Given the consumption function c=a+by(with a>0;0<b<1): (A) Find its marginal function and its average function. (B) Find the income elasticity of consumption £C, and determine its sign, assuming Y>0. (C) Show that this consumption function is inelastic at all positive income levels.

The marginal consumption function is the derivative of the consumption function with respect to income (Y).

$$\text{MPC} = \frac{dC}{dY} = \frac{d}{dY}(a + bY)$$ $$\text{MPC} = b$$

The marginal consumption function is $\boxed{b}$.

The average consumption function is total consumption divided by total income.

$$\text{APC} = \frac{C}{Y} = \frac{a + bY}{Y}$$ $$\text{APC} = \frac{a}{Y} + b$$

The average consumption function is $\boxed{\frac{a}{Y} + b}$.

B) Find the income elasticity of consumption $\epsilon_C$, and determine its sign, assuming Y>0.

The formula for income elasticity of consumption is $\epsilon_C = \frac{dC}{dY} \times \frac{Y}{C}$.
We know $\frac{dC}{dY} = b$ and $C = a + bY$.

$$\epsilon_C = b \times \frac{Y}{a + bY}$$ $$\epsilon_C = \frac{bY}{a + bY}$$

The income elasticity of consumption is $\boxed{\frac{bY}{a + bY}}$.

Given $a > 0$, $0 < b < 1$, and $Y > 0$.
Since $b > 0$ and $Y > 0$, the numerator $bY$ is positive.
Since $a > 0$, $b > 0$, and $Y > 0$, the denominator $a + bY$ is positive.
Therefore, $\epsilon_C = \frac{\text{positive}}{\text{positive}}$, which means $\epsilon_C$ is positive.

C) Show that this consumption function is inelastic at all positive income levels.

A function is inelastic if the absolute value of its elasticity is less than 1 ($|\epsilon_C| < 1$).
From Step 4, we know $\epsilon_C$ is positive, so $|\epsilon_C| = \epsilon_C$. We need to show $\frac{bY}{a + bY} < 1$.
Given $a > 0$, $0 < b < 1$, and $Y > 0$.
Consider the inequality:

$$\frac{bY}{a + bY} < 1$$

Since $a > 0$, $b > 0$, and $Y > 0$, the denominator $(a + bY)$ is positive. We can multiply both sides by $(a + bY)$ without changing the direction of the inequality:

$$bY < a + bY$$

Subtract $bY$ from both sides:

$$0 < a$$

This statement ($0 < a$) is true, as it is given that $a > 0$.
Since the final inequality is true based on the given conditions, the original inequality $\frac{bY}{a + bY} < 1$ is also true.
Thus, the consumption function is inelastic at all positive income levels.