Question

Optimal control theory can be used to generalize the model of intertemporal consumption choice contained in Example 23.1 . Consider the following simple life cycle model: An individual receives wages (w) each period and a return on his or her invested capital. Let \(k=\) capital, \(r=\) market interest rate at which the individual can borrow or lend. During each period, the individual chooses consumption (c) to maximize \\[ f(c) e-p^{\prime} d t \\] where \(p\) is the individual's rate of time preference. Given these assumptions, the intertemporal budget constraint for this problem is \\[ k=w+r k-c \\] with constraints on initial and final \(k\) of the form \(k(0)=k(T)=0\) a. What are the necessary conditions for a maximum for this problem? b. Under what conditions would optimal consumption rise over time? When would con sumption fall over time? c. Suppose \(U(c)=\ln (c),\) what is the optimal pattern of consumption? d. More generally, suppose \\[ U(c)=j \quad 8<1 \\] What is the optimal time pattern for consumption? How does this compare to the special case in part (c)? e. How does the optimal time pattern for consumption in this problem determine this in dividual's measured wealth at various points in the life cycle?

Short answer

Answer: Yes, the optimal consumption pattern depends on the relationship between the interest rate and the individual's rate of time preference, as well as the shape of the utility function.

Step-by-step solution

Calculation of Necessary Conditions

To find the necessary conditions for a maximum for this problem, we must set up the individual's optimization problem, i.e., maximizing the utility function subject to the budget constraint. Write the Hamiltonian function for this problem: \\[ H(c,k,\mu)= f(c)e^{-pt} + \mu(w+rk-c) \\] where \(\mu\) is the costate (Lagrange multiplier). Now, find the first-order conditions (FOC) by differentiating the Hamiltonian with respect to \(c\), \(k\), and \(\mu\): 1. \(\frac{\partial H}{\partial c} = 0 \Rightarrow \frac{\partial f(c)}{\partial c} e^{-pt} -\mu = 0\) 2. \(\frac{\partial H}{\partial k} = \dot{\mu} \Rightarrow -\frac{\partial}{\partial t} (\mu) = r\mu\) 3. \(\frac{\partial H}{\partial \mu} = 0 \Rightarrow w+rk-c=k\) Lastly, we need to consider the boundary conditions for the problem, given as \(k(0)=k(T)=0\).

Conditions for Optimal Consumption Patterns

Intertemporal consumption would rise over time if the optimal consumption growth outpaces the growth in income and wealth. This will occur when \(\frac{d}{dt}c > 0\). Conversely, consumption would fall over time if \(\frac{d}{dt}c < 0\). The optimal consumption pattern depends on the relationship between the interest rate and the individual's rate of time preference, as well as the shape of the utility function.

Optimal Consumption with Logarithmic Utility Function

When the utility function is logarithmic (\(U(c)=\ln(c)\)), we can derive the optimal pattern of consumption by following the previous steps. Given that \(U(c) = \ln(c)\), we have that \(f(c) = c\). Plug this into the FOC equation: \\[ ce^{-pt} - \mu = 0 \\] Solving for the optimal consumption pattern, we find: \\[ c^*(t) = \mu e^{pt} \\]

Optimal Consumption with General Utility Function

Let's now consider the general utility function given by \(U(c)=j\), where \(j \quad 8 < 1\). We can follow the same steps as above to derive the optimal consumption pattern. Given that \(U(c) = j\), we have that \(f(c) = c^j\). Plug this into the FOC equation: \\[ c^je^{-pt} - \mu = 0 \\] Solving for the optimal consumption pattern, we find: \\[ c^*(t) = \left(\mu e^{pt}\right)^\frac{1}{j} \\] Comparing this with the part (c) result, we can see that as \(j\) approaches \(1\), the optimal consumption pattern converges to the logarithmic utility function case.

Measured Wealth Over the Life Cycle

The optimal time pattern for consumption affects the measured wealth at various points in the life cycle. To analyze this, we need to make use of the intertemporal budget constraint: \(k = w + rk - c\). As the individual consumes less in the early stages of their life, they save more and invest, causing their wealth to increase. If their consumption pattern is such that it increases over time, their wealth will start to decrease as they age. This is because they will be drawing down their savings to finance their higher consumption levels. On the other hand, if their consumption pattern shows a decreasing trend over time, their wealth will continue to increase throughout their life. This is because they will continue to save and invest while consuming less.

