Utility Function

Question

The utility function is u (x1; x2) = 5 min(x1; x2) + 2x₂. Suppose that the constraints are as follows: I>= p₁x₁ + p₂x₂; x₁>= 0; x₂ >= 0 where I is income, p₁ is the price of good 1 and p₂ is the price of good 2.

a) Find the demand functions for good 1 and good 2.

b) Plot the demand functions for good 1 and good 2.

Answer 1

The key to this question is to see that the Utility curve is “quasi-L-shaped” where the elbow is at x1 = x2. For x1 > x2, u = 7*x2, and for x1 < x2, u = 5*x1 + 2*x2. For example, if you graph the curve for u=7 you’ll see the x2 intercept is at 7/2 (from the budget constraint x2 = (u/2) - 5/2 * x1) and the utility curve is a straight line that slopes down to x1 = x2 = 1, and then is flat for x1 > x2 at x2 = 1.  

So, to maximize utility, you have 2 solutions where the budget line would be tangent depending upon the slope of the budget line p1/p2: if p1/p2 > 5/2 then the corner solution at x1 = 0 is the answer; if p1/p2 < 5/2, then the solution is at the elbow x1 = x2; if p1/p2 = 5/2 then anywhere along the upper line is tangent, but for simplicity x1 = x2 is sufficient.

Once you see the 2 solutions (x1=0, or x1=x2 depending upon p1/p2) you can plug these into the budget constraint to get the demand curves.  

For example, the demand curve for x1: 

if p1/p2 > 5/2, then demand is x1 = 0;  

if p1/p2 <= 5/2, then x1 = x2, so I = p1*x1 + p2*x2 = x1 * (p1 + p2)

and demand is x1 = I / (p1 + p2)

Similar for the x2 demand curves:

if p1/p2 > 5/2, then x1 = 0 then demand is x2 = I / p2

if p1/p2 <= 5/2, then x1 = x2 then demand is x2 = I / (p1 + p2)