Question

2. Continuation of the in-class problem. There are two goods. The first good is sold at price Pi if the amount is less than or equal to> 0, and is sold at a discount (1 - X) E (0,1) if the amount is bigger than xi. Assume m/P1 > Ti- (a) Carefully draw a budget line. (b) Derive analytical expression for the budget line

Answer 1

Solution:

Let the two goods be 1 and 2, and their respective prices be p1 and p2. Denoting quantities of the two goods by x1 and x2. We are also given that if quantity of good 1 exceeds x1(bar), it's price falls to p1(1-d) (denoting lambda by d for ease of writing); where (1-d) < 1.

a) The slope of the budget line initially = p1/p2

New slope is p1(1-d)/p2

Since, p1(1-d) < p1

p1(1-d)/p2 < p1/p2, so the budget line will be a discontinuous one, because as the threshold of x1(bar) is crossed, price on all units of good 1 falls. The discontinuity will occur where quantity of good 1 will equal x1(bar). Also, before the discontinuity, budget line will be steeper, while after the kink it becomes flatter (as price of good 1 falls on exceeding the threshold making the price ratio fall, as already seen).

Following is the graph for the budget line:

Good 2, x2 m/p2 [m - p1(1-d)*x1(bar)1/p2 [m pl*x1(bar)] /p2 - - (Discontinu÷us) Budget Line 0 x1(bar m/p1) m/(p1(1-d)) Good 1, x1

b) Since, the budget line is discontinuous one, we can break the derivation of budget line in two. One when quantity of good 1 is less than or equal to x1(bar), and other when quantity of good 1 is greater than x1(bar).

Equation of the budget line is:

p1*x1 + p2*x2 = m, when x1 <= x1(bar)

p1(1-d)*x1 + p2*x2 = m when x1 > x1(bar)

Note that according to the question, by buying too big quantity of good 1 (beyond x1(bar)), per unit price on all units fall (including the units lying below the x1(bar) threshold), and thus, the budget line is discontinuous.

Had the bulk discounting been in a way that price for only units beyond the threshold would have fallen, a kinked budget line would have generated, not the discontinuous one, as under this case.