Question

a consumer utility function
is u=√EF. Find the marshalian demand function
for E and F, find the compensated demand
function. now let the budget be m=100. prices
are 1,1. what are the demand quantities? what
is the utility level? Let the price of F rise to 2.
what are the demand quantities? what is the
utility level. What is the income compensation
necessary to put the consumer back to his
original utility level after the price change?.
Assume the utility function is u= lnE + lnF. How
does this new utility function change your
results from the beginning?

Answer 1

Question#62561 - Economics - Microeconomics | Completed

Question

A consumer utility function is $u = \text{VEF}$. Find the Marshalian demand function for E and F, find the compensated demand function. Now let the budget be $m = 100$. Prices are 1,1. What are the demand quantities? What is the utility level?

Let the price of F rise to 2. What are the demand quantities? What is the utility level? What is the income compensation necessary to put the consumer back to his original utility level after the price change?

Assume the utility function is $u = \ln E + \ln F$. How does this new utility function change your results from the beginning?

Answer

$$u = \sqrt{ (E * F) }, \quad MU_E = \sqrt{ (F) / 2 * \sqrt{ (E) } }, \quad MU_F = \sqrt{ (E) / 2 * \sqrt{ (F) } }$$ $$p_E = 1, \quad p_F = 1, \quad m = 100$$ $$MU_E / MU_F = p_E / p_F \Rightarrow \left( \sqrt{ (F) * 2 * \sqrt{ (F) } } \right) / \left( \sqrt{ (F) * 2 * \sqrt{ (E) } } \right) = p_E / p_F \Rightarrow F / E = 1 \Rightarrow E = F$$ $$m = p_E * E + p_F * F \Rightarrow 100 = p_E * E + p_F * E \Rightarrow E * = 50, \quad F * = 50$$ $$u(E, F) = \sqrt{ (50 * 50) } = 50$$ $$p_E = 1, \quad p_F = 2, \quad m = 100$$ $$MU_E / MU_F = p_E / p_F \Rightarrow \left( \sqrt{ (F) * 2 * \sqrt{ (F) } } \right) / \left( \sqrt{ (F) * 2 * \sqrt{ (E) } } \right) = p_E / p_F \Rightarrow F / E = 1 / 2 \Rightarrow E = 2F$$ $$m = p_E * E + p_F * F \Rightarrow 100 = p_E * 2F + p_F * F \Rightarrow 100 = 2F + 2F \Rightarrow F * = 25, \quad E * = 50$$ $$u(E, F) = \sqrt{ (50 * 25) } = 35.4$$

income compensation: $\sqrt{ (E * F) } = \sqrt{ (2F * F) } = 50 \Rightarrow 2F^2 = 2500 \Rightarrow F = 35.4, \quad E = 70.8 \Rightarrow$

$\Rightarrow 1 * 70.8 + 2 * 35.4 = 141.6 \Rightarrow$ income compensation (for the previous utility level 50) = (141.6 - 100) = 41.6

$$u = \ln(E) + \ln(F) - \text{because of concavity max}(u): \quad E = F = 50 \quad (p_E = 1, \quad p_F = 1, \quad m = 100)$$

$u = \ln(50) + \ln(50) = 7.82$ – with such type of function general utility is much lower

https://www.AssignmentExpert.com