Consider the national income of a country, which consists of...

Consider the national income of a country, which consists of consumption, investment, and government expenditures. Here we assume the government expenditure to be constant, at $G_{0},$ while the national income $Y(t),$ consumption $C(t),$ and investment $I(t)$ change over time. According to a simple model, we have \[ \begin{array}{l} Y(t)=C(t)+I(t)+G_{0} \\ C(t+1)=\gamma Y(t) \\ I(t+1)=\alpha(C(t+1)-C(t)) \end{array} | \begin{array}{l} (0<\gamma<1) \\ (\alpha>0) \end{array} \] where $\gamma$ is the marginal propensity to consume and $\alpha$ is the acceleration coefficient. (See Paul E. Samuelson, "Interactions between the Multiplier Analysis and the Principle of Acceleration," Review of Economic Statistics, May $1939,$ pp. $75-78 .$ a. Find the equilibrium solution of these equations, when $Y(t+1)=Y(t), C(t+1)=C(t),$ and \[ I(t+1)=I(t) \] b. Let $y(t), c(t),$ and $i(t)$ be the deviations of $Y(t)$ $C(t),$ and $I(t),$ respectively, from the equilibrium state you found in part (a). These quantities are related by the equations \[ \begin{array}{l} y(t)=c(t)+i(t) \\ c(t+1)=\gamma y(t) \\ i(t+1)=\alpha(c(t+1)-c(t)) \end{array} | \] (Verify this!) By substituting $y(t)$ into the second equation, set up equations of the form \[ \left|\begin{array}{l} c(t+1)=p c(t)+q i(t) \\ i(t+1)=r c(t)+s i(t) \end{array}\right| \] c. When $\alpha=5$ and $\gamma=0.2,$ determine the stability of the zero state of this system. d. When $\alpha=1$ (and $\gamma$ is arbitrary, $0<\gamma<1$ ), determine the stability of the zero state. e. For each of the four sectors in the $\alpha-\gamma$ -plane, determine the stability of the zero state. Discuss the various cases, in practical terms.

Consider the national income of a country, which consists of consumption, investment, and government expenditures. Here we assume the government expenditure to be constant, at $G_{0},$ while the national income $Y(t),$ consumption $C(t),$ and investment $I(t)$ change over time. According to a simple model, we have
\[
\begin{array}{l}
Y(t)=C(t)+I(t)+G_{0} \\
C(t+1)=\gamma Y(t) \\
I(t+1)=\alpha(C(t+1)-C(t))
\end{array} | \begin{array}{l}
(0<\gamma<1) \\
(\alpha>0)
\end{array}
\]
where $\gamma$ is the marginal propensity to consume and $\alpha$ is the acceleration coefficient. (See Paul E. Samuelson, "Interactions between the Multiplier Analysis and the Principle of Acceleration," Review of Economic Statistics, May $1939,$ pp. $75-78 .$
a. Find the equilibrium solution of these equations, when $Y(t+1)=Y(t), C(t+1)=C(t),$ and
\[
I(t+1)=I(t)
\]
b. Let $y(t), c(t),$ and $i(t)$ be the deviations of $Y(t)$ $C(t),$ and $I(t),$ respectively, from the equilibrium state you found in part (a). These quantities are related by the equations
\[
\begin{array}{l}
y(t)=c(t)+i(t) \\
c(t+1)=\gamma y(t) \\
i(t+1)=\alpha(c(t+1)-c(t))
\end{array} |
\]
(Verify this!) By substituting $y(t)$ into the second equation, set up equations of the form
\[
\left|\begin{array}{l}
c(t+1)=p c(t)+q i(t) \\
i(t+1)=r c(t)+s i(t)
\end{array}\right|
\]
c. When $\alpha=5$ and $\gamma=0.2,$ determine the stability of the zero state of this system.
d. When $\alpha=1$ (and $\gamma$ is arbitrary, $0<\gamma<1$ ), determine the stability of the zero state.
e. For each of the four sectors in the $\alpha-\gamma$ -plane, determine the stability of the zero state.
Discuss the various cases, in practical terms.

Key Concepts

Equilibrium in Dynamic Systems

This concept refers to a state in which the system's variables remain constant over time because the forces within the model balance each other. In dynamic economic models, finding the equilibrium means determining the levels of output, consumption, and investment that do not change from one period to the next, providing a reference point around which the system's behavior can be analyzed.

Discrete-Time Difference Equations

These are mathematical expressions that describe how a variable evolves over distinct time periods. In the context of economic models, they capture the relationships between variables from one period to the next, allowing analysts to study the dynamic behavior of the economy over time.

Stability Analysis

Stability analysis involves determining whether a system will return to its equilibrium state after a small disturbance. In dynamic economic models, stability of the equilibrium indicates that the economy will self-correct after shocks, whereas instability may signal the potential for divergent or volatile behavior.

Linearization and Deviation Analysis

Linearization is the process of approximating a nonlinear system by a linear one near an equilibrium point, making it easier to analyze. By examining the deviations of variables from their equilibrium values, analysts can use linear techniques (often involving matrices and eigenvalues) to understand the system’s local dynamics and stability.

Multiplier and Acceleration Effects in Economics

The multiplier effect describes how an initial change in spending leads to a larger change in income, while the acceleration principle explains how investment levels respond to changes in income. Together, these concepts form the backbone of many macroeconomic models that explore how consumption, investment, and government spending interact over time.

Eigenvalue Analysis in Assessing Stability

This technique uses the eigenvalues of a system’s matrix (obtained after linearization) to determine the stability of the equilibrium. In economic dynamic models, if all eigenvalues lie within a certain range (for example, have absolute values less than one in discrete-time systems), the equilibrium is considered stable; otherwise, it may be unstable or exhibit complex dynamics.

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