1. **Problem statement:** A consumer buys goods $x$ and $y$ with prices $p_x=400$ and $p_y=500$, and income $m=10000$. We need to: a. Draw the budget line and find intercepts and opportunity cost of $x$. b. Draw the new budget line if prices double but income stays the same. c. Find the income change that yields the same budget line as in (b) with original prices. 2. **Budget line formula:** The budget line equation is $$p_x x + p_y y = m$$ where $x$ and $y$ are quantities of goods. 3. **Part (a): Original budget line** - Find $x$-intercept by setting $y=0$: $$400x + 500\times 0 = 10000 \Rightarrow x = \frac{10000}{400} = 25$$ - Find $y$-intercept by setting $x=0$: $$400\times 0 + 500y = 10000 \Rightarrow y = \frac{10000}{500} = 20$$ - Opportunity cost of one more unit of $x$ is how many units of $y$ must be given up: $$\text{Opportunity cost} = \frac{p_x}{p_y} = \frac{400}{500} = 0.8$$ So, consuming one more unit of $x$ costs 0.8 units of $y$. 4. **Part (b): Prices double, income fixed** - New prices: $p_x' = 800$, $p_y' = 1000$, income $m=10000$. - New $x$-intercept: $$800x + 1000\times 0 = 10000 \Rightarrow x = \frac{10000}{800} = 12.5$$ - New $y$-intercept: $$800\times 0 + 1000y = 10000 \Rightarrow y = \frac{10000}{1000} = 10$$ - New budget line is steeper, intercepts halved. 5. **Part (c): Equivalent income change with original prices** - We want $m'$ such that: $$400x + 500y = m'$$ has the same intercepts as in (b): $x$-intercept $=12.5$, $y$-intercept $=10$. - From $x$-intercept: $$12.5 = \frac{m'}{400} \Rightarrow m' = 12.5 \times 400 = 5000$$ - From $y$-intercept: $$10 = \frac{m'}{500} \Rightarrow m' = 10 \times 500 = 5000$$ - Both agree $m' = 5000$. **Final answers:** - (a) Budget line: $$400x + 500y = 10000$$ with intercepts $x=25$, $y=20$, opportunity cost of $x$ is 0.8 units of $y$. - (b) New budget line: $$800x + 1000y = 10000$$ with intercepts $x=12.5$, $y=10$. - (c) Equivalent income: $m' = 5000$ to get same budget line with original prices.