Question

The CARA and CRRA utility functions are both members of a more general class of utility functions called harmonic absolute risk aversion (HARA) functions. The general form for this function is \(U(W)=\theta(\mu+W / \gamma)^{1-\gamma}\), where the various parameters obey the following restrictions: \(\bullet$$\gamma \leq 1\) \(\bullet$$\mu+W / \gamma > 0\) \(\bullet$$\theta[(1-\gamma) / \gamma] > 0\) The reasons for the first two restrictions are obvious; the third is required so that \(U^{\prime} > 0\) a. Calculate \(r(W)\) for this function. Show that the reciprocal of this expression is linear in \(W\). This is the origin of the term harmonic in the function's name. b. Show that when \(\mu=0\) and \(\theta=[(1-\gamma) / \gamma]^{\gamma-1},\) this function reduces to the CRRA function given in Chapter 7 (see footnote 17 ). c. Use your result from part (a) to show that if \(\gamma \rightarrow \infty\), then \(r(W)\) is a constant for this function. d. Let the constant found in part (c) be represented by \(A\). Show that the implied form for the utility function in this case is the CARA function given in Equation 7.35 e. Finally, show that a quadratic utility function can be generated from the HARA function simply by setting \(\gamma=-1\) f. Despite the seeming generality of the HARA function, it still exhibits several limitations for the study of behavior in uncertain situations. Describe some of these shortcomings.

Short answer

Answer: Some limitations of the HARA utility function when studying behavior in uncertain situations include its inability to represent all possible risk preferences, the assumption of a specific functional form that may not accurately represent individual preferences, and the lack of capturing changes in preferences over time, which could be important for long-term decision-making.

Step-by-step solution

Calculating the first derivative of the utility function

To find absolute risk aversion, first, we need to find the first derivative of the utility function with respect to \(W\). Differentiate \(U(W)\) with respect to W: \(U^{\prime}(W)= \theta (1-\gamma)(\mu + W / \gamma)^{-\gamma}\)

Calculating the second derivative of the utility function

Now, we find the second derivative of the utility function with respect to \(W\): \(U^{\prime\prime}(W)= \theta (1-\gamma)\gamma (\mu + W / \gamma)^{-(\gamma+1)}\)

Finding \(r(W)\)

Absolute risk aversion, \(r(W)\), is defined as the ratio of the second derivative of the utility function to the first derivative. Thus, we find \(r(W)\) by dividing the second derivative by the first derivative: \(r(W) = \frac{U^{\prime\prime}(W)}{U^{\prime}(W)} = \frac{\theta (1-\gamma)\gamma (\mu + W / \gamma)^{-(\gamma+1)}}{\theta (1-\gamma)(\mu + W / \gamma)^{-\gamma}} = \frac{\gamma}{\mu+\frac{W}{\gamma}}\) As seen in the last step, the reciprocal of \(r(W)\) has the form \(aW+b\), where a and b are constants. This makes it linear in \(W\). #b. The CRRA Function#

Substituting values for \(\mu\) and \(\theta\)

Now we show that when \(\mu=0\) and \(\theta=[(1-\gamma) / \gamma]^{\gamma-1},\) the HARA utility function reduces to the CRRA function. Substitute these values into the HARA function \(U(W)=\theta(\mu+W/\gamma)^{1-\gamma}\): \(U(W)=\left(\frac{1-\gamma}{\gamma}\right)^{\gamma-1} (W/\gamma)^{1-\gamma}\)

Simplifying the utility function

Simplify the utility function by multiplying the two exponents: \(U(W) = W^{1-\gamma}\) This is the same as the CRRA utility function given in the question. #c. Constant Risk Aversion as \(\gamma \rightarrow \infty\)

Finding the limit of \(r(W)\)

Find the limit of the absolute risk aversion as \(\gamma\) approaches infinity: \(\lim_{\gamma \rightarrow \infty} \frac{\gamma}{\mu+\frac{W}{\gamma}}\)

Applying L'Hopital's rule

Using L'Hopital's rule, we find the limit: \(\lim_{\gamma \rightarrow \infty} \frac{\gamma}{\mu+\frac{W}{\gamma}} = \lim_{\gamma \rightarrow \infty} \frac{1}{\frac{-W}{\gamma^2}} = 0\) When \(\gamma \rightarrow \infty\), \(r(W)\) approaches a constant value. #d. The CARA Utility Function#

