# Need help with Edgeworth box and Robinson Crusoe problems

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• Mar 20th 2013, 11:49 AM
lalalulu
Need help with Edgeworth box and Robinson Crusoe problems
hi everybody,
i am looking for some help to solve these 2 problems.

FIRST

Consider a pure-exchange, private-ownership economy, consisting in two consumers, denoted by i = 1, 2, who
trade two commodities, denoted by l = 1, 2. Each consumer i is characterized by an endowment vector wi, a consumption set. Initial endowments are given by w1(1/2,1/2) and w2(3/2,1/2).
Individual utility functions are:
u1(x11, x21) = x11 + x21
u2(x12, x22) = min (x12, x22)

Tasks:

1. Draw the Edgeworth Box for this economy, plotting the endowment point w and the indifference curves
passing through it for both consumers.

2. Show graphically the set of Pareto efficient allocations, that is the set of allocations for which there is no
other allocation indifferent for one consumer and strictly preferred by the other consumer. (Hint 1: given
the specific preferences assumed, you cannot impose the tangency condition between the two consumers’ indifference curves.)

3. Find the competitive equilibrium prices and allocations. Draw the equilibrium in the Edgeworth Box (Hint 2: It is a boundary equilibrium).

SECOND

Consider a “Robinson Crusoe Economy”, i.e. a private-ownership, competitive economy with only one consumer (I = 1), one producer (J = 1), and two commodities (L = 2). The consumer is characterized by a consumption set X ={ x = (x1, x2) in R2+ } where x1 and x2 denote the quantities consumed of ”leisure time” and ”consumers good”, respectively, and a Cobb-Douglas utility function u (x1, x2) = x1^(1/2)* x2^(1/2) .

The consumer owns the endowments w*= (L*, 0) where L* = 24 denotes the time units the consumer has at his disposal (time can be used as“leisure time”or “working time”); the endowment of consumers good is nil. The consumer owns the producer, getting the latter’s profits entirely. The producer is characterized by a single-output technology, with production set:

Y={ y=(-z,q)in R2 | F(y)=F((-z,1))=q-f(z)<=0 and z>=0}

where z is the quantity of input (“labor time”), q is the quantity of output (“consumers good”), F(y) is the transformation function and f (z) = 2z^(1/2) is the production function.

Tasks:

1. Compute the (unique) Pareto-efficient allocation of the ”Robinson Crusoe Economy“ , denoted ((x1*,x2*),(-z*,q*)).

2. Let p and w be respectively the price of the consumers good and the wage rate, i.e., the price of both the”leisure time” and the ”working time”. By taking the consumers good as the numeraire, thereby setting p =1, compute the shadow prices implicit in the Pareto-efficent allocation, (w*, 1), and the corresponding profits, pi*. Show that, at those prices and by assigning those profits to the consumer, the Pareto-efficient allocation can be obtained as a Walrasian competitive equilibrium.

3. Illustrate the producer’s and the consumer-worker’s problems in one and the same diagram.

I would be very thankful if somebody could give me some additional hint how to solve part of the above microeconomics problems.
bye