Question

A consumer's Total Utility Function of two goods X and y are as follows:

TUx=50x-5x²

TUy=32y-4y²

Price of X=Rs.5

Price of y: Rs.8

Consumer income : 120

i) Derive consumer's budget constraint

ii) Derive the marginal utility X

iii) Derive marginal utility Y

iv) Find out the consumer optimum combination of good X and Y at the market.

Answer 1

$(i)\\TUx = 50x - 5x^2\\ TUy = 32y - 4y^2\\ Px = 5\\ Py = 8\\ Income = 120$

The budget line refers to the combination of two goods that a consumer can buy in the given budget or income.

$Px\times X + Py\times Y = M\\ 5X + 8Y = 120$

$(ii)\\TU \space of \space X = 50x−5x^2\\TU \space of \space Y =32y−4y^2\\MU_x = \frac{∂TUx}{∂x}\\MU_x = 50 − 10x\\(iii)\\MU_y =\frac{ ∂TUy}{∂y}\\MUy = 32 − 8y$

Optimal Bundle is achieved where $MRS = \frac{Px}{Py}$

$MRS = \frac{MU_x}{MU_y}\\therefore\frac{ MU_x}{MU_y} = \frac{Px}{Py}\\\frac{50 − 10x}{32−8y} = \frac{5}{8}\\ 8\times (50 − 10x) = 5\times (32 − 8y)\\400 − 80x = 160 − 40Y\\−80x = −40y − 240\\80x = 40y + 240\\2x = y + 6\\x = \frac{y+6}{2}$

Now Put the Value of X in budget constraint we get.

$5X + 8Y = 120\\5\times (\frac{y+6}{2}) + 8Y = 120\\\frac{5y − 30}{2} + 8Y = 120\\5y + 30 + 16Y = 240\\21y = 210\\y = 10\\we\space know\space x = \space{y+6}{2}\\x = \frac{10+6}{2}\\x = 8$

therefore the optimal bundles are (8,10)