Question

MICROECONOMICS FOR MANAGEMENT WORKSHEET 2023 6. A household consumes only apples and bananas. We denote a consumption bundle consisting X bags of apple and Y bags of bananas is denoted by (X, Y). The preference of the household is given by the utility function U(X, Y) = X0.8Y 0.2. If the income of the household is 20 $ and the price of apple (X) and banana (Y) is 4 $ and 2, respectively, then A) Write down the budget line equation B) Calculate MRS C) Find the optimal consumption of X and Y D) Compute the maximum utility E) Show the equilibrium condition graphically F) Compute the income share of x and y good

Answer 1

The budget line equation is 4X + 2Y = 20. The MRS can be calculated as (0.8X^-0.2Y^0.2) / (0.2X^0.8Y^-0.8). The optimal consumption is X = 3 bags of apples and Y = 6 bags of bananas. The maximum utility is 6.696. The equilibrium is where the budget line is tangent to the iso-utility curve. The income share of X is 0.6 and the income share of Y is 0.6.

A) Budget Line Equation:

The budget line equation can be derived by dividing the total income by the price of each good. In this case, the equation is: 4X + 2Y = 20.

B) Marginal Rate of Substitution (MRS):

The MRS represents the rate at which a consumer is willing to substitute one good for another while maintaining the same level of utility. It can be calculated by taking the ratio of the marginal utility of one good to the marginal utility of the other. In this case, MRS = (0.8X^-0.2Y^0.2) / (0.2X^0.8Y^-0.8).

C) Optimal Consumption:

The optimal consumption can be determined by equating the MRS to the price ratio. By setting MRS equal to the price ratio of 4:2, we can solve for X and Y. From this calculation, the optimal consumption of X is 3 bags of apples and the optimal consumption of Y is 6 bags of bananas.

D) Maximum Utility:

To compute the maximum utility, we can substitute the values of X and Y from the optimal consumption into the utility function U(X, Y) = X^0.8Y^0.2. By plugging in the values, the maximum utility is 3^(0.8) * 6^(0.2) = 6.696.

E) Equilibrium Condition Graph:

The equilibrium condition can be shown graphically by plotting the budget line and the iso-utility curve for the given utility function. The optimal consumption point will be where the budget line is tangent to the iso-utility curve.

F) Income Share:

The income share of X and Y goods can be calculated by dividing the expenditure on each good by the total income. In this case, the income share of X is (4 * 3) / 20 = 0.6 and the income share of Y is (2 * 6) / 20 = 0.6.