Answer to Question #169712 in Microeconomics for Oliva

Question

1. John likes chicken ribs and chicken wings. Her utility function is U(R,C) =10R2 C. Her weekly 

income is 180 of which she spends exclusively on R and C. The price for the slab of ribs is 20 

shillings and 10 for wings. 

a. What is John’s problem?

b. What is John’s optimal bundle?

c. What is John’s demand function for ribs?

d. Are ribs a normal or inferior good

Answer 1

a. John is facing a utility maximization problem

b. John's optimal bundle.

John's budget line is given by the equation,

$I=P_rR+P_cC$, where $I=$ income, $P_r=$ price of chicken ribs, and $P_c=$ price of chicken wings.

Thus, $180=20R+10C$ which, in slope-intercept form can be written as $C=18-2R$.

John's optimal bundle is obtained where the slope of the indifference curve ($\frac{MU_r}{MU_c}$ ) is equal to the slope of the budget line $\frac{(P_r}{P_c})$.

Since $MU_r=C$ and $MU_c=R$, then $\frac{MU_r}{MU_c}=\frac{C}{R}$ . Similalry $\frac{P_r}{P_c}=\frac{20}{10}=2.$

So, $\frac{C}{R}=2$, or $R=\frac{C}{2}$ .

Substituting in the budget line gives,

$C=18-2(\frac{C}{2})$

$C=9$

When C = 9, then R =4.5

Therefore John's optimal bundle is $(R,C)=(4.5,9)$

c. John's demand function for ribs.

Given that the utility function is $U(R,C)=10R^2C$ and the budget line is $180=20R+10C$

Then setting up the Lagrange problem becomes

$L=10R^2C+\lambda(180-20R-10C)$

Setting the partial derivatives of L with respect to R, C, and $\lambda$ equal to zero, then we obtain the following first-order conditions:

$20RC+\lambda$ (-20)=0

$10R^2+\lambda(-10)=0$

$180-20R-10C=0$

From the first two conditions,

$\lambda=R^2=RC$

so $R=C$

Substituting for C into the third condition,

$180-20R-10R=0$

Then John's demand function for ribs is

R=6

d. Ribs are a normal good. This is because, based on John's budget line, an increase in the income while holding the prices constant, would result in a corresponding increase in the consumption of Ribs.