Question

Malachi only consumes 2 goods: DVD rentals and coffee. His utility function is ????????(????????, ????????) = ????????0.75????????0.25, where R is the number of rentals and C is cups of coffee. Malachi has $16 in his pocket right now, and he plans to spend all of the $16 on DVD rentals and coffee today. a. The price of one rental is $4 and the price of coffee is $2 per cup. Solve for Malachi’s optimal bundle. b. Suppose that Malachi signs up for a membership that reduces the price of a rental to $2, while leaving his income unchanged. Find the substitution effect, the income effect, and the total effect of the decrease in the price of video rentals on Malachi’s consumption of rentals and coffee. c. From your answer to part (b), are DVD rentals and coffee normal or inferior goods for Malachi? Explain.

Answer 1

(a)The optimal bundle is Rental = 3 and Coffee = 2

(b)Rental = 6 and Coffee = 2

(c)They are both the goods are normal

Step-by-step explanation:

(a)The optimal bundle is Rental = 3 and Coffee = 2

The Langrangian formula for solving utilization problem is stated as:

`L=R^(0.75) C^(0.25)  + λ(16-4R-2C),\\`

where 16=4R+2C is the consumer budget limit.

Using differentiation forthe Lagrangian formula with R and C, we will get
`\beta DL/DR= 0.75C^(0.25) - 4λ = 0 .......(1)\\\\\\DL/DC= 0.25R^(0.75) - 2λ = 0 .......(2)\\\\\\\Divide equation 1 by equation 2. you will get 3C/R= 2......3`

To solve Equation 3 and the budget limit =

R=3 and C=2

(b)Rental = 6 and Coffee = 2

In order for us to solve the substitution effect. we shall calculate the income at which the initial bundle is affordable.

which is =
`RP_(r) + CP_(c)= 2*3+ 2*2= 10R = 3*10 / 4*2 = 3.75, C = 10 / 4*2 = 1.25\\`

1. Substitution effect :R -R = 3.75- 3 = 0.75 and C' - C = 1.25 - 2 = -0.75
2. Total effect = R - R = 6-3 = 3 and C- C = 0
3. Income effect = total effect - substitution effect

(c)They are both the goods are normal

The demand for rental and coffee is given as
`R = 3M/4P_(r) \\C = M/4P_(c) \\`. The increase in income also increases the demand for rental and coffee. So, they two goods are both normal.