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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?

My attempt:

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?

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The way I would set up the budget constraint is as follows:

$$p_c c = (1-t_f)p_f(E-f)$$

where $t_f$ is the tax rate on food and $E$ is the endowment. The motivation is, the more of an endowment is consumed the less of it can the farmer supply to the market to get an income. Hence the farmer's supply of food is $E-f$ (i.e. endowment minus consumption). Farmer's income then $(1-t_f)p_f(E-f)$ since income the farmer earns is equal to the supply of produce times price and times the one minus the tax rate. Budget constraint will hold with equality assuming the agent is rational.

Note this is equivalent to (you can do little algebra to prove it):

$$p_cc+(1−t_f)p_ff=(1−t_f)p_fE$$

The motivation for tax being applied to 'purchase' of food is that in this case the person already has the food, so purchase 'price' is the opportunity cost to farmer from consuming the food instead of bringing it to the market. Since in the market the farmer gets $(1-t_f)p_f$ that is the opportunity cost.

After you substitute the parameters you get:

$$10c +0.5 f = 50$$

Next you can set up lagrangian:

$$\mathcal{L} = \ln f + c - \lambda (10c+0.5f-50) $$

From which we can derive 3 FOC's:

$$ \frac{1}{f} = \lambda 0.5$$

$$ 1 = 10 \lambda$$

$$10c +0.5 f = 50$$

Combining the FOC's we get: $\lambda = \frac{1}{10}$, $f^*= 20$ and $c^*=4$.

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