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ECON7790 Assignment 6

Fall 2023

Q1. Suppose a firm produces clothes and requires two input factors: capital and labor. The firm's production function is

where k represents capital input, and l represents labor input. Suppose the price per unit of capital and labor is $40 and $10, respectively.

a. (20 point) If the firm decides to produce y pieces of clothing, what is the minimum cost to be incurred?

b. (5 point)Suppose the firm is in a perfectly competitive product market, how many pieces of clothing should this firm produce if the market price of the clothes it produces is p dollars each? ( hint: the number of clothes is a function of p )

c. (15 point) We further assume that there are 800 firms in that perfectly competitive market,  all  of which  produce  the  same  clothes  and  all  of which  have  the  same production   technology,    i.e.,    each    of   them    firm's    production    function    is f(k, l) = k1/4l1/4  . Meanwhile, the market inverse demand function for the clothes is p = 800 - D /10 , where D denotes the market's demand. Find the market price of the dress when the market is under equilibrium. What is the quantity of clothes sold by each firm? (Hint: the market is under equilibrium when demand = supply)

Q2. (Walrasian Equilibrium)Consider an economy with two goods: x and y, and two consumers: Alice and Benson. Let Px  and Py be the prices for x and y. Alice’s utility function is  U A (x, y) = xy and her initial endowment is eA   = (0,4) , (zero x and 4 y). Bob’s utility function is UB (x, y) = ln x + ln y  and his initial endowment is eB   = (2,0) .

a. (20 point) Calculate the Marshallian demand for both Alice and Benson in the form of [x A , y A ),(x B , y B)]

b. (20 point) Find the total demand for each good x and good y.

c. (20 point) Calculate the price ratio under general equilibrium.(Hint: the market clears when Demand = Supply)