Suppose a certain economy's consumption function is as follows, where C(x) and x are measured in billions of dollars.
C(x) = 0.76x1.1 + 18.19
Find the marginal propensity to consume (dC/dx) when x = 7. (Round your answer to three decimal places.)
To find the marginal propensity to consume (MPC), we need to differentiate the consumption function C(x) with respect to x, which gives us dC/dx.
In this case, the consumption function is C(x) = 0.76x^1.1 + 18.19.
To differentiate this function, we use the power rule of differentiation. For a function of the form f(x) = cx^n, the derivative is given by f'(x) = ncx^(n-1).
Applying the power rule, we have:
dC/dx = d/dx (0.76x^1.1 + 18.19)
= 1.1 * 0.76 * x^(1.1-1)
= 0.836 * x^0.1
To find the MPC when x = 7, we substitute x = 7 into the derivative expression:
dC/dx = 0.836 * 7^0.1
≈ 0.836 * 1.260
≈ 1.052
Therefore, the marginal propensity to consume (dC/dx) when x = 7 is approximately 1.052, rounded to three decimal places.