Answer to Question #243916 in Microeconomics for Sabs

Question

 Assume U(x, y) = √4 xy.

a) Derive the indirect utility function v(p, Y ).

b) Now assume a consumer is considering a gamble that has a 50% chance of returning 121% of the initial investment (a gain of 21%) and a 50% chance of returning 81% (a loss of 19%). If a consumer’s initial budget is 100, what is the expected outcome of this gamble?

c) Now using the indirect utility function from a), the gamble from b) and assuming prices are equal to 1, find an optimal allocation of consumer’s initial wealth between risky asset from gamble in b) and a risk-free asset. Assume the return on risk-free assets is 1 (no gain/loss).

Answer 1

a:

Utility feature:$U = In x_1 + In x_2 ............(1)$

Budget Line (Constraint):$p_1x_1 +p_2x_2 =m ............(2)$

b

Slope of price range line $= \frac{dx_2}{dx_1}$

$X_2 =\frac{ m}{p_2} – \frac{p_1x_1}{p_2}$

Slope of price range line =$\frac{dx_2}{dx_1}= \frac{p_1}{p_2}$

Slope of an Indifference curve = Marginal Rate of Substitution( MRS)

                         = Marginal Utility (MU) of X_I$\div$ Marginal Utility of X_I

                         $= \frac{(\frac{du}{dx_1})}{ (\frac{du}{dx_2})}\\ =\frac{ (\frac{1}{x_1})}{ (\frac{1}{x_2})}\\ = \frac{x_2}{x_1}$

c;

Individual's Marshallian call for: Slope of price range line = Slope of an Indifference Curve

             $\frac{p_1}{p_2} = \frac{x_2}{x_1} ............(3)\\ x_2=\frac{ p_1x_1}{p_2} ............(4)$

Put Eqn (4) into Eqn (3): $p_1x_1 + p_2 (\frac{p_1x_1}{p_2}) = m$

                    $2p_1x_1= m\\ x_1* = \frac{m}{2p_1} ..........(5)$

Similarly from Eqn(3):$x_1= \frac{p_2x_2}{p_1} ...............(6)$              

Put (6) into (2):$p_1 (\frac{p_2x_2}{p_1}) + p_2x_2 = m$

              $2p_2x_2 = m\\ x_2* = \frac{m}{2p_2} ...................(7)$

Now positioned both (5) and (7) into (1): $U = In(\frac{m}{2p_1})+ In(\frac{m}{2p_2})$

                                 $U= In [(\frac{m}{2p_1}) (\frac{m}{2p_2})]$            [Because $In x + In y = In (xy)]$

        (Indirect Utility Function) $U = In [\frac{m_2}{4p_1p_2}]$