. Consumers are named using real numbers, x, between 0 and 1. Each consumer is interested in buying at most one unit of a good. The reservation price that consumer x has for this good when fraction z of the population buys the good is given by r(x)f(z) where r(x)=1-x and f(z)=10z. Suppose that the price of the good is 0.9.There are two possible non-zero equilibrium fractions of the population purchasing the good.

a. The smallest non-zero equilibrium fraction of the population purchasing the good is: (Enter your answer in the format x/x)

To determine the smallest non-zero equilibrium fraction of the population purchasing the good, we need to find the value of x at which the quantity demanded equals the quantity supplied. In other words, we need to find the value of x for which the reservation price of consumer x equals the price of the good.

The reservation price of consumer x is given by r(x)f(z), where r(x) = 1 - x, and f(z) = 10z. So, the reservation price for consumer x is (1 - x) * 10z.

Given that the price of the good is 0.9, we can set up the equation:

(1 - x) * 10z = 0.9

To find the equilibrium fraction of the population, z, we need some additional information.

To find the smallest non-zero equilibrium fraction of the population purchasing the good, we need to find the value of x when the reservation price equals the price of the good.

The reservation price of consumer x is given by r(x)f(z), where r(x) = 1 - x and f(z) = 10z. The price of the good is given as $0.9.

Setting the reservation price equal to the price of the good, we have:

r(x)f(z) = $0.9

Substituting the values for r(x) and f(z), we get:

(1 - x)(10z) = 0.9

Now, we can solve for x.

Dividing both sides by 10z:

(1 - x) = 0.09 / z

Next, we isolate x:

x = 1 - (0.09 / z)

Since x must be between 0 and 1, we can set the inequality:

0 < x ≤ 1

Substituting the expression for x:

0 < 1 - (0.09 / z) ≤ 1

Simplifying this inequality, we get:

0 < (z - 0.09) / z ≤ 1

Since z is a fraction of the population, it must be greater than 0.09 to satisfy the inequality.

Therefore, the smallest non-zero equilibrium fraction of the population purchasing the good is 0.09/1, which can be simplified to 9/100.