About Marginal Propensity to Consume and the Multiplier

What is the marginal propensity to consume when consumption changes from 7 to 6 and disposable income changes from 5 to 3?

According to the material, the ratio of the change in consumption (ΔC) to the change in disposable personal income (ΔYd) is the marginal propensity to consume (MPC). The Greek letter delta (Δ) is used to denote “change in.”

The marginal propensity to consume (MPC)=(ΔC)/(ΔYd)

The change in consumption (ΔC) is from 7 to 6, so the change is -1 (If we focus on the "change", not the direction of positive or negative, that answer is 1)

The change in disposable personal income (ΔYd) is from 5 to 3, the change is -2 (If we focus on the "change", not the direction of positive or negative, that answer is 2)

The marginal propensity to consume (MPC)=(ΔC)/(ΔYd)=(-1)/(-2)=0.5

If disposable personal income is 10 and consumption is 12, what is personal savings? 

According to the material, personal saving is disposable personal income not spent on consumption during a particular period. ( Personal saving = disposable personal income - consumption )

If disposable personal income is 10 and consumption is 12, then the personal saving will be 10 minus 12, that equals minus 2 (10-12=-2)

What does this mean?

In this case, the consumption exceeds disposable personal income, so we get a negative value for saving and the excess must have come from saving accumulated in the past, from selling assets that earned in the past, or even from borrowing.

It also means that consumption choices could be affected by expectations of income and almost all consumption choices could be affected by it over a very long period. 

What is the multiplier when the change in the equilibrium level of real GDP in the aggregate expenditures model is 9, and change in autonomous aggregate expenditures is 3?

Suppose that :

ΔYeq = The change in the equilibrium level of real GDP

ΔA ̄ = The change in autonomous aggregate expenditures

MPC = marginal propensity to consume 

MPS = marginal propensity to save

And, the multiplier is the number by which we multiply an initial change in aggregate demand to get the full amount of the shift in the aggregate demand curve. 

So, the multiplier = ΔYeq/ΔA ̄ 

The relationship between a change in autonomous aggregate expenditures and the change in the equilibrium level of real GDP.

The multiplier = ΔYeq/ΔA ̄ = 9/3 = 3

According to the material, a change in autonomous aggregate expenditures changes equilibrium real GDP by a multiple of the change in autonomous aggregate expenditures. The size of the multiplier depends on the slope of the aggregate expenditures curve. The steeper the aggregate expenditures curve, the larger the multiplier; the flatter the aggregate expenditures curve, the smaller the multiplier.

What is the multiplier when the marginal propensity to save is 1/3?

The multiplier = ΔYeq/ΔA ̄ = 1/(1-MPC) = 1/MPS = 1/(1/3) = 3

What would happen to the marginal propensity to save when a tax cut was enacted causing the multiplier to change to 5?

Suppose that :

MPS = marginal propensity to save

MPC = marginal propensity to consume

If the multiplier = ΔYeq/ΔA ̄ = 1/(1-MPC) = 1/MPS = 5

then, MPS = 1/5 = 0.2 

Reference

https://my.uopeople.edu/pluginfile.php/588647/mod_resource/content/1/TEXT%20macroeconomics-principles-v2.0.pdf