Example: 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.
To derive the equation of the budget line, we first need to determine the quantity of goods X and Y that the consumer can afford with their $100 budget.
Let's assume the consumer can purchase x units of good X and y units of good Y.
The total cost of good X will be the price of X multiplied by the quantity of X: Cost of X = $3x.
The total cost of good Y will be the price of Y multiplied by the quantity of Y: Cost of Y = $5y.
Since the consumer has a budget of $100, we can set up the following equation:
Cost of X + Cost of Y = Total Budget.
$3x + $5y = $100.
Now, we can rearrange the equation to solve for y in terms of x:
$5y = $100 - $3x
y = (100 - 3x) / 5.
This is the equation of the budget line. It represents all the combinations of goods X and Y that the consumer can afford with their $100 budget.
To sketch the graph, we can plot several points on the X-Y plane to represent different combinations of goods X and Y that the consumer can purchase.
Let's choose some values for x and plug them into the equation to find the corresponding y values:
When x = 0, y = (100 - 3(0)) / 5 = 100 / 5 = 20.
When x = 10, y = (100 - 3(10)) / 5 = 70 / 5 = 14.
When x = 20, y = (100 - 3(20)) / 5 = 40 / 5 = 8.
Now we can plot these points on the graph and draw a line passing through them to represent the budget line.
To derive the equation of the budget line, we need to determine the different combinations of goods X and Y that can be purchased with the given budget.
Let's assume the consumer purchases x units of good X and y units of good Y. The prices of goods X and Y are $3 and $5 respectively.
The equation of the budget line can be derived as follows:
The total amount spent on good X (3x) plus the total amount spent on good Y (5y) should be equal to the total budget available (100).
Therefore, the equation of the budget line is:
3x + 5y = 100
To sketch the graph of the budget line, we can plot a line in the coordinate plane using this equation.
First, let's choose some values for x and find the corresponding values for y:
- For x = 0, 3(0) + 5y = 100, which gives y = 20.
- For x = 10, 3(10) + 5y = 100, which gives y = 14.
- For x = 20, 3(20) + 5y = 100, which gives y = 8.
- For x = 30, 3(30) + 5y = 100, which gives y = 2/5.
Now, we can plot these points on the graph and connect them to form the budget line.
The graph should have the x-axis representing the quantity of good X and the y-axis representing the quantity of good Y. The budget line will be a straight line connecting these points.