Suppose a certain economy's consumption function is as follows, where C(x) and x are measured in billions of dollars.
C(x) = 0.76x^1.1 + 18.19
Find the marginal propensity to consume (dC/dx) when x = 7. (Round your answer to three decimal places.)
d(C(x))/dx = .836x^.1
when x = 7 , I get 1.01558...
or 1.016 , correct to 3 decimals
To find the marginal propensity to consume (MPC), we need to take the derivative of the consumption function with respect to x, denoted as dC/dx.
Given the consumption function:
C(x) = 0.76x^1.1 + 18.19
To find the derivative, we apply the power rule of differentiation, which states that for a function f(x) = ax^n, the derivative is given by f'(x) = nax^(n-1).
Using this rule, let's differentiate the consumption function term by term:
dC/dx = d/dx (0.76x^1.1 + 18.19)
= 1.1 * 0.76 * x^(1.1 - 1)
Notice that when taking the derivative of a constant (e.g., 18.19), it evaluates to zero, so we only consider the derivative of the term involving x.
dC/dx = 1.1 * 0.76 * x^0.1
Now, we have the derivative of the consumption function. To find the MPC when x = 7, substitute x = 7 into the derivative expression:
dC/dx = 1.1 * 0.76 * 7^0.1
Calculate the result:
dC/dx = 1.1 * 0.76 * 7^0.1 = 0.967
Therefore, the marginal propensity to consume (dC/dx) when x = 7 is approximately 0.967.