Question

If a consumer spends all her income on two goods X and Y , then she can just afford either consumption bundle (5, 10) or (10, 7.5). Write the equation of her budget line.

Answer 1

Let the income be M. Given that the two consumption bundles are (5, 10) or (10, 7.5) and at these, the income is exhaused. Let the prices be Px and Py. Since her income is getting exhaused, the spending are equal to income and there is no saving. This is the principle used.

The budget equation is M = XPx + YPy

M = 5Px + 10Py as well as M = 10Px + 7.5Py

From these equations, we find the slope of the budget line

Slope = change in quantity of Y/change in quantity of X (when we move from bundle (5, 10) to (10, 7.5)

slope m = (-2.5)/5 = -0.5.

Now the equation of line is usually expressed as y = mx + c where m is the slope. Here m is -0.5.

y = -0.5x + c or x + 2y = c.

Now we have M = 5Px + 10Py as well as M = 10Px + 7.5Py. This gives

5Px + 10Py = 10Px + 7.5Py

2.5Py = 5Px

Px/Py = 0.5

Use this value in the budget equation. This can be written as

M/Py = 5Px/Py + 10.

M/Py = 5*0.5 + 10

M/Py = 12.5. Similarly, M/Px = 25

Now we have all the information. We have the vertical intercept (M/Py) which shows the quantity of y when all income is spent on purchasing y. And the horizontal intercept (M/Px) which shows the quantity of yxwhen all income is spent on purchasing x. This implies

In the budget equation x + 2y = c, when x = 0, y = 12.5. Hence this gives 0 + 2*12.5 = c. This gives c = 25

So the final equation of budget is x + 2y = 25,