Setting the constant as A

Let the constant found in part (c) be represented by \(A\). Substitute this value into the HARA utility function: \(U(W)=\theta(\mu+W/\infty)^{1-\infty}\)

Simplifying the utility function

This simplifies to the CARA utility function: \(U(W)=-e^{-AW}\) #e. Quadratic Utility Function#

Setting \(\gamma=-1\)

Set \(\gamma=-1\) in the HARA utility function: \(U(W)=\theta(\mu+W/\gamma)^{1-\gamma}\)

Simplifying the utility function

Simplify the function by plugging in \(\gamma = -1\): \(U(W)=\theta(\mu+W/(-1))^{2}\) This is a quadratic utility function. #f. Limitations of the HARA utility function# Some limitations of the HARA utility function for the study of behavior in uncertain situations include: 1. HARA utility functions cannot represent all possible risk preferences. 2. HARA utility functions assume a specific functional form, which may not accurately represent individual preferences in all situations. 3. HARA utility functions may not capture changes in preferences over time, which could be important in the context of long-term decision-making.

Key concepts

Absolute Risk Aversion

When it comes to understanding an individual's attitude toward risk, absolute risk aversion plays a crucial role. It is a measure of how much one's satisfaction (or utility) decreases as the risk in a situation increases. In more technical terms, it is defined by the negative of the second derivative of the utility function with respect to wealth (\(W\)), divided by the first derivative of the utility function with respect to wealth.

Absolute risk aversion is critical because it shows us how risk tolerance changes with varying levels of wealth. For instance, with a high level of absolute risk aversion, a small increase in risk causes a significant reduction in utility, suggesting that the individual is risk-averse. Conversely, if absolute risk aversion is low, the individual is more tolerant of risk. This concept is reflected in the utility function's curvature: the greater the curvature, the higher the level of risk aversion.

CARA Utility Function

The Constant Absolute Risk Aversion (CARA) utility function is a specific type of utility model where the degree of risk aversion remains the same irrespective of changes in wealth. It is a pivotal concept in economics and finance because it allows for the simple modeling of risk-averse behavior without the complexity that varying risk aversion with wealth entails.

In a typical CARA utility function, represented by the equation \(U(W)=-e^{-AW}\), the parameter \(A\) stands for the level of absolute risk aversion, and it stays constant as wealth, \(W\), changes. This implies that whether the individual is wealthy or not, their dislike for taking risks remains the same. This simplicity makes the CARA model convenient for certain economic analyses, although it may not accurately represent all individuals' behavior.

CRRA Utility Function

Contrastive to the CARA model, the Constant Relative Risk Aversion (CRRA) utility function describes a situation where an individual's risk aversion is related to their relative wealth position, not the absolute amount. Here, the individual’s dislike for risk is constant in percentage terms.

The CRRA utility function, which is given by \(U(W) = W^{1-\gamma}\), where \(\gamma\) is the coefficient of relative risk aversion, is particularly useful for describing behaviors where the individual's risk aversion changes alongside proportional changes in wealth. For example, if a person's wealth doubles, their risk aversion in terms of proportional wealth remains unchanged. CRRA models are widely used in financial studies because they offer a more realistic depiction of how people's risk preferences might vary with different levels of wealth.

Risk Preferences

Risk preferences are at the core of financial decision-making, guiding how individuals make choices under uncertainty. They define an individual’s willingness to take or avoid risk and are shaped by psychological factors, economic circumstances, and even cultural background.

Understanding an individual's risk preferences is crucial for predicting behavior in financial markets, personal savings, insurance, and investment decisions. They are deeply ingrained characteristics that influence how a person perceives the trade-off between the potential for higher returns and the willingness to accept the associated risks. Economists categorize risk preferences into three primary types: risk-averse (preferring certainty over gamble for the same expected return), risk-neutral (indifferent between certainty and gamble), and risk-seeking (preferring the gamble over certainty). Each of these preferences can be mathematically described by specific types of utility functions, which help in constructing financial models and theories.