Question

Krishna only spends his money on clothes, c, and food, f. His utility function is quasilinear: U(f, c) = 10ln(f) + c. His income is $2000 per month. The price of clothes is $10 while the price of food is $5.

1. Derive his demand function for clothes and food. Given the prices and income, state quantity demanded of clothes and food.

2. Due to a new free trade agreement, the price of clothes falls to $5. What is the new quantity demanded of clothes and food? Are they substitutes or complements?

3. The price of both goods remain at $5 onwards. The economy is not doing well, and Krishna got a cut of 20% on his income. What is the quantity demanded of clothes and food after the paycheck cut? Do they change or remain the same? Why?

4. Draw Engel curve for food and clothes.

5. The government needs to raise tax revenues. It is debating between 3 options: 1) income tax of 20 %, 2) revenue-equivalent of quantity tax of tf on food, or 3) revenue-equivalent of quantity tax of tc on clothes. Find tf and tc that would raise the same amount of revenue with the income tax from Krishna?

6. Under which tax scheme would Krishna be the most well-off? Explain why in 2-3 sentences.

Answer 1

Krishna's utility functin is , income (I) is $2000, and the given price of clothes is and food is .

1. The demand function will be found by the combination of c and f, such that they maximize utility with the given constraint. The budget constraint is here . The lagrangian function to maximize the utility subjected to the constraint is or . The FOC's equated to zero can be found as below.

or or . Equating , we have , which is the budget constraint itself.

or or . Equating , we have or or .

or or . Equating , we have or or .

Equating the last two FOC's having lambda in RHS, we have or . Putting it in the first FOC , we have or or or or .

The demand functions are hence and . At I=2000, pc=10 and pf=5, the quantity demanded of c and f is or and or .

2. Due to new trade agreement, the pc falls to 5. The new quantity demanded are or and or . Whether the goods are substitute or compliments is a bit tricky. Yet the change in price of c induced change in both as fall in price of c leads to increase in consumption of c and decrease in consumption of f - but not much, and the change in price of f would lead to no change in demand of c. Hence, the goods are compliments. One might plot the indifference curve to see that yet it isn't the perfect L-shaped (and hence goods are not perfect complement), but they are indeed slightly L-shaped.

3. Krishna got a cut off of 20% on his I, ie it reduced from $2000 to $1600. The quantity demanded of food is or and or . The change in income induced change in consumption of c only, but not f. This is because the income elasticiy of demand (IED) for c is positive, while for f, IED is zero.

4. The engel curve shows change in consumption of a good with respect to change in income. The combined graph is given as below.

Note: Two things to note are- the graph is scaled; engel curves should be drawn seperately.