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

To find the marginal propensity to save from the given consumption function, we need to differentiate the savings function, S(x), with respect to x.

First, let's rewrite the consumption function C(x) as a linear function in the form of y = mx + b, where m is the slope and b is the y-intercept.
C(x) = 0.701x + 99.34

Then, we can rewrite the savings function S(x) as:
S(x) = x - C(x)

Now, let's find the derivative of S(x) with respect to x using the power rule of differentiation. The derivative of x with respect to x is simply 1, and the derivative of C(x) is the derivative of the linear term, which is just the slope m.
So, dS/dx = 1 - dC(x)/dx

To find dC(x)/dx, we differentiate the consumption function C(x) with respect to x to get the slope, which is the marginal propensity to consume (MPC) in this case. In other words, dC(x)/dx represents the change in consumption corresponding to a 1-unit change in income.

Differentiating C(x) = 0.701x + 99.34 with respect to x gives:
dC(x)/dx = 0.701

Substituting this back into the derivative of S(x), we have:
dS/dx = 1 - dC(x)/dx
= 1 - 0.701

Therefore, the marginal propensity to save is 1 - 0.701, which equals 0.299.

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