Question

What is the relationship between the consumption function, the saving function, and the "income function" or \(45^{\circ}\) - line?

Short answer

The relationship between the consumption function, saving function, and the income function (45-degree line) can be illustrated through the Keynesian cross diagram. The consumption function (\(C = C_0 + cY\)) represents the relationship between a household's consumption spending and their disposable income, while the saving function (\(S = Y - C\)) represents the relationship between a household's disposable income and their saving. The 45-degree line represents all points where aggregate expenditure (AE) equals aggregate income (Y). Equilibrium in the economy occurs when the aggregate expenditure (AE) equals aggregate income (Y), at which point, the consumption function (C) and the saving function (S) intersect the 45-degree line. At equilibrium, \(C + S = Y\), indicating that all income generated by the economy is either consumed or saved.

Step-by-step solution

Consumption and Saving function

The consumption function represents the relationship between a household's total consumption spending and their disposable income, which is the income remaining after deducting taxes and adding government transfer payments. It assumes that as disposable income increases, so too will consumption spending, but spending will increase at a lesser rate than the income growth. The saving function, on the other hand, is the relationship between a household's disposable income and their saving, which is essentially the disposable income that is not dedicated to consumption. In mathematical terms, let's denote disposable income with the symbol Y, consumption with C, and saving with S, we can represent the consumption function as \(C = C_0 + cY\), where \(C_0\) is the autonomous consumption (minimum consumption irrespective of income), and c is the marginal propensity to consume (0 < c < 1). The saving function can be represented as \(S = Y - C\).

Income function (45-degree line)

The 45-degree line, also known as the income function, is a line in the Keynesian cross diagram that represents all points where aggregate expenditure (AE) equals aggregate income (Y). In the Keynesian cross model, it is given by the equation \(AE = Y\). At any point on the 45-degree line, the economy is in equilibrium as the total spending in the economy is equal to the total income produced by the economy.

Relation between Consumption, Saving function and Income function

The relationship between the consumption function, saving function, and the income function can be illustrated through the Keynesian cross diagram. On the vertical axis, we have aggregate expenditure (AE), while on the horizontal axis, we have aggregate income (Y). The consumption function is an upward sloping line as it shows that the consumption expenditure increases with an increase in disposable income. Similarly, the saving function is also an upward sloping curve to represents the relationship between income and saving as income increases. The 45-degree line is plotted by the equation \(AE = Y\), signifying where the aggregate expenditure equals the aggregate income. This line cuts the Consumption function at a point. At that point, the distance of the point to the x-axis represents consumption, and the difference between Y and Consumption represents the Saving function. The equilibrium in the economy occurs when the aggregate expenditure (AE) equals aggregate income (Y), at which point, the consumption function (C) and the saving function (S) intersect the 45-degree! f=line. Specifically, at equilibrium, \(C + S = Y\), indicating that all income generated by the economy is either consumed or saved.

Key concepts

Consumption Function

Imagine you have a certain amount of money to spend after paying all your taxes. The big question is, how much of that money will you spend, and how much will you tuck away for a rainy day? In economics, this spending behavior is captured by the consumption function.

It's a simple yet powerful idea that shows us the relationship between the income people have available to spend and the amount they actually do spend on goods and services. Typically, as people earn more, they also spend more — but not all of it! There's a tendency to save a bit of that extra cash.

The formula that represents this is (C = C_0 + cY), where C stands for consumption, C_0 for autonomous consumption, which is the amount consumed regardless of income, and c is the marginal propensity to consume or the fraction of extra income that is spent. This marginal propensity to consume is between 0 and 1, which makes sense because you won't spend more than the additional dollar you earn, right?

Why is this important?

Understanding the consumption function is crucial because it helps us predict how changes in income levels can affect spending and thus the overall economy. A government, for instance, might use this information to make decisions about taxes and spending.

Saving Function

While the consumption function looks at spending, the saving function is all about figuring out those dollars that are stashed away instead of spent. It shows the relationship between disposable income — the money left after taxes — and savings.

Remember that whatever is not consumed is considered saved. So, essentially, the saving function complements the consumption function. If we denote saving by S, and again represent disposable income as Y, the saving function formula gets pretty straightforward: (S = Y - C). When earnings increase, so does the capacity to save, assuming consumption doesn't gobble up the whole increase.

What's the takeaway?

By understanding the saving function, we get insights into the impact of various economic policies and individual decisions on a household's saving habits. For example, if interest rates rise, people might be tempted to save more since they would get more return on their savings. This could ultimately influence a country's investment levels and economic growth.

Aggregate Expenditure

Now, let's piece everything together with the concept of aggregate expenditure (AE), which is a fancy term for the total amount of spending in the economy. This includes spending by households, businesses, the government, and even foreigners buying our exports, minus the money spent on imports.

The key thing to remember here is that aggregate expenditure equals the total production of goods and services, which we also refer to as Y, in a closed economy without trade. The Keynesian cross diagram represents this beautifully with its 45-degree line, where every point on the line is where AE equals income (Y).

Connecting the Dots with Equilibrium

In an ideal scenario, the economy is at equilibrium when aggregate expenditure is equal to the output of the economy. This means that all income generated is either being spent (consumption) or saved — nothing is wasted. Understanding this balance is critical for both policymakers and businesses as it influences decisions that affect economic stability, inflation, and employment rates.