Question 6.14: Given the consumption function C = 20 + 3Y^0.4 (a) Write dow......
Given the consumption function C = 20 + 3Y^{0.4}
(a) Write down the equations for MPC and MPS.
(b) Write down the equations for APC and APS.
(c) Verify the inequalities APC > MPC and MPS > APS by comparing the values of MPC, APC, MPS and APS at Y = 10.
(a) C = 20 + 3Y^{0.4}
MPC = \frac {dC}{dY} = 0 + 3(0.4Y^{0.4-1})
MPC = 1.2Y^{-0.6} = \frac {1.2}{Y^{0.6}}
MPS = 1 − MPC = 1 − \frac {1.2}{Y^{0.6}}
(b) C = 20 + 3Y^{0.4}
APC = \frac {c}{Y} = \frac {20}{Y} + \frac {3Y^{0.4}}{Y}
= \frac {20}{Y} + \frac {3}{Y^0.6}
APS = 1 − APC = 1 − \frac {20}{Y} + \frac {3}{Y^{0.6}}
(c) Now evaluate MPC, APC, MPS, APS at y = 10.
Marginal function at Y = 10
\mathrm{MPC}={\frac{1.2}{(10)^{0.6}}}=0.30
MPS = 1 − MPC = 1 − 0.30 = 0.70
Average function at Y = 10
A P C={\frac{20}{10}}+{\frac{3}{(10)^{0.6}}}=2.75
APS = 1 − APC = 1 − 2.75 = −1.75
These results verify the given inequalities APC > MPC and MPS > APS