Key concepts

Intertemporal Consumption

Intertemporal consumption is a crucial aspect of understanding how individuals spread their consumption over time. Imagine you earn a certain amount today, but you want to decide how much to consume now versus saving it for the future. This consumption decision requires considering the future benefits and costs of consuming today rather than tomorrow.

In the context of optimal control theory, intertemporal consumption is represented through an individual's utility maximization problem. This problem involves maximizing an individual's lifetime utility by allocating resources optimally across different periods. The individual's choices depend on factors like income, savings, interest rates, and personal preferences for present versus future consumption. This creates a dynamic optimization scenario which is often modeled using the Euler's consumption equation, linking consumption growth to the difference between the interest rate and an individual's rate of time preference.

• If the interest rate is higher than the rate of time preference, consumption is likely to increase over time, as individuals postpone consumption to gain from higher future returns.
• Conversely, if the rate of time preference is higher, consumption might decrease as people prefer present gratification.

Hamiltonian Function

In optimal control theory, the Hamiltonian function helps to solve optimization problems that involve dynamic systems. It's a powerful tool used to find optimal policies over time. For intertemporal consumption, the Hamiltonian can be thought of as a bridge between current consumption decisions and future outcomes.

The Hamiltonian function combines the instantaneous utility function and the budget constraint through the use of a Lagrange multiplier, often denoted by \(\mu\). In the provided exercise, the Hamiltonian was expressed as:\[H(c,k,\mu) = f(c)e^{-pt} + \mu(w + rk - c)\]
**Key Aspects of the Hamiltonian Function:**

• The function accounts for immediate consumption utility \(f(c)e^{-pt}\), emphasizing the discounting of future utility.
• The multiplier \(\mu\) reflects the shadow price of capital, guiding decisions on resource allocation between consumption and investment.
• Through first-order conditions derived from partial derivatives of the Hamiltonian (w.r.t. \(c, k, \mu\)), we derive the equations that characterize optimal consumption paths and savings decisions over time.

Utility Function

A utility function represents an individual's preferences over a range of goods and services they can consume. Generally, it reflects the satisfaction or happiness derived from consuming these goods. In intertemporal choice models, utility functions are used to express the trade-off between consumption today and consumption in the future.

The simpler form given in the exercise was the logarithmic utility function \(U(c) = \ln(c)\), widely used due to its desirable mathematical properties, such as diminishing marginal utility. It allows for relatively straightforward computation of optimal consumption plans.

More complex utility functions like \(U(c) = j ext{ where } j<1\) present other dynamics. For each of these, the function \(f(c) = c^j\) incorporated into the Hamiltonian leads to different first-order conditions and thereby different consumption plans. These utility functions highlight various degrees of risk aversion and intertemporal elasticity of substitution.

• Higher values of \(j\) (close to 1) indicate lesser risk aversion and a consumption pattern close to logarithmic behavior.
• Lower values of \(j\) (significantly less than 1) reflect greater risk aversion and a stronger inclination to smooth consumption over time.

Life Cycle Model

The life cycle model is a theoretical framework used to explain the financial behavior of individuals throughout their lives. It describes how people plan their consumption, savings, and work decisions over their lifetime to maximize their utility.

In the exercise, the life cycle model explores how individuals receive wages and returns on their capital, which they allocate for consumption and savings to maximize their utility over their entire life. The constraint that capital at the beginning and end of the life cycle \((k(0) = k(T) = 0)\) implies that individuals optimize their resources to neither leave debts nor unused wealth.

**Key insights from the Life Cycle Model:**

• Individuals seek to smooth consumption over time, balancing income fluctuations and saving for retirement.
• In early life, they may save to invest or borrow against future earnings. They draw down these savings as they approach retirement.
• Consumption patterns and wealth accumulation are driven by the discount rate on future utility, interest rates on savings, and predicted life span.

This model is fundamental for understanding how economic agents plan for both systematic (predictable) and idiosyncratic (random) life events.