2. In this exercise, we'll practice how to derive individual and market demand curves from utility functions and consumers' preferences: a. 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: U(x,y) = 2y (x + 1) 2/3 U(x, y)2 = 3y1 x2 + 1)1/3 Assume that the price of good x is Px, and the price of good y is Py. Let the income of high-income individuals bel and the income of low-income individuals be l,. Answer the following questions: 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 Ih). 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). 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 1. 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. 8. At what price would you expect to see a kink in the market demand curve for good x? Explain your answer. C. 2. In this exercise, we'll practice how to derive individual and market demand curves from utility functions and consumers' preferences: a. 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: U(x,y) = 2y (x + 1) 2/3 U(x, y)2 = 3y1 x2 + 1)1/3 Assume that the price of good x is Px, and the price of good y is Py. Let the income of high-income individuals bel and the income of low-income individuals be l,. Answer the following questions: 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 Ih). 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). 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 1. 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. 8. At what price would you expect to see a kink in the market demand curve for good x? Explain your answer. C