Suppose a certain economy's consumption function is as follows, where C(x) and x are measured in billions of dollars.
C(x) = 0.822x1.1 + 17.35
Find the marginal propensity to consume (dC/dx) when x = 7. (Round your answer to three decimal places.)
$ 
 
Incorrect: Your answer is incorrect.
 billion per billion dollars

The marginal propensity to consume (MPC) is the derivative of the consumption function $ C(x) $ with respect to $ x $. To find the marginal propensity to consume at $ x = 7 $, we need to compute $ \frac{dC}{dx} $, then substitute $ x = 7 $.

Given the consumption function:
\[
C(x) = 0.822x^{1.1} + 17.35
\]
we need to differentiate it with respect to $ x $.

### Step 1: Differentiate $ C(x) $
The derivative of $ C(x) = 0.822x^{1.1} + 17.35 $ is:

\[
\frac{dC}{dx} = 0.822 \cdot \frac{d}{dx}(x^{1.1}) + \frac{d}{dx}(17.35)
\]

Using the power rule for differentiation $ \frac{d}{dx}(x^n) = n \cdot x^{n-1} $, we get:

\[
\frac{dC}{dx} = 0.822 \cdot 1.1 \cdot x^{0.1} + 0
\]
\[
\frac{dC}{dx} = 0.9042 \cdot x^{0.1}
\]

### Step 2: Evaluate $ \frac{dC}{dx} $ at $ x = 7 $

Now, substitute $ x = 7 $ into the derivative:

\[
\frac{dC}{dx} = 0.9042 \cdot 7^{0.1}
\]

Let's calculate this value.

The marginal propensity to consume (MPC) at $ x = 7 $ is approximately $ 1.098 $ billion per billion dollars.

Would you like more details or further clarifications?

Here are 5 related questions to expand your understanding:
1. How is the marginal propensity to consume related to the overall consumption function?
2. What economic implications can be drawn if the MPC is greater than 1?
3. How would the marginal propensity to consume change if the consumption function had a different exponent?
4. What is the significance of the constant 17.35 in the consumption function?
5. Can the marginal propensity to consume be interpreted differently in various economic models?

**Tip:** In economics, a marginal value represents the rate of change of one variable with respect to another, providing insight into incremental effects.

Suppose a certain economy's consumption function is as follows, where C(x) and x are measured in billions of dollars. C(x) = 0.822x^1.1 + 17.35 Find the marginal propensity to consume (dC/dx) when x = 7. (Round your answer to three decimal places.)

Learn how to find the marginal propensity to consume (MPC) from the consumption function C(x) = 0.822x^1.1 + 17.35 when x = 7. The solution involves applying the power rule for differentiation and calculating the rate of change of consumption with respect to income. Suitable for undergraduate economics or calculus students.

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