \rightarrow \infty} \beta^i_t = \bar{\beta}^I. \]
Thus, \(\bar{\beta}^I\) is the limiting discount factor for these farmer dynasties, not a maximum over every feasible initial discount factor.
5.4.1 Proof (not in the paper)
We want to show that \(\beta^i_{t+1} = \phi(\beta^i_t,R^1)\) has a unique steady state on \([\hat{\beta},1]\). This amounts to showing that \(\bar{\beta}^I = \phi(\bar{\beta}^I,R^1)\) for a unique \(\bar{\beta}^I\) in this interval. Define
\[ G(\beta) = \phi(\beta,R^1) - \beta. \]
The assumptions imply
\[ G(\hat{\beta}) = \phi(\hat{\beta},R^1) - \hat{\beta} > 0 \]
and
\[ G(1) = \phi(1,R^1)-1 < 0. \]
Continuity therefore guarantees a root \(\bar{\beta}^I\in(\hat{\beta},1)\). Moreover,
\[ G''(\beta)=\phi_{\beta\beta}(\beta,R^1)<0, \]
so \(G\) is strictly concave. Once a strictly concave function falls from the positive value \(G(\hat{\beta})\) to a root, its subsequent secant slopes are negative; it therefore remains below zero and cannot cross the horizontal axis again. The root \(\bar{\beta}^I\) is thus unique. For \(\beta^i_0\in(\hat{\beta},\bar{\beta}^I]\), the transmission sequence is nondecreasing and, because \(\phi\) is increasing, bounded above by \(\bar{\beta}^I\). Its limit must satisfy the fixed-point equation and hence equals \(\bar{\beta}^I\).
5.5 Evolution of traits over time
Lastly, suppose that, at time \(t=0\), the population has different levels of time preference. We assume that initial traits are characterised by a density \(\eta(\beta^i_0)\) with support \([0,\bar{\beta}^I]\). Furthermore, we normalise the initial generation to be of size one: \(L_0 = 1.\) Equivalently,
\[ L_0 = \int_0^{\bar{\beta}^I} \eta(\beta^i_0)\,\mathrm{d}\beta^i_0 = 1. \]
We also know that all individuals whose \(\beta^i_0 \leq \hat{\beta}\) decide to use the endowment technology, while the remaining individuals opt for the investment technology. Therefore, the sizes of the hunter (E) and farmer (I) groups are given by
\[ L^E_0 = \int_0^{\hat{\beta}} \eta(\beta^i_0)\,\mathrm{d}\beta^i_0, \]
\[ L^I_0 = \int_{\hat{\beta}}^{\bar{\beta}^I} \eta(\beta^i_0)\,\mathrm{d}\beta^i_0. \]
The number of individuals evolves according to each group’s fertility rate: \[ L_t^E = L_0^E(n^E)^t = \left(\frac{\gamma}{\tau}R^0\right)^tL_0^E, \]
\[ L_t^I = L_0^I(n^I)^t = \left(\frac{\gamma}{\tau}R^1\right)^tL_0^I, \]
and total population is \(L_t = L_t^E + L_t^I.\)
Finally, notice that the distribution of \(\beta^i_t\) does not change within the endowment group, whose members have \(\beta^i_t \leq \hat{\beta}\): they all have the same number of children, and each child inherits the parent’s trait. Therefore, the average value \(\bar{\beta}^E_t\) is constant over time. In contrast, the average value \(\bar{\beta}^I_t\) increases toward \(\bar{\beta}^I\). At any period \(t\), the overall average value of the discount factor is given by
\[ \bar{\beta}_t = \theta^E_t \bar{\beta}^E_t + (1-\theta^E_t)\bar{\beta}^I_t, \]
where \(\theta^E_t\) is the fraction of individuals who engage in the endowment production process and \(\bar{\beta}_t\) is the population-weighted average discount factor.
\[ \theta^E_t = \frac{L_t^E}{L_t^E + L_t^I} = \frac{(R^0)^t}{(R^0)^t+(R^1)^t\frac{L^I_0}{L^E_0}}. \]
Hence, as time advances, the share of the population engaged in the endowment production process approaches zero:
\[ \lim_{t\rightarrow\infty}\theta^E_t = 0. \]
This process reflects the endowment group’s lower reproductive success.