Consider the following case of constrained optimisation U(x, y) = X²Y and P_x=2, P_Y=l. Total income, I=200.

where consumer’s utility (U) is dependent upon the two goods x and y. I is the total income of the consumer. The consumer spends all the income on spending over two goods x and y. Maximise the utility given the income constraint. Find out the optimal consumption bundle (x*, y*) and the maximum utility of the consumer.

To solve this constrained optimization problem, we need to maximize the utility function U(x, y) = x^2 * y, subject to the income constraint I = Px * x + Py * y and given the prices Px = 2 and Py = 1, where I = 200.

The Lagrangian for this problem is given by:
L(x, y, λ) = x^2 * y + λ * (I - Px * x - Py * y)

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