A consumer has $100 to spend on two goods X and Y with prices $3 and $5
respectively. Derive the equation of the budget line and sketch the graph.
The budget line equation can be derived by setting up the equation that represents the total expenditure within the given budget of $100.
Let's assume the consumer spends x amount of money on good X and y amount of money on good Y.
The total cost of good X would be 3x, and the total cost of good Y would be 5y.
According to the given information, the consumer has $100 to spend, so the equation representing the total expenditure is:
3x + 5y = 100
This equation represents the budget line. Now, let's sketch the graph based on this equation.
To plot the graph, we need to find two points on the line.
Point 1: When the consumer spends all the money on good X, y = 0
Substituting y = 0 into the equation:
3x + 5(0) = 100
3x = 100
x = 100/3
So, point 1 is (100/3, 0)
Point 2: When the consumer spends all the money on good Y, x = 0
Substituting x = 0 into the equation:
3(0) + 5y = 100
5y = 100
y = 100/5
y = 20
So, point 2 is (0, 20)
Now, let's plot these points and draw a line passing through them.
The graph of the budget line is as follows:
|
20 |
|
| .
| .
|
| .
10 | .
--------------------------------------------------
20 33.33 100
In this graph, the x-axis represents the quantity of good X, the y-axis represents the quantity of good Y, and the line represents the budget line. The line passes through points (100/3, 0) and (0, 20).
Note that the slope of the budget line is -3/5, which represents the rate at which good X can be exchanged for good Y while keeping the budget constant.