Problem Code: 060609MICRO

[Amit Goyal]

A consumer has a utility function of the form
U(x, y) = u(x) + y,
Good x is a discrete good; the only possible levels of consumption of good x are x = 0 and x = 1. For convenience, assume that u (0) = 0 and p(y) = 1.
Question: The consumer will definitely choose x = 1 if p(x) is strictly less than what?

p(i) is price of good i in {x, y}

[namrata]

px must be less than py.that is <1

[Amit Goyal]

We need two conditions for x = 1 to be chosen definitely
i) Affordability: 1 unit of x should be affordable. i.e. p(x)(1) ≤ M
ii) Optimality: Spending on 1 unit of x should give you higher utility then not spending anything on it. Hence, we compare the two situations:
Spending on 1 unit of x and remaining money on y will give you utility = u(1) + (M - p(x))/p(y)
Spending everything on y will give utility = M/p(y)
i.e. u(1) + (M - p(x))  >  M   (Since p(y) =1)
Combining i) and ii),
p(x) < u(1) and p(x) ≤ M