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▎1. Construct Budget Equation
The budget equation can be constructed using the prices of the commodities and the total amount of money available to the consumer.
Let:
• P_X = 20 (price of commodity X)
• P_Y = 30 (price of commodity Y)
• I = 600 (total income)
The budget equation is given by:
P_X ⋅ X + P_Y ⋅ Y = I
Substituting the values:
20X + 30Y = 600
▎2. Determine the Optimal Consumption of X and Y
To maximize utility, we need to find the values of X and Y that satisfy the budget constraint and maximize the utility function U(X, Y) = 10X^(0.6)Y^(0.4) .
We can use the method of Lagrange multipliers or the marginal utility per dollar spent approach. First, we find the Marginal Utility (MU) for both goods:
MU_X = ∂ U / ∂ X = 10 ⋅ 0.6X^(-0.4)Y^(0.4) = 6X^(-0.4)Y^(0.4)
MU_Y = ∂ U / ∂ Y = 10 ⋅ 0.4X^(0.6)Y^(-0.6) = 4X^(0.6)Y^(-0.6)
The consumer maximizes utility when:
MU_X / P_X = MU_Y / P_Y
Substituting in the values:
6X⁽-0.4)Y⁽0.4) / 20 = 4X⁽0.6)Y⁽-0.6) / 30
Cross-multiplying gives:
6 ⋅ 30 ⋅ Y^(0.4) = 4 ⋅ 20 ⋅ X^(0.6) ⋅ X^(-0.4)Y^(-0.6)
180Y¹ = 80X^(0.2)
This simplifies to:
Y = 80 / 180X^(0.2) = 4 / 9X^(0.2)
Now, substitute Y back into the budget equation:
20X + 30((4 / 9X^(0.2))) = 600
Solving this equation will give you the optimal values for X and Y .
▎3. Find Total Utility
Once you find the optimal values of X and Y , substitute them back into the utility function:
U(X, Y) = 10X^(0.6)Y^(0.4)
▎4. Show Consumer Equilibrium Graphically
To show consumer equilibrium graphically, plot the budget line from the budget equation 20X + 30Y = 600 and the indifference curves derived from the utility function U(X, Y) = k . The point where the budget line is tangent to an indifference curve represents consumer equilibrium.
▎5. Discuss Main Goals and Objectives of Macroeconomics
Macroeconomics has several primary goals and objectives, which include:
• Economic Growth: Aiming for sustained increases in national output (GDP), which enhances living standards and overall prosperity.
• Full Employment: Striving to achieve a low level of unemployment where all individuals willing and able to work can find jobs.
• Price Stability: Maintaining stable prices to avoid inflation or deflation, which can disrupt economic planning and savings.
• Balance of Payments Stability: Ensuring that a country can meet its international financial obligations and maintain a stable currency.
• Equitable Distribution of Income: Working towards reducing income inequality to promote social justice and cohesion.
These objectives guide government policy decisions, including fiscal and monetary policies, to foster a stable economic environment conducive to growth and development.
Mar 09, 2025