经济问题集代做 Economics 426代写 经济学作业代写代做

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

Spring 2020

Points Possible: 100

Due: Wednesday (in class), February 19.

I. 经济问题集代做

Robin Crusoe is endowed with 112 labor-hours per week. There is a production function for the output of oysters

q = F(L, W) := L^1/2W,                      (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 p^o = 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(L^d , W) − wL^d = q^s − wL^d ,                  (3)

where q^s is oyster supply and L^d 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(L^d , W) − wL^d

consistent with the production function. Robin then faces the budget constraint wR + c = Y = Π⁰ + 112w.

A. 经济问题集代做

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]

B.

Suppose W = 1. Solve for Robin’s optimal choice of c and R given the production function and time constraint. [10 pts]  

C.

Suppose W = 1. Solve for the general equilibrium. [10 pts]

D.

Suppose w > w^∗ (the equilibrium wage found in part C) is set at a disequilibrium level. Then L^d + R ≠ 112 and q^s ≠ 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, L^d > 112−R. How would you expect w to adjust? [4 pts]

E. 经济问题集代做

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]

F.

Redo part C with W = 2. What happens to the wage, output and employment with this better weather? Explain intuitively.

[20 pts]

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(L^x , T^x ) := 1 + (L^xT^x )^1/2 ,

where L^x is L used to produce x, T^x is T used to produce x. f(L^x , T^x ) ≥ 0 for L^x ≥ 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(L^y , T^y ) := 1 + (L^yT^y )^1/2 ,

where L^y is L used to produce y, T^y is T used to produce y. g(L^y , T^y ) ≥ 0 for L^y ≥ 0, T^y ≥ 0; g(0, 0) = 0. The price of good x is p^x . The price of good y is p^y . Profits of firm x are Π^x = p^x f(L^x , T^x )−wL^x−rT^x . Profits of firm y are Π^y = p^y g(L^y , T^y )−wL^y−rT^y .

Robinson’s income then is wL + rT + Π^x + Π^y . 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(L^x , T^x )] + b[y − g(L^y , T^y )] + c(L^o − L^x − L^y ) + d(T⁰ − T^x − T^y ).

A.

Derive and graph the production possibility frontier. [14 pts]

B.

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]

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