Answer to Question #187895 in Microeconomics for DALHAT MAITAMA GAR

Question

The consumer utility function for mango tree and orange tree is given U=q1q2, the price for mango tree is $40.00 and orange is $20.00. The consumer income for the period is $120.00. Determine the quantities of mango tree and orange tree which should be purchased in order to maximize derived utility?.

Answer 1

Solution:

The optimal consumption bundle is one where the slope of the indifference curve $\frac{MUq1}{MUq2}$ is equal to the slope of the budget line.

Derive MRS_q1q2 = $\frac{MUq1}{MUq2}$

MU_q1 = ​$\frac{∂U}{∂q1}$ = q₂

MU_q2 = ​$\frac{∂U}{∂q2}$ = q₁

MRS_q1q2 = $\frac{q_{2} }{q_{1} }$

MRS_q1q2 = $\frac{Pq1}{Pq2}$

$\frac{q_{2} }{q_{1} }$ = $\frac{40 }{20 }$

q₂ = 2q₁

Derive utility function:

M = Pq₁q1+ Pq₂q2

120 = 40q1 + 20q2

Therefore:

120 = 40q1 + 20(2q₁)

120 = 40q1 + 40q1

120 = 80q1

q1 = $\frac{120}{80 }$ = 1.5

q1 = 1.5

Plug this figure into the q₂ function:

q₂ = 2q₁

q₂ = 2(1.5)

q2 = 3

Maximizing Utility (1.5, 3)

The quantities of the mango tree to be purchased to maximize derived utility = 1.5

The quantities of the orange tree to be purchased to maximize derived utility = 3