Question

The Query to Example 17.2 asks how uncertainty about the future might affect a person's savings decisions. In this problem we explore this question more fully. All of our analysis is based on the simple two-period model in Example 17.1 a. To simplify matters, assume that \(r=\delta\) in Equation \(17.15 .\) If consumption is certain, this implies that \(u^{\prime}\left(c_{0}\right)=u^{\prime}\left(c_{1}\right)\) or \(c_{0}=c_{1} .\) But suppose that consumption in period 1 will be subject to a zero-mean random shock, so that \(c_{1}=c_{1}^{p}+x,\) where \(c_{1}^{p}\) is planned period- 1 consumption and \(x\) is a random variable with an expected value of \(0 .\) Describe why, in this context, utility maximization requires \(u^{\prime}\left(c_{0}\right)=E\left[u^{\prime}\left(c_{1}\right)\right]\) b. Use Jensen's inequality (see Chapters 2 and 7 ) to show that this person will opt for \(c_{1}^{p}>c_{0}\) if and only if \(u^{\prime}\) is convex-that is, if and only if \(u^{\prime \prime \prime}>0\) c. Kimball \(^{14}\) suggests using the term "prudence" to describe a person whose utility function is characterized by \(u^{\prime \prime \prime}>0\) Describe why the results from part (b) show that such a definition is consistent with everyday usage. d. In Example 17.2 we showed that real interest rates in the U.S. economy seem too low to reconcile actual consumption growth rates with evidence on individuals' willingness to experience consumption fluctuations. If consumption growth rates were uncertain, would this explain or exacerbate the paradox?

Short answer

Answer: A person will opt for higher planned consumption in period 1 than in period 0 (i.e., \(c_1^p > c_0\)) if the third derivative of the utility function is positive (\(u^{\prime\prime\prime}>0\)). This is because the positive third derivative signifies prudence, making the person more inclined to save in the face of uncertainty, consuming less in period 0 and more in period 1.

Step-by-step solution

a. Utility Maximization in the Presence of Random Shock

If consumption in period 1 will be subject to a zero-mean random shock, we have \(c_1 = c_1^p + x\), where \(c_1^p\) is the planned period-1 consumption and \(x\) is a random variable with an expected value of \(0\). In this situation, to maximize utility, we need to take into consideration the expected value of \(u^{\prime}(c_1)\) instead of \(u^{\prime}(c_1^p)\). Mathematically, the utility maximization condition becomes: $$u^{\prime}(c_0) = E[u^{\prime}(c_1)]$$ This is because in the presence of the random shock, the consumer's actual consumption will be different from the planned consumption, and we need to compare the marginal utility of consumption in period 0 with the expected marginal utility of consumption in period 1.

b. Applying Jensen's Inequality

To determine whether the person will opt for \(c_1^p > c_0\), we can use Jensen's inequality which states that for a convex function \(\phi\), the expected value of the function is greater than the function of the expected value: $$E[\phi(x)] \geq \phi(E[x])$$ In this case, we want to examine the convexity of the function \(u^{\prime}\), which is determined by the sign of \(u^{\prime \prime \prime}\). If \(u^{\prime \prime \prime} > 0\), the function is convex, and by Jensen's inequality we have: $$E[u^{\prime}(c_1)] \geq u^{\prime}(E[c_1])$$ From part (a), we know that for utility maximization, \(u^{\prime}(c_0) = E[u^{\prime}(c_1)]\). Combining this with the inequality from Jensen's inequality, we get: $$u^{\prime}(c_0) \geq u^{\prime}(E[c_1])$$ Since \(u^\prime\) is a decreasing function, we have: $$c_0 \leq E[c_1] = c_1^p$$ Therefore, if \(u^{\prime \prime \prime} > 0\), the person will opt for \(c_1^p > c_0\). The opposite is true: if \(u^{\prime \prime \prime} \leq 0\), the person will opt for \(c_1^p \leq c_0\).

c. Prudence Definition Consistency

Prudence is the tendency to save in anticipation of future uncertainties, which is a rational response to uncertainty. The results from part (b) show that when \(u^{\prime \prime \prime}>0\), the person will opt for \(c_1^p > c_0\), meaning they will save more and consume less in period 0. This decision is consistent with the everyday usage of the term "prudent" because a prudent person is careful in making decisions, especially under uncertain circumstances (such as the random shock to consumption in period 1). This result supports the idea of naming this property of the utility function as "prudence".

d. Uncertain Consumption Growth Rates and Paradox

In Example 17.2, the paradox refers to the observation that real interest rates seem too low to reconcile actual consumption growth rates with evidence on individuals' willingness to experience consumption fluctuations. If consumption growth rates were uncertain, this would, in fact, exacerbate the paradox. This is because, under consumption uncertainty, individuals with a positive third derivative of utility function (\(u^{\prime \prime \prime} > 0\)) will tend to save more, due to their prudent behavior, and consume less in period 0. This higher savings rate would lead to even lower equilibrium interest rates, which would make the paradox more severe, as the difference between observed interest rates and what would be expected based on consumption growth rates and risk aversion would be even larger.

