# Suppose you have a budget of $1,300 to spend of beer and pizza this year. Your utility function... ## Question: Suppose you have a budget of$1,300 to spend of beer and pizza this year. Your utility function is {eq}U = 200B + 150P - 0.5B^2 - 0.25P^2 {/eq}. The price of a beer is $5 and the price of a pizza is$10.

a. Using calculus, set up a constrained optimization problem using the Lagrangian multiplier that derives the optimal consumption of pretzels and beer. Find the optimal values.

b. At the optimum, what are the marginal utilities of both pizza and beer? Does the Utility-Maximizing Rule hold?

c. Does the utility function indicate a relationship between pizza and beer?

## Utility-optimization:

Any rational consumer in the market who is consuming goods and services produced by the producer looks for that particular bundle of goods which maximizes its satisfaction or utility. The utility can be maximized through Lagrange 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.