Question

4. In a two-good world, suppose a consumer views the goods as perfect substitutes. Draw a graph of the consumer's choice problem, with a budget constraint and a few indifference curves. (Assume the slope of the indifference curves is different from the slope of the budget constraint.) What is notable about the consumer's preferred bundle?

Answer 1

I assume a utility function of the form:

U(x,y) = x + y

and let the budget constraint be:
P₁x + P₂y = M, where M is consumer income, P₁ is price of good x and P2 is price of good y.

Let P₁ = $1 per unit, P₂ = $2 and M = $100.

Thus a consumer wants to maximize her utility function U(x,y) = x + y subject to a constraint 1*x + 2*y = 100 => x + 2y = 100

Slope of the utility function is constant and it is equal to -1:

dy/dx =-1/1 =-1 Oy

The slope of budget line is dy/dx = -1/2

The same can be shown in the below diagram -

Good y 100 U3 100 75 U2 75 50 U1 50 50 75 100 Good x

In the above diagram, it can be seen that 3 utility functions are plotted against the budget line (in black). The utility function is maximized when X = 100 and Y = 0. This is a corner solution which happens since slope of utility function and indifference curve are not equal. Thus, the notable fact about consumer bundle is that it can be a corner solution where tangency doesn't even hold true.