Marginal propensity to consume (MPC) can be defined as the change in consumption that results from changes in national income, assuming all income is either saved or consumed.

It is the first derivative of the consumption function.

To obtain consumption function from marginal propensity to consume, we simply integrate the marginal propensity to consume and the procedures are as follows:

**STEPS TO OBTAIN CONSUMPTION FUNCTION FROM MARGINAL PROPENSITY TO CONSUME**

1. Given the marginal propensity to consume function

2. Integrate the marginal propensity to consume function

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

4. Insert the value of c on the original consumption function.

__Illustrative Example__

Obtain the consumption function given that MPC=0.3+0.0075Y and consumption is 75 when national income is 100

**Solution:**

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

$MPC=0.03+0.0075Y$

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

Using power rule of integration

$C=\int (0.03+0.0075Y) dy$

$C=\frac{0.03Y}{1}+\frac{0.0075Y^2}{2}+c$

$$C=0.03Y+0.00375Y^2+c$$

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

Since C=75 when Y=100

$75=0.03(100)+0.00375(100)^2+c$

$75=3+37.5+c$

$c=75-40.5=34.5$

**Step 4:** *Insert the value of c in the original consumption function*.

$C=0.03Y+0.00375Y^2+c$

$C=0.03Y+0.00375Y^2+34.5$

Hence, the consumption function is $C=0.03Y+0.00375Y^2+c$

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