Question

suppose that the utility function of a consumer is given by TU(x,y)=3x2 y and the price of x and y are $1 and $2 per unit, respectively. if the income of the consumer is $600 and if he spends all of his income on the consumption of commodities of x and y, find the optimum amount of x and y that the consumer will consume at equilibrium and find MRTSx,y.

Answer 1

The utility function is given as follows:

$U=3x^2y$

The budget constraint is as follows:

$Y = P_xx + P_yy\\600= x + 2y$

Using Lagragian function, the optimal value of x and y is determined as follows:

$∅ = U + λ(Budget \space constraint)\\=3x^2y + λ(600−x − 2y)\\\frac{∂∅}{\delta x}= 6xy − λ (1) \\\frac{∂∅}{\delta y} =3x^2 −2 λ(2)\\\frac{∂∅}{\delta \lambda}= 600−x − 2y (3)$

Put equation (1), (2) and (3) equal to 0:

$\frac{∂∅}{∂x}=0\\6xy − λ=0\\ 6xy =\lambda(4)\\\frac{∂∅}{∂y}=0\\3x^2 − 2λ=0\\\frac{ 3x^2}{2} =λ(5)\\\frac{∂∅}{∂\lambda}=0\\ 600 − x − 2y =0 (6)$

From equation (4) and (5):

$6xy =\frac{ 3x^2}2 \\4y=x (7)$

Substitute equation (7) in equation (6):

$600 − x − 2y =0\\600 −4y − 2y=0\\ 600 y =6y \\y=100$

Substitute value of y in equation (7):

$x= 4y\\=4(100)\\=400$

Therefore, optimal combination is 400 units of x and 100 units of y.

The marginal rate of substitution is the amount of good x consumer is willing up to consume one extra unit of good y.

Mathematically, it is ratio of marginal utilities and is calculated as follows:

$MRS =\frac{\frac{∂U}{∂x}}{\frac{∂U}{∂y}}\\=\frac{6xy}{3x^2}\\=\frac{2y}{x}$