Consider a consumer who buys two goods, x and y with utility function u(x, y) = 2√x + y. The consumer’s income is 20 and price of y is 4. Compute the optimal consumption bundle when the price of x is equal to 1 using constrained optimisation

To compute the optimal consumption bundle for a consumer who buys two goods, x and y, with the utility function u(x, y) = 2√x + y, given an income of 20 and a price of y equal to 4, when the price of x is equal to 1, we can use constrained optimization.

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The consumer’s problem can be stated as follows:

Maximize u(x, y) = 2√x + y

Subject to the budget constraint: 1x + 4y = 20

Here, x represents the quantity of good x consumed, and y represents the quantity of good y consumed.

To solve this problem, we can use the method of Lagrange multipliers. We set up the Lagrangian as follows:

L(x, y, λ) = 2√x + y – λ(1x + 4y – 20)

Now, we need to find the partial derivatives with respect to x, y, and λ and set them equal to zero:

1. ∂L/∂x = 1/√x – λ = 0

2. ∂L/∂y = 1 – 4λ = 0

3. ∂L/∂λ = 1x + 4y – 20 = 0

From equation (1), we get:

1/√x = λ

From equation (2), we get:

4λ = 1

Λ = ¼

Now, we can solve for x and y:

From equation (1):

1/√x = ¼

√x = 4

X = 16

From equation (3):

1x + 4y = 20

16 + 4y = 20

4y = 20 – 16

4y = 4

Y = 1

So, the optimal consumption bundle is x = 16 and y = 1 when the price of x is equal to 1. This means the consumer should buy 16 units of x and 1 unit of y to maximize their utility given their budget and the prices of the goods.

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