Understanding Consumption Functions

The question asks for the specific form of the consumption function ($C$) when the Marginal Propensity to Consume (MPC) is equal to the Average Propensity to Consume (APC) at all income levels ($Y$).

Defining MPC and APC

• Marginal Propensity to Consume (MPC): This measures the change in consumption resulting from a unit change in income. Mathematically, $MPC = \frac{\Delta C}{\Delta Y}$. For a linear consumption function, $MPC$ is the slope.
• Average Propensity to Consume (APC): This measures the proportion of income that is spent on consumption. Mathematically, $APC = \frac{C}{Y}$.

Analyzing the Condition: MPC = APC

We are given the condition that $MPC = APC$ for all income levels ($Y$). Let's analyze the options:

1. If $C = a + bY$:
   • $MPC = \frac{dC}{dY} = b$ (constant)
   • $APC = \frac{C}{Y} = \frac{a + bY}{Y} = \frac{a}{Y} + b$
   • For $MPC = APC$, we need $b = \frac{a}{Y} + b$, which implies $\frac{a}{Y} = 0$. This only holds if $a = 0$ (and $Y \neq 0$).
2. If $C = bY$:
   • $MPC = \frac{dC}{dY} = b$ (constant)
   • $APC = \frac{C}{Y} = \frac{bY}{Y} = b$ (constant)
   • In this case, $MPC = APC = b$ for all income levels. This satisfies the given condition.
3. Other forms like $C = a + \frac{b}{Y}$ or $C = bY^2$ yield MPC and APC values that change with income, thus not satisfying the condition $MPC = APC$ for all levels of $Y$.

Conclusion

The consumption function where MPC equals APC for all income levels is the one where the constant term is zero. Therefore, the correct consumption function is $C = bY$.