Question

Suppose that my income is $100 and I go a supermarket to buy good X and good Y. My utility function is U(X, Y)=X^0.5+Y^0.5, and Px=1 and Py=2. Determine in which good I will use my first peso, my second peso, my third peso and my fourth peso. In the end what will be the final consumption bundle?

Answer 1

Ans. Utility function, U = X^0.5 + Y^0.5

Marginal utility of x, MUx = dU/dx = 0.5*X^(-0.5)

Marginal utility of y, MUy = dU/dy = 0.5*Y^(-0.5)

=> Marginal rate of substitution, MRS = MUx/MUy = Y/X

At utility maximizing level,

MRS = Px/Py

Y/X = 1/2

=> X = 2Y ---> Eq1

Substituting Eq1 in the budget constraint, X + 2Y = 100, we get,

2Y + 2Y = 100

=> Y = 25 unit and from Eq1,X = 50 units

Thus, optimal consumption bundle is (50,25)

Which good will be  consumed with the first peso will be the one which gives higher per dollar marginal utility,

So, For fist peso, X will give higher marginal utility of 0.5 while Y will give utility of 0.25

For second peso, X will give marginal utility per dollar of 0.5*(2^(-0.5)) = 0.3535 or on Y which will give utility of 0.25, so, X will be consumed.

For the third peso, X will give marginal utility per dollar of 0.5*(3^(-0.5)) = 0.288675 or on Y which will give utility of 0.25, so, X will be consumed.

For fourth peso, X will give marginal utility per dollar of 0.5*(4^(-0.5)) = 0.25 or on Y which will give utility of 0.25, so, X and Y will be indifferent.

For fifth peso, X will give marginal utility per dollar of 0.5*(5^(-0.5)) = 0.2236 or on Y which will give utility of 0.25 (if fourth peso was spent on X), so, Y will be consumed.