ECON5007 经济练习题代写 Week 9 review questions 1.What is the role of an arbitrageur in the context of futures markets? 2.How does a futures contract differ in relation to Week 9 review ques...View details
Economics 426: Problem Set 1 – Robinson Crusoe
经济问题集代做 I. Robin Crusoe is endowed with 112 labor-hours per week. There is a production function for the output of oysters
Points Possible: 100
Due: Wednesday (in class), February 19.
Robin Crusoe is endowed with 112 labor-hours per week. There is a production function for the output of oysters
q = F(L, W) := L1/2W, (1)
where L is labor applied to oyster harvesting and W is “weather.” W takes on one of two values: 1 (bad weather) or 2 (good weather). W is known by Robin when harvesting oysters and is also taken as given. Robin’s leisure, R, is determined by
R = 112 − L. (2)
Her utility function is
u(c, R) := ln c + 2 ln R,
where c is Robin’s consumption of oysters (ln is the natural or base e logarithm). Let production be organized in a firm, and let consumption and labor supply decisions occur in the household. Let oysters act as numeraire with price fixed at po = 1. The wage rate w is expressed in oysters per labor-hour.
Planned profits of the oyster harvesting firm then are 经济问题集代做
Π = F(Ld , W) − wLd = qs − wLd , (3)
where qs is oyster supply and Ld is labor demanded. Robin is the sole owner of the oyster harvester. Her income Y may most easily be thought of as the value of her labor endowment plus her profits:
Y = w · 112 + Π. (4)
She spends her income Y on the (re)purchase of leisure R and on the purchase of oysters c, giving the budget constraint
Y = wR + c. (5)
As a household, Robin is a price-taker and profit-taker; she regards w and Π parametrically. Given her income from (4) and her budget constraint (5), she chooses c and R to maximize u(c, R) subject to (5). At wage rate w, the firm chooses the production plan giving the highest profit
Π0 = F(Ld , W) − wLd
consistent with the production function. Robin then faces the budget constraint wR + c = Y = Π0 + 112w.
Define fully a general competitive equilibrium. What does equilibrium require for w? What is required of c,R, q, and L? Clearly describe firm behavior, household behavior, and market-clearing conditions. [10 pts]
Suppose W = 1. Solve for Robin’s optimal choice of c and R given the production function and time constraint. [10 pts]
Suppose W = 1. Solve for the general equilibrium. [10 pts]
Suppose w > w∗ (the equilibrium wage found in part C) is set at a disequilibrium level. Then Ld + R ≠ 112 and qs ≠ c.
1. Does Walras’ Law hold at the disequilibrium w? Why or why not? [4 pts]
2. At the disequilibrium wage rate w, the firm’s plans for its profits cannot be fulfilled. Does this affect the household budget at w? [4 pts]
3. Suppose at the disequilibrium wage rate w, Ld > 112−R. How would you expect w to adjust? [4 pts]
Suppose the economy achieves a wage rate w∗ that gives the economy a general competitive equilibrium, as defined in part A.
1. Show that the equilibrium allocation is identical to the solution of the problem in part A. [4 pts] The choices (utilty maximizing and profit maximizing)
2. What can you then conclude about the allocative efficiency of the market mechanism? [4 pts]
3. The comparison in part 1. is sometimes described as comparing centralized and decentralized allocation mechanisms. Explain this interpretation. [4 pts]
Redo part C with W = 2. What happens to the wage, output and employment with this better weather? Explain intuitively.
II. A simple economy with two factor inputs and two outputs. 经济问题集代做
Let there be two factor inputs: land denoted T and labor denoted L. The resource endowment of T is T0 = 8; the resource endowment of L is L0 = 8. Let there be two goods: x and y. Robinson has a utility function
u(x, y) := xy.
The prevailing wage rate of labor is w, and the rental rate on land is r. Good x is produced in a single firm by the production function
x = f(Lx , Tx ) := 1 + (LxTx )1/2 ,
where Lx is L used to produce x, Tx is T used to produce x. f(Lx , Tx ) ≥ 0 for Lx ≥ 0, Tx ≥ 0; f(0, 0) = 0. Good y is produced in a single firm by the production function 经济问题集代做
y = g(Ly , Ty ) := 1 + (LyTy )1/2 ,
where Ly is L used to produce y, Ty is T used to produce y. g(Ly , Ty ) ≥ 0 for Ly ≥ 0, Ty ≥ 0; g(0, 0) = 0. The price of good x is px . The price of good y is py . Profits of firm x are Πx = px f(Lx , Tx )−wLx−rTx . Profits of firm y are Πy = py g(Ly , Ty )−wLy−rTy .
Robinson’s income then is wL + rT + Πx + Πy . Assume f, g, u, to be strictly concave, differentiable. Assume all solutions are interior solutions. Subscripts denote partial derivatives. An efficient allocation in the economy is characterized by maximizing the Lagrangian, V with Lagrange multipliers a, b, c, d,
V = u(x, y) + a[x − f(Lx , Tx )] + b[y − g(Ly , Ty )] + c(Lo − Lx − Ly ) + d(T0 − Tx − Ty ).
Derive and graph the production possibility frontier. [14 pts]
Solve for the Pareto efficient allocation x∗ and y∗ along with the allocation of land and labor. Hint: there is symmetry in preferences and production for the two goods. [12 pts]