The marginal propensity to save (MPS) is simply the change in national savings that result from a change in national income.

Because marginal propensity to consume is the first derivative of the consumption function, it follows that consumption function is the anti-derivative of the marginal propensity to consume.

**Steps To Obtain Savings Functions From Marginal Propensity To Save (Mps)**

1. Given the marginal propensity to save function

2. Integrate the marginal propensity to save function

3. Obtain the value of the arbitrary constant, c, by inserting the value of national income and savings

4. Insert the value of c in the original function to get your savings function

__Illustrative example__

Obtain the savings function given that $MPS=0.97-0.0075Y$ and saving is 25 when national income is 100.

**Solution:**

**Step 1:** Given the marginal propensity to save function

$MPS=0.97-0.0075Y$

**Step 2:** Integrate the marginal propensity to save function

Using the power rule of integration

$S=\int (0.97-0.0075Y) dY$

$S=0.97Y-\frac{0.0075Y^2}{2}+c$

$$S=0.97Y-0.00375Y^2+c$$

**Step 3: **Obtain the value of the arbitrary constant, c, by inserting the value of national income and savings.

S=25 when Y=100

$S=0.97Y-0.00375Y^2+c$

$25=0.97(100)-0.00375(100)^2+c$

$25=97-37.5+c$

$c=25-59.5=-34.5$

**Step 4:** Insert the value of c in the original function to get your savings function.

$S=0.97Y-0.00375Y^2-34.5$

Hence, the savings function is $S=0.97Y-0.00375Y^2-34.5$

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