# Galor and Özak, 2016

## Introduction

In this paper, Galor and Özak show empirically how differences in time preference (how much people discount the future) has an agricultural origin. This is important for development, because those who save more are the individuals who are more patient, this is, who are more future-oriented. Although the major contribution of the paper is empirical, the authors develop an OLG model from which they derive implications that are tested.

The theoretical model shows how the composition of a population can be modeled using the OLG framework. The most influential model in that regard is Bisin, A., & Verdier, T. (2001). The Economics of Cultural Transmission and the Dynamics of Preferences. Journal of Economic Theory, 97(2), 298–319. However, the model in Galor and Özak is relatively simple and illustrates well some population dynamics.

## The model

We work under the OLG framework, and we assume that the economy is agricultural and at the very early stages 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. Consumption during this period is provided by parents.
- In the second period and third periods, individuals work
- All individuals can choose between two modes of production:
- Endowment mode: it provides an equal pay-off during the second and third periods of live. For instance, individuals may be hunters.
- Investment mode: it pays little during the second period of live, but the pay-off during the third period is much larger. This represents farmers: they must seed and wait for crops to grow.

Lastly, a crucial assumption of the mode is the **lack of financial markets and long-term storage technology**.
This implies that individuals **cannot** transfers consumption between periods two and three.
Hence, production in the second period has to be consumed in the second period; and consumption in the third period must be consumed in the third period.

### Production modes

All adults must decide between the *endowment mode* or the *investment mode*.
The endowment mode provides a constant level of output,$R^0>1$ in each of the working periods.
If investment is instead chosen, it requires an investment during the first working period, which implies that less 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: $$(y_{i,t}, y_{i,t+1}) = \begin{cases} (R^0, R^0) & \mathrm{if\, endowment} \\\ (1, R^1) & \mathrm{if\, investment} \end{cases}.$$

## Preferences

A key aspect of the model to generate dynamics in the evolution of individual traits is the fertility decision.
This approach is common: typically, fertility is linked to a trait through income.
This is, individuals with more income will be able to have more children.
If the trait is transmitted from parents to children, then the trait that allows generating more income will become more and more prevalent in the economy.^{1}
In any case, preferences are really important in this type of models, because fertility decisions are derived from them.

Every period $t$, a generation of size $L_t$ becomes economically active, this is, reaches the second period of life.
Those individuals were born in period $t-1$.
At this stage, each individual will live for two periods.
Remember that financial markets do not exist, and also that it is **impossible** to transfer resources between periods by storing them.

We assume that, during the second period of life, individuals only consume what they produce.
Lastly, during the third and last period individuals consume and also 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 the levels of consumption in the second and third periods of life and $n_{i,t}$ is the number of children.
It is important to comment on $\beta^i_t \in (0,1]$: it represents *individual i*’s discount factor, this is, how much he values the future with respect to the present.
The larger $\beta^i_t$, the more the value of future and, hence, the more patient the individual is.
Notice that $\beta^i_t$ changes over time and also by individual.

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 last period individuals can trade-off utility from consumption and utility from children.
The paper assumes that each child costs $\tau$ units of consumption, which gives rise the last-period budget constraint:
$$y_{i,t+1} = c_{i,t+1} + \tau n_{i,t+1}.$$

Considering the 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: $$v_{i,t} = \ln y_{i,t} + \beta^i_t [\ln y_{i,t+1} + \xi],\quad \xi \equiv \gamma \ln\left(\frac{\gamma}{\tau} \right) + (1-\gamma)\ln(1-\gamma).$$

## Hunters or farmers

Since individuals can decide on their mode of production, they are free to choose to become either hunters or farmers. This is, each individual will decide the mode of production (endowment or investment) that maximises lifetime utility. Hence,: $$v_{i,t} = \begin{cases} \ln R^0 + \beta^i_t (R^0 + \xi) & \mathrm{if\, endowment} \\\ \ln 1 + \beta^i_t ( R^1 + \xi) & \mathrm{if\, investment} \end{cases}.$$

An individual is indifferent between modes of production if he obtains the same utility from both.
This is, the individual with $\beta^i_t = \hat{\beta}$ is indifferent between becoming a hunter or a farmer if and only if:
$$\ln R^0 + \hat{\beta} (R^0 + \xi) = \ln 1 + \hat{\beta} \ln (R^1+\xi).$$
Solving for $\hat{\beta}$ allows us to identify such individual:^{2}
$$\hat{\beta} = \frac{\ln R^0}{\ln R^1 - \ln R^0} \in (0,1).$$

Lastly, all individuals with $\beta^i_t < \hat{\beta}$ will 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: $$\frac{\partial \hat{\beta}}{\partial R^1} = \frac{-\ln R^0}{R^1(\ln R^1 - \ln R^0)^2} < 0,$$ this is, as agriculture becomes more and more profitable, more individuals will find it optimal to become hunters.

Hence, we can rewrite the income of an individual as a function of his $\beta^i_t$: $$(y_{i,t}, y_{i,t+1}) = \begin{cases} (R^0, R^0) & \mathrm{if\, \beta^i_t \leq \hat{\beta}} \\\ (1, R^1) & \mathrm{if\, \beta^i_t > \hat{\beta}} \end{cases}.$$

Of course, since income in the last period of life is different, hunters and farmers will have different number 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.

