This is a question and answer questions.
There are five questions?.?Please answer all the questions.
1. Suppose an economy has two consumers, Ellen and Kerry, and two commodities X and Y.
Ellen's utility function and initial endowment are:
UE = min {XE, YE}
ωE = (ωEX, ωEY) = (a, 5a)
Kerry's utility function and initial endowment are:
UK = min {XK, YK}
ωK = (ωKX, ωKY) = (5a , a) where a is a positive constant.
(a) Please draw the Edgeworth box.
(b) Please label Ellen's and Kerry's initial endowment points in the Edgeworth box.
(c) Please draw at least three Ellen's and Kerry's indifference curves in the Edgeworth box.
(d) Please draw the contract curve in the Edgeworth box, and explain how you got this contract curve.
Hint: This problem set is similar to your assignment question. The only difference is that you are not given numbers to work with but random constants, parameters, or exogenous variables.
However, it takes the same method to solve this type of questions.
2. Suppose an economy economy has two consumers, A and B, and two commodities X and Y. A's utility function and initial endowment are:
UA ( XA, YA )= XA1/3 YA1/3
ωA =(ωAX, ωAY )=(30 , 20)
B's utility function and initial endowment are:
UB = (XB , YB )= XB1/2 YB1/2
ωB = (ωB X , ωBY ) = (30, 18 0)
(a) What is A's Marshallian demand function for XA ?
(b) What is A's Marshallian demand function for YA ?
(c) What is B's Marshallian demand function for XB ?
(d) What is B's Marshallian demand function for YB ?
(e) In the general competitive equilibrium, what is PX/PY?
(f) Find ( XA, YA, XB, YB ) in the general competitive equilibrium.
Hint: Please leave your answers as fractions, and do not attempt to convert them to decimals. Please be careful at each step of your calculation.
3. Robinson Crusoe obtains utility from the quantity of fish he consumes in one day (F), the quantity of coconuts he consumes that day (C), and the hours of leisure time he has during the day (H) according to the utility function:
U = F1/5 C1/5H3/5
Robinson's production of fish is given by
F = LC1/2
(where LC is the hours he spends fishing), and his production of coconuts is determined by
C = LC1/2
(where LC is the time he spends picking coconuts).
Assuming that Robinson decides to work a 10-hour day (that is, H=14):
(a) Graph his production possibility curve for fish and coconuts.
(b) Show his optimal choices of those goods.
4. Suppose the production possibility frontier for guns (X) and butter (Y) is given by
12X2 + 3Y2 = 300
(a) Graph this frontier.
(b) If individuals always prefer consumption bundles in which Y =2 X , how much X and Y will be produced?
(c) At the point described in part (b), what will be the slope of the production possibility frontier, and what price ratio will cause production to take place at that point?
(d) Please identify your solution on the figure from part (a).
5. A firm producing hockey sticks has a production function given by Q= AK1/2L1/2 where A is a parameter for technology and held as a positive constant, K is capital, and L is labor. In the short run, the firm's amount of capital equipment is fixed at K=K¯ (where K¯ is the targeted production level set by its manager, and is undoubtedly greater than zero. ) The rental rate for capital is r, and the wage for labor is w.
(a) Please use your words to explain why Q= AK1/2L1/2 is a Cobb-Douglas production function. Also, does it exhibit increasing, decreasing or constant returns to scale?
(b) Please write down the firm's short-run total cost function (STC).
(c) Please write down the firm's short-run average cost function (SAC).
(d) Please write down the firm's short-run marginal cost function (SMC).
(e) What does the SMC curve intersect the SAC curve? That is, at what level of hocky sticks does average cost reach a minimum? What is the average cost at this level output? What is the marginal cost at this level of output?
# Hint: Again, your answers will end up with no numbers, but endogenous or exogenous variables. Please be careful about your calculation.