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.

Here's the breakdown of the consumption function: A) Find its marginal function and its average function. The consumption function is given by C = a + bY. Step 1: Find the marginal consumption function (MPC). The marginal consumption function is the derivative of the consumption function with respect to income (Y). MPC = (dC)/(dY) = (d)/(dY)(a + bY) MPC = b The marginal consumption function is b. Step 2: Find the average consumption function (APC). The average consumption function is total consumption divided by total income. APC = (C)/(Y) = (a + bY)/(Y) APC = (a)/(Y) + b The average consumption function is (a)/(Y) + b. B) Find the income elasticity of consumption _C, and determine its sign, assuming Y>0. Step 3: Calculate the income elasticity of consumption. The formula for income elasticity of consumption is _C = (dC)/(dY) × (Y)/(C). We know (dC)/(dY) = b and C = a + bY. _C = b × (Y)/(a + bY) _C = (bY)/(a + bY) The income elasticity of consumption is (bY)/(a + bY). Step 4: Determine the sign of _C. 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, _C = positivepositive, which means _C is positive. C) Show that this consumption function is inelastic at all positive income levels. Step 5: Show that the consumption function is inelastic. A function is inelastic if the absolute value of its elasticity is less than 1 (|_C| < 1). From Step 4, we know _C is positive, so |_C| = _C. We need to show (bY)/(a + bY) < 1. Given a > 0, 0 < b < 1, and Y > 0. Consider the inequality: (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 (bY)/(a + bY) < 1 is also true. Thus, the consumption function is inelastic at all positive income levels. That's 2 down. 3 left today — send the next one.