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Choice Continued Interpretation At Z: x2 Z p1 MRS ( x1, x 2) p2 X At Y: p1 MRS ( x1, x 2) p2 Y x1 Consumer Demand Consumer’s demanded bundle: optimal choice of goods 1 and 2 for given prices and income. Consumer’s demand functions: x1 x1 p1, p2, m x2 x 2 p1, p2, m Cobb-Douglas Demand function for good 1: x2 m x1 c p1 x2 * Demand function for good 2: * x1 x1 m x 2 1 c p2 Perfect Substitutes x2 Demand function for good 1: 1 x1 m / p1 p1 p2 x1 0 if p1 p 2 if p1 p 2 x1 0, m / p1 * x1 x1 if p1 p 2 Perfect Complements Optimal choice: x2 x2 x1 x2 x1 Budget line: p1x1 p2 x2 m x2 * Demand function for goods 1 and 2: * x1 x1 m x1 x 2 p1 p 2 Example: Choosing a Tax Q: Suppose that the government wants to raise a certain amount of revenue. Is it better to raise it via a quantity tax or an income tax? Before the Tax x2 Budget line: p1x1 p2 x2 m p1 p2 x2 Well-behaved preferences o Consumer chooses: o x1 x1 x1 , x2 o o Quantity Tax Budget line with quantity tax: x2 p1 t x1 p2 x2 m p1 t p2 * x 2o x2 Tax revenues: R tx1 * * x1 o x1 x1 * Income Tax New budget line: p1 x1 p 2 x 2 m R where R tx1 * * * Questions Q1: What is the slope of the budget line with the income tax? Q2: Which kind of tax is this income tax? Q3: Can the consumer afford X * when he is paying the income tax? Answers A1: Slope of the budget line with income tax is: p1 A2: Lump-sum. p2 A3: Yes. X * Is still Affordable Budget line with income tax: p1 x1 p 2 x 2 m tx * 1 Substitute X * : p1x1 p2 x2 m tx1 * Rearrange: * p1 t x1 * * p2 x 2 m * On the Graph Budget line with income tax: x2 Budget line with quantity tax: X * X o p1 p2 x1 * Q: Is X Optimal with the Income Tax? x2 X * X o x1 A: No, it is not. The x2 budget line is not tangent X * X to indifference curve at o X X * . Optimal choice: x1 X Q: Which Tax Does the Consumer Prefer? x2 X * X o X x1 A: Income Tax Gives Higher Utility to the Consumer Caution: A uniform income tax for all consumers is not necessarily better than a uniform quantity tax. Assumption that income tax is lump-sum is key. Another Question Suppose that the government can use a uniform value tax on goods 1 and 2 to raise revenue (e.g. a sales tax). Does the consumer still prefer an income tax? Answer With a uniform value tax, the budget consumer line reads 1 t p1x1 1 t p2 x2 m Rearranging: m p1 x1 p 2 x 2 1 t Answer Continued Rearranging: m p1 x1 p 2 x 2 1 t Budget line with income tax: p1 x1 p 2 x 2 m R t * * * R tp1 x1 tp2 x 2 m 1 t * Answer Continued Budget line with income tax: m p1 x1 p 2 x 2 1 t Same as budget line under value tax: m p1 x1 p 2 x 2 1 t