Question

Nadia likes pork Ribs (R) and Chicken wings (C). Her utility function is U(R,C)=10R²C. Her weekly income is $90 which she spends exclusively on R and C. The price for a slab of ribs is $10 and $5 for a piece chicken. Note that MU_R=20RC and MU_c=10R²

1. State in words and in math Nadia's consumer problem.
2. What is Nadia's optimal bundle?
3. What is her demand function for ribs?
4. Are ribs a normal or inferior good?
5. The price of ribs falls to $5. Draw the income and substitution effects of this price change graphically. Assume ribs are on the horizontal axis.
6. What would Nadia's optimal bundle be if her utility function was given by U = R^0.5+C^0.5? Note that MU_R=0.5R^-0.5 and MU_c=0.5C^-0.5. Assume the price for a slab of ribs is $10 and $5 for chicken.

Answer 1

$$Solution$$

1. Nadia wants to choose the bundle $(R,C)$ that maximizes her utility subject to the budget constraint.

$Max_{10R^2C}\ S.t\ 90 \ge10R+5C$

2.A the U-function is of the Cobb-Douglas type, we know that the ICS are 'nice and convex'; Hence the optimum bundle satisfies the slope condition and is also on the budget line.

Slope condition$\frac{MU_R}{MU_C}=\frac{P_R}{P_C}\implies\frac{2C}{R}=2\\$

Budget line $90=10R+5C\implies10R+5R=90\implies15R=90R^*=6,C^*=6$

$3.\frac{2C}{R}=\frac{P_R}{5}\implies\ C=\frac{P_RP}{10}\\ 90=P_R+5\frac{P_RR}{10}\implies\ P_RR+\frac{1}{2}P_RR=90\implies\frac{3}{2}P_RR=90\implies\ R(P_R)=\frac{60}{P_R}$

4.Normal, this is because demand increases as income increases

5.

$6. MRS=\frac{\frac{1}{2}R-\frac{1}{2}}{\frac{1}{2}c-\frac{1}{2}}=\frac{C}{R}=4\implies\ c=4R\\ 90=10R+5(4R)\implies90=30R\implies\ R^*=3, C^*=12$