Suppose C(x) measures an economy's personal consumption expenditure and x the personal income, both in billions of dollars. Then the following function measures the economy's savings corresponding to an income of x billion dollars.

S(x) = x - C(x) (Income minus consumption)
The quantity dS/dx below is called the marginal propensity to save.

For the following consumption function, find the marginal propensity to save.
C(x) = 0.701x + 99.34
$ billion per billion dollars

according to your definition ...

S(x) = x - .701x - 99.34
then dS(x)/dx = 1 - .701 = .299

To find the marginal propensity to save, we need to find the derivative of the savings function, S(x) = x - C(x), with respect to x.

First, let's substitute the consumption function into the savings function:
S(x) = x - (0.701x + 99.34)

Now, let's simplify the expression:
S(x) = x - 0.701x - 99.34
S(x) = (1 - 0.701)x - 99.34
S(x) = 0.299x - 99.34

Now, we will take the derivative of S(x) with respect to x:
dS/dx = d/dx (0.299x - 99.34)

The derivative of 0.299x with respect to x is 0.299, and the derivative of -99.34 with respect to x is 0 (since it's a constant).

Therefore, the marginal propensity to save, dS/dx, is 0.299.

So, the marginal propensity to save is 0.299 billion dollars per billion dollars.

To find the marginal propensity to save (dS/dx), we need to differentiate the savings function S(x) = x - C(x) with respect to x.

First, let's substitute C(x) = 0.701x + 99.34 into the equation for S(x):

S(x) = x - (0.701x + 99.34)

Next, simplify the equation:

S(x) = x - 0.701x - 99.34

To differentiate the equation, we treat x as the variable and differentiate each term separately.

The derivative of x with respect to x is 1.

The derivative of -0.701x with respect to x is -0.701.

The derivative of -99.34 with respect to x is 0, as it's a constant term.

Now we can put it all together:

dS/dx = 1 - 0.701 - 0

Simplifying further:

dS/dx = 0.299

Therefore, the marginal propensity to save for the given consumption function C(x) = 0.701x + 99.34 is 0.299 billion per billion dollars.

This is a optimization problem !!