Question

Let the national-income model be: \\[ \begin{array}{ll} Y=C+l_{0}+C \\ C=a-b\left(Y-T_{0}\right) & (a>0, \quad 0

Short answer

(a) Endogenous: \(Y\) and \(C\). (b) \(g\) relates government spending to income. (c) \(Y = \frac{a + bT_0 + l_0}{1 + b - g}\). (d) \(1 + b > g\).

Step-by-step solution

Identify Endogenous Variables

Endogenous variables are those whose values are determined by the model. In this national income model, the endogenous variables are \(Y\) (national income) and \(C\) (consumption), as they are both determined within the system of equations provided.

Economic Meaning of Parameter g

The parameter \(g\) in the equation \(G=gY\) represents the government's expenditure as a fraction of national income \(Y\). It shows how government spending is directly related to the national income. If \(g\) increases, government spending rises proportionally to any change in \(Y\).

Find Equilibrium National Income

At equilibrium, total output equals total expenditure. The expression for national income \(Y\) can then be derived by combining the given equations. Consider the equilibrium condition: \[ Y = C + l_0 + G \]Substitute for \(C\) and \(G\):\[ Y = (a - b(Y - T_0)) + l_0 + (gY) \]Re-arrange and solve for \(Y\):\[ Y = a - bY + bT_0 + l_0 + gY \]Collect like terms:\[ Y(1 + b - g) = a + bT_0 + l_0 \]Solving for \(Y\):\[ Y = \frac{a + bT_0 + l_0}{1 + b - g} \]

Parameter Restriction for Solution

For a solution to exist, the denominator in the equilibrium expression for \(Y\) must be non-zero to avoid division by zero. Thus:\[ 1 + b - g eq 0 \]Further, since \(0 < b < 1\) and \(0 < g < 1\), a logical restriction is \(1 + b > g\), ensuring that the denominator remains positive.

Key concepts

Endogenous Variables

In the context of a national income model, endogenous variables are those that are determined within the model itself. These variables depend on the interactions described by the equations provided in the model. In our specific exercise, the endogenous variables are the national income ( Y ) and consumption ( C ). This means that both these variables are determined by the interplay of the given equations. Consumption ( C ) is described as a function of the income and taxes, making it inherently linked to changes in national income. Hence, C is affected by variations in income, while the national income ( Y ) is a primary outcome influenced by various components such as consumption and government spending. Recognizing which variables are endogenous is crucial as it helps in understanding the dynamics of the entire economic model.

Equilibrium National Income

The equilibrium national income is a fundamental concept in understanding economic stability. Equilibrium occurs when total output in an economy equals total expenditure. In simpler terms, this means that all production is being bought, implying no surplus or shortage of goods and services. To find this equilibrium, we set the left side of the income equation equivalent to total spending:

• The equation is initially given by Y = C + l_0 + G, where Y is national income, C is consumption, l_0 is a constant, and G is government spending.
• Substituting the expressions for C and G, we get Y = (a - b(Y - T_0)) + l_0 + (gY).
• By isolating Y, we rearrange terms to find that Y equals \( \frac{a + bT_0 + l_0}{1 + b - g} \).

This expression answers what the national income will be when the economy is in a state of balance, with no tendencies to change.

Economic Parameters

Economic parameters are constants in a model that define specific attributes and interactions, they are not influenced by the model itself but are rather set externally. In the national income model, several parameters shape its structure and outcome. The parameter a represents autonomous consumption, which is the level of consumption when income is zero. b is the marginal propensity to consume, indicating the fraction of additional income that will be spent.
Furthermore, the constant l_0 represents other fixed influences within the system, such as baseline consumption not influenced by direct income factors. Each of these components needs to be understood in context to grasp their influence on the overall model results. Without properly defined parameters, interpreting the results from any economic model becomes challenging, as these parameters guide us to how sensitive the model is to changes in income or consumption rates.

Government Expenditure

In the economic model discussed, government expenditure is denoted by G = gY . Here, g is a parameter that indicates the fraction of national income ( Y ) that the government spends. This suggests a responsive relationship where government spending scales with changes in national income. If g were to increase, it would lead to higher government expenditures as national income increases.
Understanding the role of government expenditure is critical because it reflects how government activities can influence aggregate demand and, thus, overall economic activity. By linking government spending with income, it sheds light on fiscal policy implications and the potential for stimulating or dampening economic growth through budget adjustments.