Problem 12 - Advanced Macroeconomics

Question

The equity premium and the concentration of aggregate shocks. (Mankiw, 1986.) Consider an economy with two possible states, each of which occurs with probability \(\frac{1}{2} .\) In the good state, each individual's consumption is \(1 .\) In the bad state, fraction \(\lambda\) of the population consumes \(1-(\phi / \lambda)\) and the remain der consumes \(1,\) where \(0<\phi<1\) and \(\phi \leq \lambda \leq 1 . \phi\) measures the reduction in average consumption in the bad state, and \(\lambda\) measures how broadly that re duction is shared. Consider two assets, one that pays off 1 unit in the good state and one that pays off 1 unit in the bad state. Let \(p\) denote the relative price of the bad-state asset to the good-state asset. (a) Consider an individual whose initial holdings of the two assets are zero, and consider the experiment of the individual marginally reducing (that is, selling short) his or her holdings of the good-state asset and using the proceeds to purchase more of the bad-state asset. Derive the condition for this change not to affect the individual's expected utility. (b) since consumption in the two states is exogenous and individuals are ex ante identical, \(p\) must adjust to the point where it is an equilibrium for individuals' holdings of both assets to be zero. Solve the condition derived in part ( \(a\) ) for this equilibrium value of \(p\) in terms of \(\phi, \lambda, U^{\prime}(1)\) and \(U^{\prime}(1-(\phi / \lambda))\) (c) Find \(\partial p / \partial \lambda\) (d) Show that if utility is quadratic, \(\partial p / \partial \lambda=0\) (e) Show that if \(U^{\prime \prime \prime}(\bullet)\) is everywhere positive, \(\partial p / \partial \lambda<0\)

Step-by-Step Solution

Answer

(a) -U'(1) + pU'(1-\frac{\phi}{\lambda}) = 0. (b) p = \frac{U'(1)}{U'(1-\frac{\phi}{\lambda})}. (c) \frac{\partial p}{\partial \lambda}. (d) \frac{\partial p}{\partial \lambda} = 0 with quadratic utility. (e) Do \(U'''(\bullet) > 0 \to \frac{\partial p}{\partial \lambda} < 0\).

Step 1: Set Up the Expected Utility

An individual receives different consumptions in the two states. Their utility in the good state is given by \(U(1)\) and in the bad state by \(U\left(1 - \frac{\phi}{\lambda}\right)\). When selling the good-state asset and buying the bad-state asset, let their holdings in the good state be \(x\) and in the bad state be \(y\). Thus, the expected utility is \(\frac{1}{2}U(1 - x) + \frac{1}{2}U\left(1 - \frac{\phi}{\lambda} + y\right)\). We want the change in this expected utility to be zero.

Step 2: Marginal Change in Utility

A marginal change involves altering the holdings by \(dx\) and \(dy\). The change in expected utility due to this shift is \(\frac{dy}{dp} \left( -U'(1) + pU'\left(1 - \frac{\phi}{\lambda}\right) \right) dx \). This should equal zero for expected utility to remain unaffected. Thus, the condition becomes \(-U'(1) + pU'\left(1 - \frac{\phi}{\lambda}\right) = 0\).

Step 3: Solve for the Equilibrium Price

From the condition \(-U'(1) + pU'\left(1 - \frac{\phi}{\lambda}\right) = 0\), solve for \(p\). This gives \(p = \frac{U'(1)}{U'\left(1 - \frac{\phi}{\lambda}\right)}\), which is the price ratio that makes holding both assets zero in expectation.

Step 4: Derivative of Price with Respect to \(\lambda\)

The next task is to find \(\frac{\partial p}{\partial \lambda}\). Differentiating \(p = \frac{U'(1)}{U'\left(1 - \frac{\phi}{\lambda}\right)}\) with respect to \(\lambda\) gives us \( \frac{\partial p}{\partial \lambda} = \frac{U'(1) \cdot U''\left(1 - \frac{\phi}{\lambda}\right) \cdot \frac{\phi}{\lambda^2}}{\left(U'\left(1 - \frac{\phi}{\lambda}\right)\right)^2} \).

Step 5: Examine Quadratic Utility

For a quadratic utility \(U(c) = a - bc + \frac{c^2}{2}\), the second derivative \(U''(c) = -b\) is constant. Hence, \(U''\left(1 - \frac{\phi}{\lambda}\right)\) doesn't vary with \(c\), leading to \(\frac{\partial p}{\partial \lambda} = 0\).

Step 6: Effect of Positive Third Derivative

If \(U'''(\bullet) > 0\), increasing the concavity of utility makes \(U'(\bullet)\) decrease faster as \(\lambda\) increases. This results in \(\frac{\partial p}{\partial \lambda} < 0\), implying the equilibrium price of the bad-state asset decreases as \(\lambda\) broadens the impact of shocks.

Key Concepts

Aggregate Shocks

Aggregate shocks refer to sudden, unpredictable events that impact the economy as a whole. These shocks can be positive or negative.
When a negative shock occurs, it can result in a decrease in overall economic productivity or consumption. Imagine a severe weather event affecting agricultural output, leading to a reduced supply of food.
In such cases, the economy experiences hardship that affects nearly everyone, though not necessarily equally.An important aspect of aggregate shocks is how they spread across the population. This is often represented by the parameter \( \lambda \) in economic models.
It determines how widespread the effect is.
A higher \( \lambda \) means that a larger portion of the population shares the economic downturn, albeit with less severity per individual.Understanding how aggregate shocks propagate can help policymakers create strategies that mitigate these effects, ultimately stabilizing the economy for everyone.

Utility Function

A utility function represents an individual's preference, showcasing how different levels of consumption or wealth affect their overall happiness or satisfaction.
The function assigns a numerical value to each possible outcome, making it easier to compare them.In this context, the utility function is denoted by \( U(c) \), where \( c \) is the level of consumption.
Often, utility functions are increasing, meaning that more consumption typically leads to higher utility.
However, they may also be concave, indicating diminishing returns as consumption increases.Economists use utility functions to predict how individuals make choices under uncertainty. For example, a person might choose to limit consumption in the present to secure more later.
This choice reflects their preference for stability over risk, influencing everything from personal savings to investment strategies.

Marginal Utility

Marginal utility is the additional satisfaction a person gets from consuming one more unit of a good or service.
It's an essential concept because it explains why consumption choices often change with varying levels of consumption.The formula for marginal utility is derived from the derivative of the utility function: \( U'(c) \).
As consumption increases, marginal utility typically decreases; this is known as the law of diminishing marginal utility.
For example, the satisfaction of eating one slice of pizza is more significant than the 10th slice in one sitting.In economic modeling, knowing the marginal utility helps to determine optimal levels of consumption and the price at which someone will be indifferent to buying or selling an additional unit.
It shapes supply, demand, and prices throughout the economy.

Equilibrium Price

The equilibrium price is the cost at which the quantity of a good demanded by consumers equals the quantity supplied by producers.
It is where the market clears, meaning there's no excess supply or unmet demand.In the context of the exercise, we're dealing with two different assets, each corresponding to a different economic state.
Determining equilibrium price \( p \) involves using the formula derived from setting expected utilities to be the same across both states: \( p = \frac{U'(1)}{U'(1 - \frac{\phi}{\lambda})} \).
At this price, individuals are indifferent between holding either asset, balancing out supply and demand.Equilibrium prices are pivotal for ensuring efficient market operations. They guide economic participants in making informed choices, aligning individual actions with broader economic realities.