Assume the price of good Y with its quantity measured on the vertical axis is $20 and the price of good X with its quantity measured on the horizontal axis is $5. If the consumer's budget is $100, then the absolute value of the slope of the budget line is:
A) 100.
B) 20.
C) 1/4.
D) 4.

Question

Quizplus · Accepted Answer

The equation for the budget line is:

Income = Px(Qx) + Py(Qy)

where Px and Py are the prices of goods X and Y respectively, and Qx and Qy are the quantities consumed.

Using the given information, we can rewrite this equation as:

100 = 5Qx + 20Qy

Solving for Qy in terms of Qx, we get:

Qy = (100 - 5Qx)/20

This is the equation of the budget line.

The slope of the budget line is equal to the absolute value of the ratio of the change in Qy to the change in Qx:

slope = |ΔQy/ΔQx|

In this case, suppose we move from Qx = 0 (i.e. consuming no good X) to Qx = 20 (i.e. consuming all of the budget on good X). Then the change in Qy is:

ΔQy = (100 - 5Qx)/20 - 0/20 = (100 - 5Qx)/20

The change in Qx is:

ΔQx = 20 - 0 = 20

So the slope of the budget line is:

slope = |ΔQy/ΔQx| = |(100 - 5Qx)/20 / 20| = |(100 - 5Qx)/400|

To find the maximum absolute value of this slope, we want to set Qx equal to either 0 or 20, whichever produces the largest slope.

If Qx = 0, then the slope is:

slope = |(100 - 5*0)/400| = 1/8

If Qx = 20, then the slope is:

slope = |(100 - 5*20)/400| = 3/8

So the maximum absolute value of the slope is 3/8, which is equivalent to 0.375.

Finally, the answer is the reciprocal of the slope, so the answer is:

Answer: D
 The maximum absolute value of the slope of the budget line is 3/8. The absolute value of the reciprocal of this value is 8/3 or approximately 2.67. Thus, the absolute value of the slope of the budget line is 2.67, which corresponds to Choice D.

Assume the Price of Good Y with Its Quantity Measured