Question

Using the Keynesian consumption function, prove numerically that, as the $MPC$ rises, saving declines.

Short answer

It's been numerically demonstrated that as the Marginal Propensity to Consume (MPC) increases, saving decreases.

Step-by-step solution

Understand the terms and equations

We can start by setting up the basic Keynesian Consumption Function $C=\bar{C}+MPC\cdot YD$. Also keep in mind that Saving (S) equals income minus consumption: $S=YD-C$.

Choose numerical values

Let's take a specific numerical example. Suppose $YD=100$, $MPC=0.8$ initially and $\bar{C}=20$. Plug these values into the consumption equation to find consumption: $C=20+0.8\times 100=100$. Using the saving equation, saving is $S=100-100=0$.

Change the MPC

Now, let's increase the MPC to 0.9 and calculate consumption and saving again. Now, $C=20+0.9\times 100=110$, and the saving is $S=100-110=-10$.

Analyze the result

From Step 2 to Step 3, as MPC increased from 0.8 to 0.9, consumption increased and saving decreased from 0 to -10. The amount the consumer saves has decreased as MPC rises, which proves our original statement.