Suppose you have a budget of $1,300 to spend of beer and pizza this year. Your utility function...

_a_

The equation of a budget line can be written as:

{eq}\begin{align*} {P_B}B + {P_P}P &= M\\ 5B + 10P &= 1300 \end{align*}{/eq}

Set up the Lagrange function as below:

{eq}\begin{align*} L &= U\left( {B,P} \right) - \lambda \left( {{P_B}B + {P_P}P - M} \right)\\ L &= 200B + 150P - 0.5{B^2} - 0.25{P^2} - \lambda \left( {5B + 10P - 1300} \right)\\ \dfrac{{\partial L}}{{\partial B}} &= 200 - B - 5\lambda .........\left( 1 \right)\\ \dfrac{{\partial L}}{{\partial P}} &= 150 - 0.5P - 10\lambda ..........\left( 2 \right)\\ \dfrac{{\partial L}}{{\partial \lambda }} &= - 5B - 10P + 1300........\left( 3 \right) \end{align*}{/eq}

Now substitute equation (1) and (2) equal to zero:

{eq}\begin{align*} 200 - B - 5\lambda &= 0\\ \Rightarrow \dfrac{{200 - B}}{5} &= \lambda .......\left( 4 \right)\\ 150 - 0.5P - 10\lambda &= 0\\ \Rightarrow \dfrac{{150 - 0.5P}}{{10}} &= \lambda ........\left( 5 \right) \end{align*}{/eq}

Now equate equation (4) and (5),

{eq}\begin{align*} \dfrac{{200 - B}}{5} &= \dfrac{{150 - 0.5P}}{{10}}\\ 400 - 2B &= 150 - 0.5P\\ 250 &= 2B - 0.5P........\left( 6 \right) \end{align*}{/eq}

Equating equation (3) equal to 0 gives us:

{eq}\begin{align*} - 5B - 10P + 1300 &= 0\\ 5B + 10P &= 1300\\ B + 2P &= 260\\ B &= 260 - 2P.......\left( 7 \right) \end{align*}{/eq}

Substituting the value of 'B' in equation (6) gives us:

{eq}\begin{align*} 250 &= 2B - 0.5P\\ 250 &= 2\left( {260 - 2P} \right) - 0.5P\\ 4P + 0.5P &= 520 - 250\\ 4.5P &= 270\\ P* &= 60 \end{align*}{/eq}

Putting back the value of P in equation (7):

{eq}\begin{align*} B &= 260 - 2P\\ B &= 260 - 2\left( {60} \right)\\ B &= 260 - 120\\ B* &= 140 \end{align*}{/eq}

The optimal bundles of beer and pizza are (140, 60)

_b

At the optimal values obtained above, the marginal utilities of both the good is given below:

Marginal utility of beer:

{eq}\begin{align*} M{U_B} &= 200 - B\\ {\rm{At}}\;B &= 140\\ M{U_B} &= 200 - 140\\ M{U_B} &= 60 \end{align*}{/eq}

Marginal utility of Pizza:

{eq}\begin{align*} M{U_P} &= 150 - 0.5P\\ {\rm{At}}\;P &= 60\\ M{U_P} &= 150 - 0.5\left( {60} \right)\\ M{U_P} &= 120 \end{align*}{/eq}

Yes, the utility-maximization rule does hold because according to this rule:

{eq}\begin{align*} \dfrac{{{\rm{Marginal}}\;{\rm{utility}}\;{\rm{of}}{\rm{beer}}}}{{{P_B}}}& = \dfrac{{{\rm{Marginal}}\;{\rm{utility}}\,{\rm{of}}\;{\rm{pizza}}}}{{{P_P}}}\\ LHS &= \dfrac{{60}}{5} \Rightarrow 12\\ RHS &= \dfrac{{120}}{{10}} \Rightarrow 12\\ LHS &= RHS \end{align*}{/eq}

c

Yes, the utility function indicates a relationship between pizza and beer because the utility function is in the form of both these goods.