Diogos utility function is Uq1 q2 q0751 q0252 where q1 is c

Diogo\'s utility function is U(q_1, q_2) = q^0.75_1 q^0.25_2 where q_1 is chocolate candy and q_2 is slices of pie. If the price of a chocolate bar, p_1, is $1, the price of a slice of pie, p_2, is $2, and Y is $80, what is Diogo\'s optimal bundle

Solution

The equation for budget line is

Y = P₁q₁ + P₂q₂

Y is the income.

P₁ is the price of a chocolate bar.

q₁ is the quantity of a chocolate bar.

P₂ is the price of a slice of pie.

q₂ is the quantity of a slice of pie.

Putting the values of Income, Price of a chocolate bar and price of a slice of pie in the budget line equation -

80 = q₁ + 2 q₂

Now if the indifference curve and budget line are tangent they have the same slope at that point.

Slope of Indifference curve = Marginal rate of substitution_q1,q2 = - Marginal Utility of q₁ / Marginal Utility of q₂

Marginal Utility of q₁ = 0.75 q1 ^{0.75 - 1} q2 ^0.25

                                 = 0.75 q₁^{- 0.25} q₂^0.25

Marginal Utility of q₂ = 0.25 q₁^0.75 q₂^{0.25 - 1}

                                 = 0.25 q₁^0.75 q₂^{- 0.75}

Now the slope of Indifference curve is - 0.75 q₁^{- 0.25} q₂^0.25 / 0.25 q₁^0.75 q₂^{- 0.75}

                              Slope of Indifference curve = - 0.75 q₂ / 0.25 q₁

Now the slope of budget line = - P₁/ P₂

                                              = - 1/2

For optimal bundle ,

Slope of Indifference curve = Slope of the budget line

- 0.75 q₂ / 0.25 q₁ = -1 /2

1.50q₂ = 0.25 q₁

q₂ = 0.25q₁/1.50

Putting the value of q₂ into the budget line -

80 = q₁ + (2) * 0.25 q₁ / 1.50

80 = 1.50 q₁ + 0.50 q₁ / 1.50

80 = 2q₁/1.50

120 = 2q₁

q₁ = 60

So the quantity of chocolate bar in the optimal bundle is 60.

Putting value of q₁ on the budget line equation

80 = 60 + 2q₂

80 - 60 = 2q₂

20 = 2q₂

q₂ = 10

So the quantity of slice of pie in the optimal bundle is 10.

The optimal bundle for Diogo is 60 pieces of chocolate bar and 10 slices of pie.