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 find the optimal consumption bundle, we can set up the consumer’s optimization problem using constrained optimization.

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The objective is to maximize the utility function ( u(x, y) = 2\sqrt{x} + y ) subject to the budget constraint ( px + qy = M ), where ( p ) and ( q ) are the prices of goods ( x ) and ( y ) respectively, and ( M ) is the consumer’s income.

In this case:

• ( p = 1 ) (price of ( x )),
• ( q = 4 ) (price of ( y )),
• ( M = 20 ) (consumer’s income).

The optimization problem is then:
[ \max_{x, y} 2\sqrt{x} + y ]
subject to
[ x + 4y = 20 ]

Now, we can solve this problem using the Lagrange multiplier method or substitution. Let’s use substitution. The budget constraint implies ( x = 20 – 4y ).

Substitute this into the utility function:
[ u(y) = 2\sqrt{20 – 4y} + y ]

Now, find the critical points by taking the derivative of ( u(y) ) with respect to ( y ) and setting it equal to zero.

[ \frac{du}{dy} = 0 ]

Once you find the value(s) of ( y ), use them to find ( x ) from the budget constraint ( x = 20 – 4y ). This will give you the optimal consumption bundle.