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Answer :
The utility function is given by:
u (x1, x2) = x₁²x₂²
Where price of good 1 is $10/unit price of good 2 is $3/unit income = $135
We need to calculate the optimal consumption level for good 1.
The optimal consumption level can be calculated using the formula of the MRS (marginal rate of substitution) and the budget constraint equation.
MRS = ∆x₂/∆x₁ = Px₁/Px₂ where Px₁ is the price of good 1, Px₂ is the price of good 2.Budget constraint equation:
P₁x₁ + P₂x₂ = I
where P₁ is the price of good 1, P₂ is the price of good 2, and I is income.
Substitute the values in the above equations:
MRS = 2x₁x₂²/2x₁²x₂= x₂/x₁P₁x₁ + P₂x₂ = I∴ 10x₁ + 3x₂ = 135
Now, we have two equations: x₂/x₁ = MRS10x₁ + 3x₂ = 135
Substitute the value of MRS in the budget constraint equation:10x₁ + 3(x₂/x₁)x₁ = 13510x₁² + 3x₂ = 135x₁ = √((I-P₂MRS)/P₁) = √((135-3(1/3))/10) = √13.5/10 = 1.16 (approx)
Therefore, the optimal consumption level for good 1 is 1.16 units.
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