Insurance I How the Poor Deal with Risk

🔹 Slide Title: Savings for a rainy (or dry...) day: Introducing uncertainty

We’re introduced to a simple intertemporal choice model under uncertainty — a foundation for understanding how poor households deal with risk through savings (before getting into insurance or borrowing).

✅ The Scenario:

A farmer earns income in two periods:

Period 1 income (y₁) is known.
Period 2 income (y₂) is uncertain — it can be either:
- High income (yᴴ) with probability p
- Low income (yᴸ) with probability 1 - p

🎯 Objective:

The farmer wants to maximize total expected utility over the two periods.

The utility function is:

\max u(c_1) + \beta \mathbb{E}[u(c_2)]

$c_1$ : consumption in period 1
$c_2$ : consumption in period 2
$\beta$ : discount factor (how much they value future consumption relative to today)

🔁 Budget Constraints:

They can save or borrow:

$S$ : savings in period 1 (could be negative if borrowing)
$R$ : gross interest rate (i.e., $R = 1 + r$ )

So:

$c_1 = y_1 - S$ → period 1 consumption is what's left after saving
$c_2 = y_2 + RS$ → period 2 consumption is income plus returns from savings

🤔 What this means:

The farmer is choosing how much to save in period 1 in order to smooth consumption across the two periods, despite income being uncertain in period 2.

They face a trade-off:

Consume more now and risk suffering in the future if income is low.
Consume less now to cushion against bad outcomes in the future.

Great — let’s walk through the slides from page 4 to page 7, building on the model. This is where we analyze the farmer’s decision more formally and begin deriving key insights.

🔹 Page 4: Risky business

This slide highlights that future consumption $c_2$ is uncertain, while the budget constraint always holds.

From earlier:

$c_1 = y_1 - S$
$c_2 = y_2 + RS$

Since $y_2$ is uncertain, $c_2$ is too.

So:

With probability p, $c_2 = y_H + RS$
With probability 1 − p, $c_2 = y_L + RS$

🔑 Key insight: Even though the farmer doesn't know what $c_2$ will be, they know the probabilities and their total resources across the two periods are fixed.

🔹 Page 5: Euler Equation

Now we optimize consumption using the first-order condition (FOC) for utility maximization.

The utility maximization problem:

\max u(c_1) + \beta \left[ p \cdot u(y_H + RS) + (1 - p) \cdot u(y_L + RS) \right]

Taking the derivative w.r.t. S (savings), the FOC becomes:

\beta R = \frac{u'(c_1)}{\mathbb{E}[u'(c_2)]}

Which you can also write as:

\beta R = \frac{u'(y_1 - S)}{p \cdot u'(y_H + RS) + (1 - p) \cdot u'(y_L + RS)}

What does this mean?

It equates the marginal cost of saving (in terms of lost utility today) with the expected marginal benefit in the future. If saving is too painful (i.e., current utility drops too much), the farmer saves less.

🔹 Page 6: An example with quadratic utility

To make this more concrete, assume:

u(c) = ac - \frac{1}{2}bc^2 \Rightarrow u'(c) = a - bc

Then the FOC becomes:

\beta R = \frac{a - bc_1}{\mathbb{E}[a - bc_2]}

Let’s suppose:

$\beta R = 1$ , i.e., no preference between today and tomorrow
Then:
$a - bc_1 = \mathbb{E}[a - bc_2] \Rightarrow c_1 = \mathbb{E}[c_2]$

This means:
🔑 Consumption is a martingale — current consumption equals expected future consumption.

This result only holds for quadratic utility (which implies risk neutrality beyond the second moment) and βR = 1.

🔹 Page 7: Solving for optimal consumption

Now we combine the two budget constraints to solve for $c_1$ :

Start with:

$c_1 = y_1 - S$
$c_2 = y_2 + RS$

Multiply the first by $R$ :

Rc_1 = Ry_1 - RS \Rightarrow c_2 + Rc_1 = y_2 + Ry_1

Now isolate the total present value of consumption:

c_1 + \frac{c_2}{R} = y_1 + \frac{y_2}{R}

Taking expectations on both sides:

c_1 + \frac{\mathbb{E}[c_2]}{R} = y_1 + \frac{\mathbb{E}[y_2]}{R}

If $\mathbb{E}[c_2] = c_1$ (from martingale result):

c_1 + \frac{c_1}{R} = y_1 + \frac{\mathbb{E}[y_2]}{R} \Rightarrow c_1 \left(1 + \frac{1}{R} \right) = y_1 + \frac{\mathbb{E}[y_2]}{R}

So:

c_1 = \frac{y_1 + \frac{\mathbb{E}[y_2]}{R}}{1 + \frac{1}{R}} = \frac{R y_1 + \mathbb{E}[y_2]}{R + 1}

This is the optimal consumption in period 1 under perfect smoothing with saving.

🧠 Intuition:

The agent spreads consumption across both periods, weighting it by the present value of income. Risk is accounted for in the expected value of $y_2$ .

before we continue I need to understand why:
c2=y2+S

whether:
c1=y1−S

Great question! Let's clarify this carefully, because this is core to understanding intertemporal savings behavior.

