2. In this exercise, we'll practice how to derive individual and market demand curves from utility functions and

consumers' preferences:

Assume that consumers in an economy can be divided into two groups: high-income consumers and low-income

consumers. These consumers only consume two goods: x and y. The utility functions of the high-income and low-

income groups are given as follows, respectively:

a. Derive the demand function of good x for high-income consumers (Hint: Find the optimal xH function in terms

of other constants like Px, Py, and I).

b. Perform the natural log monotonic transformation on the utility function for low-income consumers to

make this problem simpler (Hint: Take the natural log function of U (x, y), and apply the natural log

properties on the exponents of the function).

c. Derive the demand function of good x for low-income consumers (Hint: Use the function in part (b) to find

the optimal x,function in terms of other constants like Px, Py, and I₁. Using the function in (b) is just easier, but

you are not obligated to use it).

Now assume in the remaining parts that the monthly income of low-income consumers is $300, and the monthly

income of high-income consumers is $6000.

d.

Write down the demand functions of good x for low-income consumers and high-income consumers at the

given incomes.

e.

At what range of prices do high-income consumers stop buying product x? At what range of prices do low-

income consumers stop buying product x?

f. Write down the market demand equation for good x.

g.

At what price would you expect to see a kink in the market demand curve for good x? Explain your answer.