Question
Let x1 ≥ 0 and x2 ≥ 0 be the levels of consumption of good 1 and good 2, respectively. Income is I = 8 and the prices of good 1 and good 2 are p1 = 1 and p2 = 1, respectively. The government decides to subsidize the consumption of good 1 in the following way. For each of the first 6 units of good 1 (i.e., x1 ≤ 6), the agent receives a per-unit subsidy of 2 dollars (i.e., s = 2). Yes, s > p1, this is not a typo. Then, there is no more subsidy for any additional unit beyond 6 units (i.e., x1 > 6). Draw the budget line and clearly label your graph.
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GPT 3.5
To draw the budget line in this scenario, we need to plot the possible combinations of consumption of good 1 and good 2 that are affordable given the income and prices.
We know that the income is I = 8, the price of good 1 is p1 = 1, and the price of good 2 is p2 = 1.
Before we consider the subsidy, let's first consider the budget line without any subsidy. The budget line is represented by the equation:
p1 * x1 + p2 * x2 = I
Substituting the values, we have:
1 * x1 + 1 * x2 = 8
Simplifying the equation, we get:
x1 + x2 = 8
Now, let's incorporate the subsidy. For each unit of good 1 up to 6 (i.e., x1 ≤ 6), the agent receives a subsidy of 2 dollars per unit. This means that for the first 6 units of consumption of good 1, the effective price is reduced by the subsidy amount. So instead of paying 1 dollar per unit, the agent effectively pays (1 - 2) = -1 dollar per unit.
Since the effective price is negative, the agent receives money for each unit of consumption up to 6 units. This results in a positive income for the agent, allowing them to consume more of both goods.
To incorporate the subsidy in the budget line equation, we can rewrite it as:
(p1 - s) * x1 + p2 * x2 = I
Substituting the values, we have:
(1 - 2) * x1 + 1 * x2 = 8
Simplifying the equation, we get:
-1 * x1 + x2 = 8
Now, let's plot this budget line on a graph. The x-axis represents the consumption of good 1 (x1) and the y-axis represents the consumption of good 2 (x2).
The coordinates of the points on the budget line can be obtained by substituting different values of x1 into the equation and solving for x2.
When x1 = 0, -1 * 0 + x2 = 8
=> x2 = 8
When x1 = 6, -1 * 6 + x2 = 8
=> x2 = 14
Now, we can plot the points (0, 8) and (6, 14) on the graph and connect them with a straight line.
The resulting line represents the budget line with the subsidy.
We know that the income is I = 8, the price of good 1 is p1 = 1, and the price of good 2 is p2 = 1.
Before we consider the subsidy, let's first consider the budget line without any subsidy. The budget line is represented by the equation:
p1 * x1 + p2 * x2 = I
Substituting the values, we have:
1 * x1 + 1 * x2 = 8
Simplifying the equation, we get:
x1 + x2 = 8
Now, let's incorporate the subsidy. For each unit of good 1 up to 6 (i.e., x1 ≤ 6), the agent receives a subsidy of 2 dollars per unit. This means that for the first 6 units of consumption of good 1, the effective price is reduced by the subsidy amount. So instead of paying 1 dollar per unit, the agent effectively pays (1 - 2) = -1 dollar per unit.
Since the effective price is negative, the agent receives money for each unit of consumption up to 6 units. This results in a positive income for the agent, allowing them to consume more of both goods.
To incorporate the subsidy in the budget line equation, we can rewrite it as:
(p1 - s) * x1 + p2 * x2 = I
Substituting the values, we have:
(1 - 2) * x1 + 1 * x2 = 8
Simplifying the equation, we get:
-1 * x1 + x2 = 8
Now, let's plot this budget line on a graph. The x-axis represents the consumption of good 1 (x1) and the y-axis represents the consumption of good 2 (x2).
The coordinates of the points on the budget line can be obtained by substituting different values of x1 into the equation and solving for x2.
When x1 = 0, -1 * 0 + x2 = 8
=> x2 = 8
When x1 = 6, -1 * 6 + x2 = 8
=> x2 = 14
Now, we can plot the points (0, 8) and (6, 14) on the graph and connect them with a straight line.
The resulting line represents the budget line with the subsidy.
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