Consider an economy with one farmer whose farm produces 100 units of food. The farmer can trade food for clothing from other countries, but he cannot produce clothing. The price of food in the market is 1, and the price of clothing is 10. The farmer has quasi-linear preferences of the form $u(f, c) = ln f + c$ where f is food consumption and c is clothing consumption.
Suppose the government imposes a 50% tax on sales of food, meaning that the farmer now receives only 50 cents per unit sold. What is the farmer’s consumption of food and clothing, and how much food does he sell?
I first compute the budget constraint: $p_c *c+p_f*f = m$, where m is income. When we have endowment, we can calculate m as 0.5*100=50.
So the budget constraint is $10c+f=50$? My idea is that when the farmer sells food, the selling price is 0.5. But when the farmer buys food, the price is still 1. Am I correct?