🧩 Where do these equations come from?

We're looking at a simple 2-period model where a household (or farmer) earns income and chooses how much to consume and save in period 1, and how to consume in period 2.

We assume:

They can save or borrow at a constant interest rate $R = 1 + r$
There are no other income sources or shocks, except the ones specified

✅ Period 1:

The household earns $y_1$ , consumes $c_1$ , and saves $S$ .

So the budget constraint for period 1 is:

c_1 = y_1 - S

This means:

If $S > 0$ , they’re saving
If $S < 0$ , they’re borrowing

✅ Period 2:

The household earns uncertain income $y_2$ , and gets back the savings $S$ with interest $R$ , i.e., they receive $RS$ .

So the budget constraint in period 2 is:

c_2 = y_2 + RS

Which you correctly wrote as:

c_2 = y_2 + S \quad \text{ONLY IF } R = 1

🔔 So be careful:
Your formula $c_2 = y_2 + S$ only holds if there’s no interest — i.e., the return on savings is zero.

In general:

c_2 = y_2 + RS

⚖️ Why do we use these two constraints?

Because the household is choosing S (savings), and those two constraints describe how their consumption in both periods depends on that choice. They want to choose $S$ to maximize total utility across both periods.

🧮 Example to make it clearer:

Say:

$y_1 = 100$ , $y_2$ could be 60 or 140 with 50% probability each
$R = 1.1$ (i.e., 10% interest rate)
They choose to save $S = 20$

Then:

$c_1 = y_1 - S = 100 - 20 = 80$
$c_2 = y_2 + RS = y_2 + 1.1×20 = y_2 + 22$

So in period 2, they’ll have:

With 50% chance: $c_2 = 60 + 22 = 82$
With 50% chance: $c_2 = 140 + 22 = 162$

✅ Summary:

Expression	Meaning
$c_1 = y_1 - S$	Consume what's left after saving
$c_2 = y_2 + RS$	Consume income + returns from savings
If $R = 1$	Then $c_2 = y_2 + S$ (no interest earned)

Excellent — let’s continue to page 9 of the slides, which introduces empirical testing of the theoretical model using real-world data.

🔹 Slide Title: Testing this model — Savings and Rainfall in Thailand

We’re now moving from theory into evidence.

The slide refers to a famous paper by Chris Paxson, who tested whether the behavior of Thai farmers aligns with the theory we just studied.

🎯 Theoretical prediction we want to test:

From earlier, the model predicts:

People consume more out of permanent increases in income.
People save more out of temporary increases in income.

So:

If income goes up temporarily, households will save most of it.
If income goes up permanently, they’ll consume most of it.

🧪 Paxson’s Empirical Strategy

She estimates the following regression:

S_{irt} = \alpha_0 + \alpha_1 Y^P_{irt} + \alpha_2 Y^T_{irt} + \text{Controls} + \varepsilon_{irt}

Where:

$S_{irt}$ : Savings of farmer $i$ in region $r$ at time $t$
$Y^P_{irt}$ : Permanent income
$Y^T_{irt}$ : Transitory (temporary) income
Controls: other variables like demographics, farm size, etc.
$\varepsilon$ : error term

🔍 What does she expect?

From theory:

$\alpha_1$ (response to permanent income): should be close to 0
- People consume most of their permanent income
$\alpha_2$ (response to temporary income): should be positive and large
- People save temporary windfalls

So:

\text{Expected: } \alpha_2 > \alpha_1 \approx 0

🌦️ Why Thailand? Why rainfall?

In the next slides, Paxson uses rainfall as a natural source of variation:

Rainfall affects rice harvests, which determines income.
But rainfall is:
- Unpredictable (not serially correlated)
- Exogenous (not controlled by farmers)

So it’s a great proxy for temporary income shocks.

Great! Let's walk through pages 10 to 13, where Paxson's method is explained in more detail, and we approach the results of her empirical test.

🔹 Page 10: Paxson – Measurement challenge

Problem:

To run the regression:

S_{irt} = \alpha_0 + \alpha_1 Y^P_{irt} + \alpha_2 Y^T_{irt} + \text{controls} + \varepsilon_{irt}

Paxson needs to separate income into two parts:

$Y^P_{irt}$ : Permanent income
$Y^T_{irt}$ : Transitory income

But she doesn’t directly observe which part is permanent or temporary. So, she must construct them.

🔹 Page 11: Paxson – Strategy using rainfall

Her key idea:

Rainfall affects income but is transitory (it’s random from year to year)
Other characteristics (like land size, household head's education) explain permanent income

So she models income as:

Y_{irt} = \beta_t + \beta_{0r} + X^P_{irt} \beta_1 + X^T_{irt} \beta_2 + \hat{\varepsilon}_{irt}

Where:

$X^P$ : variables that predict permanent income (e.g., fixed household traits)
$X^T$ : rainfall — a shock affecting only the current period

Then:

Predicted permanent income: $\hat{Y}^P_{irt} = X^P_{irt} \beta_1$
Predicted transitory income: $\hat{Y}^T_{irt} = X^T_{irt} \beta_2$
Residual: $\hat{\varepsilon}_{irt}$ , unobserved shocks

Then she plugs these into the original regression:

S_{irt} = \alpha_0 + \alpha_1 \hat{Y}^P_{irt} + \alpha_2 \hat{Y}^T_{irt} + \text{controls} + \varepsilon_{irt}

🔹 Page 12: What are the results?

