Problem 79 - Contemporary Precalculus

Question

In a simple model of the economy (by John Maynard Keynes), equilibrium between national output and national expenditures is given by the equilibrium equation $$ Y=C+I+G+(X-M) $$ where \(Y\) is the national income, C is consumption (which depends on national income), I is the amount of investment, \(G\) is government spending, \(X\) is exports, and \(M\) is imports (which also depend on national income). Solve the equilibrium equation for \(Y\) under the given conditions. $$\begin{aligned} &C=120+.9 Y, M=20+.2 Y, I=140, G=150, \text { and }\\\ &X=60 \end{aligned}$$

Step-by-Step Solution

Answer

## Short Answer In this exercise, we solved the equilibrium equation for the national income (Y) using given conditions for consumption, imports, investment, government spending, and exports. After substituting these conditions into the equilibrium equation and solving, we found that the national income at this equilibrium point is 4500.

Step 1: Write down the equilibrium equation

The equilibrium equation is given by: $$ Y = C + I + G + (X - M) $$

Step 2: Substitute the given conditions

We have the given conditions: $$ C = 120 + 0.9Y \\ M = 20 + 0.2Y \\ I = 140 \\ G = 150 \\ X = 60 $$ Substitute these conditions into the equilibrium equation: $$ Y = (120 + 0.9Y) + 140 + 150 + (60 - (20 + 0.2Y)) $$

Step 3: Simplify and solve for Y

Now, we can simplify the equation and solve for Y: $$ Y = 120 + 0.9Y + 140 + 150 + 60 - 20 - 0.2Y $$ Combine like terms: $$ Y - 0.9Y - 0.2Y = 120 + 140 + 150 + 60 - 20 $$ Simplify: $$ 0.1Y = 450 $$ Now, divide both sides by 0.1 to solve for Y: $$ Y = \frac{450}{0.1} = 4500 $$

Step 4: Interpret the solution

The solution we found, Y = 4500, indicates that the national income at this equilibrium point is 4500.

Key Concepts

Keynesian economics

Keynesian economics is a theory that arose during the Great Depression, developed by British economist John Maynard Keynes. This economic school of thought emphasizes the importance of aggregate demand in the economy and suggests that government intervention can lead to beneficial economic outcomes. According to Keynesian economics, markets do not always clear on their own, as classical economics suggested. Instead, they may require active policy measures to ensure economic stability.

Keynes proposed that high unemployment and low economic output could result from insufficient aggregate demand. To address this, the approach recommends increasing government expenditure and adjusting monetary policies to boost demand. This perspective has been foundational in shaping macroeconomic policy and offers tools to manage and predict fluctuations in the economy.

national income

National income is a crucial concept that reflects the total economic output of a country. It represents the sum of all goods and services produced over a specific period, often considered on an annual basis. In practical terms, national income measures the total value of income earned by a nation's residents and businesses.

This includes various components such as:

• Wages and salaries for labor provided.
• Profits from businesses and corporations.
• Income from investments and rental properties.

The interaction of these elements provides a comprehensive picture of a country’s economic performance.
National income is not just a measure of wealth; it is pivotal in policy formation, helping governments determine tax policies, spending, and welfare benefits. It also plays a significant role in comparing economic performance between countries.

equilibrium equation

The equilibrium equation presented in the exercise is a representation of the balance that must be achieved in an economy, as per Keynesian theory. The equation given is:
\[ Y = C + I + G + (X - M) \]
Where:

• \( Y \) is the national income.
• \( C \) is consumption, which generally increases with national income.
• \( I \) represents investments made by businesses.
• \( G \) denotes government spending, which is a tool for economic management.
• \( X - M \) is net exports, being the difference between what a country exports and imports.

Substituting the respective values and expressions into this equation helps to determine the national income necessary for economic equilibrium. Through simplification and solving, the solution demonstrates how these factors interact dynamically to sustain balance in the economy.

consumption function

The consumption function is a key concept in understanding how households decide on spending. In Keynesian economics, consumption is seen as primarily dependent on the national income level. The function generally follows the form:
\[ C = a + bY \]
Where:

• \( C \) is the total consumption.
• \( a \) is the autonomous consumption, which is the base level of consumption that occurs regardless of income.
• \( b \) is the marginal propensity to consume, reflecting the proportion of additional income that will be spent on consumption.

In this context, increases in national income naturally lead to increased consumption levels, as more income typically means more spending ability.
Understanding the consumption function is vital for predicting spending patterns and designing policies aimed at stimulating or cooling down an economy, as it directly affects aggregate demand levels and, thus, the overall economic activity.