Anna consumes good x and good y. Her utility function is given u(x, y) = 100x − x2 + y. Due to the resource constraint, Anna cannot consume more than 100 units of good x. The income is given m and the prices of goods are px and py. Let py = 1 in what follows. Suppose that Anna is the only consumer in this economy.
(1) Derive Anna’s (Marshallian) demand curve of good x.
(2) Is x inferior good? Is x ordinary good?
(3) If px = 50, then what is her demand? Calculate Anna’s consumer surplus.
(4) Similarly, if px = 80, what is her demand? Calculate Anna’s consumer surplus.
(5) Calculate the change in consumer surplus when the old price is px = 50 and the new price is p′x = 80.
(6) What is the own price elasticity of the demand for x?
(7) What is the income elasticity of the demand for x?
(8) What is the cross price elasticity of the demand for y?
(9) Suppose now that there is another consumer who is exactly the same as Anna in the economy. Derive the market demand. Calculate the own price elasticity of the market demand for x and compare it with your answer in (6).