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.
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.
To draw the budget line, we need to determine the different segments of the line based on the given information.
First, let's consider the case when the consumption of good 1, x1, is less than or equal to 6. In this case, the agent receives a per-unit subsidy of 2 dollars, which means the effective price of good 1 is reduced by 2 dollars. Therefore, the effective price of good 1, p1', is given by:
p1' = p1 - s = 1 - 2 = -1
Since the price cannot be negative, the effective price of good 1, p1', is taken as zero. The effective price of good 2, p2', remains the same, i.e., p2' = p2 = 1.
Now, let's consider the case when x1 is greater than 6. In this case, the subsidy is no longer applicable, so the original prices p1 and p2 apply.
With this information, we can now draw the budget line on a graph. Since x1 ≥ 0 and x2 ≥ 0, the budget line will pass through the points (6,0) and (0,8).
Here is the graph representing the budget line:
X2
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8 |-------------------------
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6 |-------------------/
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4 |------------/
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2 |-------/
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0---------------------------- X1
The line starts from the point (6,0) where x1 = 6 and x2 = 0, and ends at the point (0,8) where x1 = 0 and x2 = 8. The line is straight and has a negative slope, indicating the trade-off between consuming goods 1 and 2 given the limited income and prices.
To draw the budget line, we need to find the equation that represents the agent's budget constraint given the prices and the subsidy.
Given that the price of good 1 (p1) is $1 and the price of good 2 (p2) is also $1, and the income (I) is $8, we can start by setting up the equation for the budget constraint without considering the subsidy:
p1 * x1 + p2 * x2 = I
Substituting the values, we have:
1 * x1 + 1 * x2 = 8
x1 + x2 = 8
This equation represents the budget constraint without the subsidy. Now, we need to consider the subsidy for the first 6 units of good 1.
Given that the subsidy per unit (s) is $2, for each unit of good 1 up to 6 units, the agent receives an additional $2. This means that the effective price of good 1 for these units is reduced by the amount of subsidy.
Let's consider two cases:
1. For 0 ≤ x1 ≤ 6:
The effective price of good 1 is p1 - s, which is $1 - $2 = -$1 (negative because the subsidy is greater than the price). The equation of the budget constraint for this range is:
(-$1) * x1 + 1 * x2 = 8
-x1 + x2 = 8
2. For x1 > 6:
The subsidy only applies to the first 6 units of good 1. Beyond that, the subsidy is not applicable, so the effective price remains the same as the original price. The equation of the budget constraint for this range is:
1 * x1 + 1 * x2 = 8
x1 + x2 = 8
We can now plot these two ranges of the budget constraint on a graph. Let's assume x1 is on the horizontal axis and x2 on the vertical axis.
1. For 0 ≤ x1 ≤ 6:
Plot two points on the line -x1 + x2 = 8:
- When x1 = 0, x2 = 8 (point A)
- When x1 = 6, x2 = 2 (point B)
Draw a straight line passing through these two points.
2. For x1 > 6:
Plot two points on the line x1 + x2 = 8:
- When x1 = 6, x2 = 2 (point B, same as before)
- When x1 = 8, x2 = 0 (point C)
Draw a straight line passing through these two points.
The graph will show a budget line with a kink at x1 = 6, where the slope changes due to the subsidy.