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 equation of the budget line can be derived by the formula:
Px * X + Py * Y = I
where:
- Px is the price of good X
- Py is the price of good Y
- X is the quantity of good X
- Y is the quantity of good Y
- I is the available income or budget
In this case, Px = $3, Py = $5, and I = $100. Substituting these values into the equation, we get:
3X + 5Y = 100
To sketch the graph, we'll first solve the equation for Y:
5Y = 100 - 3X
Y = (100 - 3X)/5
Now we can create a table of values for X and Y:
| X | Y |
|---|---|
| 0 | 20 |
|10 | 14 |
|20 | 8 |
|30 | 2 |
|40 |-4 |
Now we can plot the points (0,20), (10,14), (20,8), (30,2), and (40,-4) on a graph with X on the x-axis and Y on the y-axis.
To derive the equation of the budget line, we need to determine the different combinations of goods X and Y that the consumer can afford given their budget of $100.
Let's assume that the quantity of good X is represented by the variable 'x', and the quantity of good Y is represented by the variable 'y'.
The price of good X is $3, so the total amount spent on good X (Px) can be calculated as Px = 3x.
Similarly, the price of good Y is $5, so the total amount spent on good Y (Py) can be calculated as Py = 5y.
The consumer has a total budget of $100, so the sum of the amounts spent on goods X and Y should equal $100.
Hence, the equation of the budget line can be written as:
3x + 5y = 100
Now, let's sketch the graph of the budget line on a coordinate plane. To do this, we need to find two points that satisfy the equation.
When x = 0, the equation becomes:
3(0) + 5y = 100
5y = 100
y = 20
So, the first point is (0, 20).
Similarly, when y = 0, the equation becomes:
3x + 5(0) = 100
3x = 100
x ≈ 33.33 (rounded to 2 decimal places)
So, the second point is (33.33, 0).
Plotting these two points and drawing a straight line passing through them will give us the graph of the budget line.
Note: The graph may differ based on the scale used for plotting the x and y axes.