## The evolution of preferences

Finally, we can compute how preferences change over time due to the differential fertility between farmers and hunters. This is, because farmers have more children than hunters, if we assume that preferences about time $\beta^i_t$ are transmitted between parents and children, the share of farmers will increase over time. The paper assumes almost that, although modifies slightly the transmission of preferences for individuals engaging in farming. In particular:

- $\beta^i_t$ is perfectly transmitted if an individual is a farmer.
- Farmers transmit a larger value of $\beta^i_t$ to their children, reflecting an acquired tolerance to waiting and delaying reward. This is: $$\beta^i_{i,t+1} = \begin{cases} \beta^i_t & \mathrm{if\, \beta^i_t \leq \hat{\beta}} \\\ \phi(\beta^i_t, R^1) & \mathrm{if\, \beta^i_t > \hat{\beta}} \end{cases},$$ with
- $\beta^i_t \leq \phi(\beta^i_t) < 1$: the transmitted $\beta$ is always more than the one the parent had,
- $\phi(\hat{\beta},R^1) > \beta^i_t$,
- $\phi_\beta(\beta^i_t,R^1)>0$: the higher the value of $\beta^i_{t+1}$, the more it increases,
- $\phi_{\beta\beta}(\beta^i_t,R^1)>0$: but at a decreasing rate,
- $\phi_R(\beta^i_t,R^1)>0$: the higher the value of $R$, the more $\beta^i_{t+1}$ increases.

Suppose an individual at the beginning of time who has $\beta^i_0 < \hat{\beta}.$ This individual will optimally decide to be a hunter, and according to the process for the transmission of preferences, his sons will inherit $\beta^i_1 = \beta^i_0.$ Because $\hat{\beta}$ is constant over time, all sons will decide to be hunters as well and transmit the same time preferences, over and over again. Hence, if $\beta^i_0 \leq \hat{\beta} \implies \lim_{t\rightarrow \infty} \beta^i_t = \beta^i_0.$

Suppose instead that $\beta^i_0 > \hat{\beta}.$ The individual will become a farmer and transmit $\phi(\beta^i_0,R^1) > \beta^i_0.$ Accordingly, all his sons will also be farmers and keep transmitting an ever-increasing value of $\beta^i_{t+1}.$ However, because $\phi_{\beta \beta}(\beta^i_t,R^1) < 0$, the transmission process has a steady-state, this is, $\lim_{t \rightarrow \infty} \beta^i_t = \bar{\beta^I}.$ Notice that $\hat{\beta^I}$ is the maximum level $\beta^i_t$ can reach.

### 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. This amount to showing that $\bar{\beta^I} = \phi(\bar{\beta^I},R^1)$ for a unique value $\bar{\beta^I}.$ Define $G(\beta) = \phi(\beta,R) - \beta.$ Since we focus on farmers, we know that the very initial one in the dynasty had $\beta^i_0 > \hat{\beta}$. So, for our purposes, the function $G$ has as domain $\beta \in [\hat{\beta},\infty).$ We know that $G(\hat{\beta}) = \phi(\hat{\beta}, R^1) - \hat{\beta} > 0$ because $\phi(\hat{\beta},R^1) > \beta^i_t.$ Moreover, the function $G$ has unique maximum at $\phi_\beta = 1$ because $\phi_{\beta \beta} < 0$, and the maximum is positive because it must be larger than $\phi(\hat{\beta},R^1)-\hat{\beta} > 0.$ After the maxima, the function continuously decreases, thus crossing only one the horizontal axis, this is, there is a unique value $\bar{\beta}$ such that $G(\bar{\beta^I}) = 0$, which constitutes the unique steady state.

## Evolution of traits over time

Lastly, suppose that initially, at time $t=0$, the initial population presents different levels of time preference. We assume that, initially, traits are characterised by some distribution of $\eta(\beta^i_0)$ with support $[0,\bar{\beta^I}].$ Furthermore, we normalise the initial generation to be of size one: $L_0 = 1.$ Alternatively: $$L_0 = \int_0^\bar{\beta^I} \eta(\beta^i_0)\mathrm{d}\beta^i_0 = 1.$$ We also know that all individuals whose $\beta^i_0 \leq \hat{\beta}$ decide to use the endowment technology, and the remaining opt for the investment technology. Therefore, the size of hunters (E) and farmers (I) is 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 the fertility rate of each group, this is, $$L_t^E = L_0^E {n^E}^t = (\frac{\gamma}{\tau}R^0)^tL_0^E$$ $$L_t^I = L_0^I {n^I}^t = (\frac{\gamma}{\tau}R^1)^tL_0^I$$ and total population is $L_t = L_t^E + L_t^I.$

Finally, notice that the distribution of $\beta^i_t$ does not change for those individuals with $\beta^i_t \leq \bar{\beta}$: in fact, they all have the same number of children, and each child inherits the trait of his parent. Therefore, the average value $\bar{\beta^E}$ is constant over time. In contrast, the average value $\bar{\beta^E_t}$ increases. In any case, at any given period $t$, the overall average value for time preference is given by: $$\bar{\beta_t} = \theta^E_t \bar{\beta^E_t} + (1-\theta^E_t)\bar{\beta_t^I},$$ where $\theta^E_t$ is the fraction of individuals who engage in the endowment production process, and $\bar{\beta_t}$ is just the weighted average. $$\theta^E_t = \frac{L_t^E}{L_t^E + L_t^I} = \frac{{R^0}^t}{{R^0}^t+{R^1}^t\frac{L^I_0}{L^E_0}}.$$

Hence, as time advances, the share of the population engaged in the endowment production process shrinks towards zero: $$\lim_{t\rightarrow\infty}\theta^E_t = 0.$$ This process reflects their lower reproductive success.