# Development Economics (and the United States) Part I: Savings Theory

During my Master's degree, I took a course in Development Economics from Cynthia Kinnan* (which had nothing to do with transportation directly). Throughout the course, I noticed that many of the concepts in the course could easily be applied to the United States. A lot of the statistics and economic concepts are simple enough to understand with the knowledge from an Econ 101 course (one of the most dangerous courses an arrogant policy maker can take in school). The data sets are often unreliable depending on the country of interest, which makes the analyses less reliable too. This first part of what will hopefully be a multipart series will focus on saving.**

**A lot of the theory I cover is based on her lecture notes.*

** *You will need Javascript enables to see the equations correctly since they require Mathjax.*

# Table of Contents

- Definitions

1.1 Poverty Trap - Reasons for Saving

2.1 Consumption Smoothing

2.2 Decisions Over Time - What Next?

# 1. Definitions

## 1.1 Poverty Trap

From Investopedia:

The poverty trap is a mechanism which makes it very difficult for people to escape poverty. A poverty trap is created when an economic system requires a significant amount of various forms of capital in order to earn enough to escape poverty. When individuals lack this capital, they may also find it difficult to acquire it, creating a self-reinforcing cycle of poverty.

## 1.2 Consumption Smoothing

Again from Investopedia:

The ways in which people try to optimize their lifetime standard of living by ensuring a proper balance of spending and saving during the different phases of their life.

# 2. Savings Theory

Before diving into the the lack of tools available to people in many low-income areas, we need to identify the traditional reasons for saving. Despite being simplistic and somewhat artificial, these categories at least provide a framework for the discussion.

## 2.1 Consumption Smoothing

### 2.1.1 Permanent Income Hypothesis (PIH)

Consumption smoothing relies on the idea that people prefer to smooth out their consumption patterns over extended periods. While there are some one time expenses, like vehicles) that this does not apply to, it makes sense that people would want to be able to consume the same amount of food and water (or possibly more) in the future as they do currently. As an example, Mette works in the tourist industry and made $1,500 this year (period I). However, she knows that she will only make around $500 next year (period II). Ignoring the time value of money for now ($1 today=$1 in the future), assuming Mette is patient and her utility curve has \(u''(c)<0\), then saving will increase her utility

$$2u(1000)>u_I(1500)+u_{II}(500).$$

Assuming Mette is maximizing her own utility and $1000 covers her necessary expenses in period I (or her household's if she lives with family), then she will clearly save $500 and spend it in the future. Before moving on to the issues, we can add some much needed nuance to the model by introducing an interest rate \(r\). We will also explicitly add a weighting of \(\beta\leq 1\) to the model which implies Mette discounts future spending. Now her total utility function (U) is:

$$U(c_I,c_II)=u(c_I)+\beta u(c_{II}).$$

Introducing a budget constraint (she can't spend more than she earns and she has no access to credit right now),

$$c_I+\frac{1}{1+r}(c_{II})=y_I+\frac{1}{1+r}y_{II}.$$

Moving slightly beyond a typical intermediate Microeconomics classes, we will just use the Lagrangian to find a maximum to the function given the constraint.

$$\mathcal{L}(c_I,c_{II}|y_I,y_{II})\equiv u(c_I)+\beta u(c_II)+\lambda \left(y_I+\frac{y_{II}}{1+r}-c_I-\frac{c_{II}}{1+r}\right)$$

Now we need some first order conditions (FOCs) which are \(u'(c_I)=\lambda\) and \(\beta u'(c_{II})=\frac{\lambda}{1+r}\)

Combining, we get the inter-temporal Euler equation.

$$u'(c_I)=\beta (1+r)u'(c_{II})=\lambda)$$

Normally, we would assume that income is equal accross time periods. Instead though, only expected marginal utility can be equalized accross periods. When \(\beta\) or \(r\) is lower, then the household chooses a higher MU in the future. The future consumption is therefore lower. When \(\lambda\) (shadow value of income) is higher MU is also higher in the future.

After all of this math, the end result is we can model (somewhat) a person's decision making process. The analysis yields the Permanent Income Hypothesis. Changes to income only affect consumption via their effect on lifetime ("permanent") wealth. Permanent and transitory income changes have different effects on consumption vs. savings

- Permanent changes in income are fully consumed
- A transitory change \(\Delta\) is like a permanent change of \(r\Delta\)
- When consumption increases by \(r\Delta\), savings increase by \((1-r)\Delta\)
- If \(r=0\), transitory changes are fully saved

### 2.1.2 Is the PIH real?

In 1992, Paxson (behind a paywall) investigated the PIH using detailed Thai household data from the Socio-Economic-Surveys (SES) from the National Statistical Office. The surveys collected information on income and expenditure by commodity type in 1976, 1981 and 1986. The study corrected for inflation bias (when there is inflation during a reporting period). The results of Paxson's analysis supported the idea that Thai farmers saved most transitory income, although the author was unable to conclusively show Thai farmers consumed the entirety of a permanent change in income.

