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
7. Suppose that Mr. John has the following Cobb-Douglas utility function: U = 9X1/3y
2/3
. The market price of
X and Y commodity are $1 and $2, respectively. If Mr. John's total income is $90, then answer the following
questions:
A) Find the utility maximizing equilibrium quantity of X and Y commodities.
B) Show the equilibrium condition graphically.
C) Calculate

(Lagrange multiplier) and interpret.

Answer 1

In microeconomics, a household's consumption choices can be analyzed using the budget constraint and utility maximization. The optimal consumption bundle can be determined by setting the Marginal Rate of Substitution equal to the price ratio and using the budget constraint. Graphical representations often include the budget line and indifference curves.

Step-by-step explanation:

Microeconomics involves the study of how households and firms make decisions to allocate their resources. In the question provided, a household aims to maximize utility with a given budget by consuming apples and bananas, reflected in a Cobb-Douglas utility function. Here are the solutions for the given microeconomics problem:

• A) Budget Line Equation: The budget line is given by 20 = 4X + 2Y, where X is the bags of apples and Y is the bags of bananas.
• B) Marginal Rate of Substitution (MRS): MRS can be calculated by taking the derivative of Y with respect to X in the utility function: MRS = - (dY/dX) * (Ux/Uy), which simplifies to -4/Y for this case.
• C) Optimal Consumption of X and Y: To find the optimal consumption, we would typically set MRS equal to the price ratio and solve for X and Y using the income constraint. However, without the complete calculation, we can't provide deterministic values.
• D) Maximum Utility: Maximum utility is the highest utility that can be achieved given the budget constraint, which is found by substituting the values of X and Y that we find from part C into the utility function.
• E) Equilibrium Condition Graphically: On a graph, the equilibrium condition is where the budget constraint line is tangent to an indifference curve representing the utility function.
• F) Income Share of X and Y Good: To compute the income share, we would divide the money spent on each good by the total income.

As for Mr. John's utility function, the outstanding questions would follow a similar approach but would result in different numeric answers based on the given prices and income.