Let a consumer selects two goods, i. e x and y for consumption having prices of Rs. 1600 and Rs. 800 respectively and fixed income with Rs. 16000. a. Derive budget line and determine equilibrium…

Model Answer: Microeconomics for Business (TU Board 2081)

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a. Budget Line and Equilibrium Point (Equal Allocation)

Given:

• Price of good $x$ ($P_x$) = Rs. 1600
• Price of good $y$ ($P_y$) = Rs. 800
• Income ($M$) = Rs. 16,000

Step 1: Derive the Budget Line Equation The budget line represents all possible combinations of $x$ and $y$ that a consumer can afford given their income and prices. The equation is: $$P_x\cdot x+P_y\cdot y=M$$ $$1600x+800y=16000$$ Simplify by dividing by 800: $$2x+y=20$$ $$y=20-2x$$

Step 2: Find Intercepts

• X-intercept (when $y=0$): $$1600x=16000⟹x=10$$
• Y-intercept (when $x=0$): $$800y=16000⟹y=20$$

Step 3: Equilibrium Point (Equal Allocation) The consumer allocates the entire budget equally between $x$ and $y$. Since the prices are different, equal allocation means spending Rs. 8,000 on each good.

• Quantity of $x$: $$1600x=8000⟹x=5$$
• Quantity of $y$: $$800y=8000⟹y=10$$

Final Equilibrium Point: $$\boxed{(x,y)=(5,10)}$$

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b. New Budget Line and Equilibrium Point (After Price Change)

Given:

• New price of $x$ ($P_x^{\prime }$) = Rs. 800
• Income remains Rs. 16,000
• New consumption: Rs. 6,400 on $x$ and Rs. 9,600 on $y$

Step 1: Derive the New Budget Line Equation $$800x+800y=16000$$ Simplify: $$x+y=20$$ $$y=20-x$$

Step 2: Find New Intercepts

• X-intercept (when $y=0$): $$800x=16000⟹x=20$$
• Y-intercept (when $x=0$): $$800y=16000⟹y=20$$

Step 3: New Equilibrium Point Given the consumer spends Rs. 6,400 on $x$ and Rs. 9,600 on $y$:

• Quantity of $x$: $$800x=6400⟹x=8$$
• Quantity of $y$: $$800y=9600⟹y=12$$

Final New Equilibrium Point: $$\boxed{(x,y)=(8,12)}$$

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c. Price Demand Curve for Good $x$

The price demand curve shows the relationship between the price of $x$ and the quantity demanded, holding income and the price of $y$ constant.

Step 1: Assume Income and Price of $y$ are Fixed

• Income ($M$) = Rs. 16,000
• Price of $y$ ($P_y$) = Rs. 800 (unchanged)

Step 2: Express Demand for $x$ as a Function of $P_x$ The budget constraint is: $$P_x\cdot x+800y=16000$$ Assuming the consumer spends all income, we can express $y$ in terms of $x$: $$y=\frac{16000-P_x\cdot x}{800}$$

Step 3: Derive Demand Function (Assuming Equal Allocation) If the consumer allocates half the income to $x$, then: $$P_x\cdot x=\frac{16000}{2}=8000$$ $$x=\frac{8000}{P_x}$$

Step 4: Plot Key Points for Price Demand Curve

Graphical Representation: The demand curve is downward sloping, showing that as the price of $x$ decreases, the quantity demanded increases.

Final Price Demand Curve: $$\boxed{x=\frac{8000}{P_x}}$$

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