Further research shows the PIH's biggest issue is the lack of consideration of liquidity constraints. We are also assuming that consumers are able to save effectively which may not be the case.

## 2.2 Decisions Over Time

Many models assume that discount rates do not change over time. Assuming a constant discount rate ignores that many people may have a desire to save over the long term, but also have a desire for hipster coffee in the short term. This is called time-inconsistency and is not in \(\beta\).

### 2.2.1 Time Inconcistency

Once again, we assume there are no liquidity constraints. Using the same utility function as previously but extended to \(T\) periods,

$$U(c_0, ..., c_T)=\sum_{t=0}^T \beta^t u(c_t).$$

Any two periods \(k\) months apart will be discounted by \(\beta^k\). However, if we let the personal discount rate change, then a new utility function will be

$$U(c_0, .., c_T)= u(c_0) + \delta\sum_{t_1}^T \beta^t u(c_t).$$

So somebody who has quasi-hyperbolic discounting values consumption \(k\) months from now at \(\delta\beta^k\).

There are a few different situations to consider here. Somebody who is self-aware of being time-inconsistent may want to commit to saving, while somebody is not aware ("naive") may not want to commit to savings. This can lead to under-investment.

### 2.2.2 SEED

In 2006, Ashraf, Karlan, and Yin created a Commitments Savings product in the Philippines called SEED (Save, Earn, Enjoy, Deposits). The product prevented withdrawing until the consumer reached a goal date or amount. Not only did they find support for time inconsistent behavior, they also found that education led to people being more away of their own time inconsistencies. Traditional financial institutions are not the only way to combat time inconsistency.

### 2.2.3 ROSCAs

Rotating Savings and Credit Associations (ROSCAs) are an especially effective way to handle time inconsistency and different forms of them are present in a wide variety of cultures. The general idea is that \(N\) people get together and put \(m\) into a common pot. Each round, one person gets the pot with size \(Nm\). In a random ROSCA, a random winner who has not won yet gets the pot. In a bidding ROSCA, bidders who have not won yet bid for the pot.

ROSCAs have no need for bookkeeping and many may prefer to default on a loan than to renege on a community ROSCA. However, there is rarely any interest and there is always the chance they will collapse if an entire community faces a shock at the same time.

As an example, assume there are two members of a household: \(W\) and \(H\). There are also two time periods.

\begin{align}

U_W&=u(c_1)+u(c_2)+\delta D\

U_H&=u(c_1)+u(c_2)

\end{align}

\(D\) is just a dummy variable that is equal to 1 if \(W\) purchases a durable good. The good has a price of 1, and the household can only purchase it in period 1. In this case:

\begin{align}

s&\geq0 \

Y&\geq c_1 + s \

Y + s &\geq c_2+D

\end{align}

\(H\) and \(W\) decide jointly whether to purchase the durable good. Then the weighted sum (where \(\gamma \) is how much \(W\)'s opinion matters) is

$$U_T=(1-\gamma )u_H+\gamma u_W.$$

Leading to \(s=\frac{1}{2}\). Now assume \(W\) wants to buy \(D\). Then

$$u(Y)+u(Y) < u(Y-\frac{1}{2})+u(Y-\frac{1}{2})+\delta D.$$

However, \(H\) does not

$$u(Y)+u(Y) < u\left(Y-\frac{1}{2}\right)+u\left(Y-\frac{1}{2}\right)+\gamma\delta D.$$

If \(\gamma\) is low, the household will not buy the good. What about a ROSCA? \(W\) will only join a ROSCA if the following two equations are true:

\begin{align}

u(Y)+u(Y) &< u\left(Y-\frac{1}{2}\right)+u\left(Y+\frac{1}{2}\right)+\delta D - T \

u\left(Y-\frac{1}{2}\right)+\gamma\delta D &> u\left(Y+\frac{1}{2}\right)

\end{align}

Considering three possibilities,

- If \(\gamma=1\), then they'll buy \(D\)
- If \(\gamma=0\), then they will go with the ROSCA.
- If \(\gamma\) is anywhere in between, then the choice is a subgame perfect strategy

## 3. What next?

That depends. I may choose to dive into some more theory about borrowing and lending at financial institutions. At the same time, in the U.S. there was talk of letting U.S. Post Offices offer basic banking services which sounds like an interesting QGIS project. Or I might finish up working on my Heroes of the Storm competitive analysis.