Chapter 5: Problem 167

Explain why the sum of the MPC and the MPS for any given change in disposable income must always be equal to 1 .

Short Answer

Expert verified

The sum of the Marginal Propensity to Consume (MPC) and the Marginal Propensity to Save (MPS) for any given change in disposable income must always be equal to 1 because disposable income (YD) is divided into two parts: the amount consumed (C) and the amount saved (S). When considering a change in disposable income (ΔYD) and dividing this change into corresponding changes in consumption (ΔC) and savings (ΔS), the following relationship holds: \[1 = \frac{ΔC}{ΔYD} + \frac{ΔS}{ΔYD}\] This relationship confirms that the sum of MPC and MPS equals 1, demonstrating the fundamental principle in macroeconomics between consumption, savings, and disposable income.

Step by step solution

Define Marginal Propensity to Consume (MPC)

The Marginal Propensity to Consume (MPC) is the fraction of an additional unit of disposable income that is spent on consumption. Mathematically, it is represented as the change in consumption (ΔC) divided by the change in disposable income (ΔYD): \[MPC = \frac{ΔC}{ΔYD}\]

Define Marginal Propensity to Save (MPS)

The Marginal Propensity to Save (MPS) is the fraction of an additional unit of disposable income that is saved. Mathematically, it is represented as the change in savings (ΔS) divided by the change in disposable income (ΔYD): \[MPS = \frac{ΔS}{ΔYD}\]

Relate MPC and MPS to disposable income

A household's disposable income (YD) is divided into two parts: the amount consumed (C) and the amount saved (S). Therefore, we can represent the relationship between disposable income, consumption, and savings as: \[YD = C + S\]

Derive the relationship between MPC and MPS

Now, let's consider a change in disposable income (ΔYD) and divide this change into the corresponding changes in consumption (ΔC) and savings (ΔS): \[ΔYD = ΔC + ΔS\] Now, divide both sides of the equation by ΔYD: \[\frac{ΔYD}{ΔYD} = \frac{ΔC}{ΔYD} + \frac{ΔS}{ΔYD}\] Since the left side of the equation is equal to 1 (as any number divided by itself is equal to 1), we can rewrite the equation as: \[1 = MPC + MPS\]

Conclusion

We have shown that the sum of the Marginal Propensity to Consume (MPC) and the Marginal Propensity to Save (MPS) for any given change in disposable income must always be equal to 1. This result is a fundamental principle in macroeconomics, and it demonstrates the relationship between consumption, savings, and disposable income.