Key concepts

Utility Maximization with Random Shock

Understanding the concept of utility maximization in the presence of a random shock can be quite intriguing. In a simplistic two-period model, individuals aim to balance their consumption over time to achieve the highest utility. Under normal circumstances, this might result in equal consumption levels across different periods because of the marginal utility of consumption being identical. But let's add a twist to this scenario.

A random shock means there is uncertainty in future consumption, and this unpredictability needs to be incorporated into the decision-making process. Imagine you're planning to spend a certain amount next year, but there's a chance something unexpected could happen that will affect that amount. The solution here involves adjusting your current consumption to align with the expected utility of your future consumption—this is where expectations come into play. Instead of equating marginal utilities directly, you equate the marginal utility of current consumption with the expected marginal utility of future consumption.

Formally, if we denote planned consumption for period 1 as \(c_1^p\) and let \(x\) be the zero-mean random shock, utility maximization requires that:
\[u'(c_0) = E[u'(c_1)]\]
Here, \(u'(c_0)\) represents the marginal utility of current consumption, and \(E[u'(c_1)]\) is the expected marginal utility of future consumption. Making decisions based on this equality ensures that individuals are optimizing their utility given the uncertainty they face.

Jensen's Inequality

The principle behind Jensen's Inequality is a fundamental one in the study of economics, insurance, and finance. It comes down to understanding how to handle averages when putting them through a function, particularly one that curves, like our utility function.

To put it simply, if you take a convex function—picture a smile-shaped curve—and you feed it a bunch of numbers, the function's average output will always be equal to or greater than the function's value at the average input. In the context of our problem, if someone's utility function's second derivative is positive, making it convex, or smile-shaped, the expected marginal utility of uncertain consumption is larger than the marginal utility at planned consumption.

Mathematically, Jensen's Inequality in this context can be stated as:
\[E[u'(c_1)] \textgreater u'(E[c_1])\]
For someone who has a convex utility function, this inequality implies that they'll opt for a higher planned future consumption than current consumption which essentially promotes saving behavior, aligning with the idea of utility maximization when faced with future uncertainties.

Prudence in Consumption

Delving into the notion of prudence in consumption brings us to the characteristic of caution or wisdom in avoiding future risks and uncertainties. In economic terms, when we say a person is prudent, we mean they are inclined to save more today to cushion against potential adverse shocks tomorrow—exactly what you might do if you think your expenses might unexpectedly rise.

Connecting Prudence to Utility Functions

The term 'prudence' is linked to the nature of the utility function itself. If the third derivative of the utility function is positive (\(u''' > 0\)), it mathematically signifies a greater level of concavity in the utility of wealth, leading to greater precautionary savings. This is aligned with the colloquial usage of 'prudence' since it reflects a behavior that takes a precautionary stance against uncertainty. A prudent consumer, facing a random shock to future consumption, chooses to save more today, leading to a consumption plan where future planned consumption exceeds current consumption, as dictated by Jensen's Inequality.

Uncertain Consumption Growth

When we look at uncertain consumption growth, we're trying to understand how consumers adjust their behavior in the face of unknowable future changes in their consumption. This uncertainty can stem from various sources like job security, health, or sudden changes in the economy. These uncertainties influence the savings and consumption decisions of individuals.

The Paradox of Low Interest Rates

In real-world scenarios like the U.S. economy, we often encounter situations where actual consumption growth rates don't fit neatly with the low levels of interest rates observed. The paradox arises because, traditionally, people should save less when interest rates are low since the reward for saving is diminished. However, if people actually expect uncertain consumption growth, we would predict even greater saving to hedge against unknown future risks, reinforcing the behavior discussed under prudence. This further exacerbates the paradox by leading to even lower interest rates—the opposite of what would be needed to reconcile actual consumption patterns with the theory of utility maximization under risk.