The slide prompts: What are the results?

We’re not shown exact numbers, but here’s what Paxson finds:

✅ Consistent with theory:

$\alpha_2$ (effect of transitory income) is large and significant → Farmers save a big chunk of temporary income windfalls
$\alpha_1$ (effect of permanent income) is small or insignificant → Farmers consume most of their permanent income

This supports the idea from the consumption smoothing model:
Households treat temporary and permanent income very differently.

🔹 Page 13: Effect of permanent and temporary shocks

This slide summarizes the theory-backed insight:

"The propensity to consume out of permanent income change should be higher than the propensity to consume out of a temporary change."

That is:

Permanent income change → mostly consumed
Temporary income change → mostly saved

Mathematically:

Propensity to save out of permanent income ≈ 0
Propensity to save out of transitory income ≈ 1 (especially with a long horizon)

This is a core prediction of intertemporal consumption theory.

✅ Recap from Pages 10–13:

Concept	What Paxson Does
Challenge	Separate income into permanent vs. transitory parts
Strategy	Use rainfall to identify transitory shocks
Method	Regression of income on rainfall + fixed traits
Test	Do people save transitory income but not permanent?
Result	Yes — strongly supports the theory

Great — now let's walk through pages 14 to 21 of the slides, which shift from the idealized savings world into real-world constraints and alternative risk-coping mechanisms used by poor households.

🔹 Page 14: Introducing borrowing constraints

What if households can’t borrow?

In theory, we assumed people could borrow and save freely. But in real life, poor households often:

Can’t borrow at all (no access to credit)
Or can only borrow at very high interest rates

So what do they do instead?

✅ They try to save during good times and spend those savings in bad times

🔁 Deaton-style “rule of thumb”:

If we define:

$x_t$ : “cash on hand” (current income + any past saved assets)

Then:

If $x_t$ is below a threshold → consume all of it
If $x_t$ is above the threshold → save some of the excess

This is a simplified way to describe how people try to self-insure by saving during good periods.

🔹 Page 15: Smoothing by saving up

Example:

Suppose income is i.i.d. (independent and identically distributed) with a mean of 100.

Then:

If $x_t < 100$ → consume everything
$c_t = x_t$
If $x_t \geq 100$ → consume only part
$c_t = x_t - (x_t - 100) \cdot 0.7$

So they save 70% of any extra income above the mean.

❗ But:

Assets can run out during bad times.
Consumption can still drop sharply.
So even with saving, there’s limited smoothing — especially when hit by repeated shocks.

This is where insurance (next lecture) comes in.

🔹 Pages 16 & 17: Buffer Savings Simulations (Deaton, 1997)

These show simulation results with different savings rules:

70% savings (Page 16) vs. 80% savings (Page 17)

🧪 These simulations show:

Some smoothing is possible, but not perfect.
Consumption still dips sharply when assets run out.
Even very cautious saving can’t fully protect against bad shocks in absence of borrowing or insurance.

🔹 Page 18: Savings constraints

Theoretical problem becomes practical:

Poor households often:

Have no access to banks
Face low returns on their savings
Store savings in inefficient forms, like:
- Jewelry
- Grain
- Livestock (e.g., bullocks)

But these are:

Hard to liquidate
Risky (animals can die, grain can spoil)

🔹 Page 19: Income smoothing (limits)

Without borrowing or formal insurance:

Consumption must fall when income does
Savings might not be enough
This motivates looking at informal risk-sharing (next lecture)

But households also use behavioral responses to cope with risk.

🔹 Page 20: Work more when wages are low

This refers to a paradox from Seema Jayachandran (2006):

If a region:

Has no banks
Is isolated (bad roads, no insurance)

Then:

People can’t save or insure
So they work more when wages are low — to maintain consumption
This leads to a more volatile labor supply and even more volatile wages

👉 Paradox: The poor have less income security, and this makes their wages more volatile, not less.

🔹 Page 21: Avoid risk ex-ante

When people can’t manage risk, they avoid it entirely — even at the cost of lower income.

Examples:

Avoid planting a new crop or fertilizer that has high expected profit but also a risk of total failure
Choose low-risk, low-return activities

This leads to persistent poverty traps, because people avoid innovation.

🧠 Summary of Pages 14–21:

Topic	Key Point
Borrowing constraints	Poor can’t borrow → rely on saving alone
Saving rules	Simple rules help but don’t fully smooth consumption
Simulation (Deaton)	Shows savings alone can't eliminate big consumption drops
Savings constraints	No banks, low returns, illiquid assets like bullocks
No borrowing or insurance	Consumption still falls in bad years
Work when wages are low	Wages become more volatile in poor, uninsurable areas
Risk avoidance	People choose safer, lower-return options → slows development

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