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Unformatted text preview: Fall 2009 Truman Bew!ey Economics 350a
Answers to Midterm Examination
Tuesday, October 27 Problem: 1) Suppose there are two commodities and that a consumer has utility function
u(x1, x2) = 2 ’x1 + In x2, where Ma 2 0 and x2 2 0. Suppose that the consumer’s endowment is e : (2, 0). Find the consumer's offer curve and draw, it accurately. Answer: The consumer’s maximization probtem is max(2J; + n(x ))
eri2 1 2 s.t.px +px =2p.
11 22 1 The Lagrangian is .L’ = 2 1/; + ln(x2) — ?t(p1x1 + p2x2) . The first order conditions are '1 =lp1andL=hp, ’x X 2
2 1
so that p1 J: = l = p x . Substituting this equation into the budget equation,
it 2 2
px +px =2p,
‘i1 2 2 1
weseethat
pixt + pt‘fi‘ : 2[01'
andhence
x + J—x‘=2.
1 t inspection shows that the solution to this equation is x = 1 . This equation defines the offer
1 %
2%
e:
E
g ‘ “ mm’m ‘ ‘wmxwawirwannwmmmnwzmwwm  armmww curve, which appears as 0 in the diagram below. If p1 = 0 or p2 = 0, there is no point in the budget set that maximizes utility. Problem: 2) Consider the Edgeworth box economy 24!: + ln(x2), 2n(x1) + ln(x ). 2 II eA = (6,0), uA(x1, x2) ll eB = (0,6), uB(x1, x2) a) Find a competitive equilibrium with the price of commodity'1 equal to 1. it) Find weights aA and aB such that aA + aB = 1 and the equilibrium allocation maximizes the welfare function a u (x , x ) + a u (x , x ) over alt feasible
A A A1 A2 B B B1 B2 allocations. Answer: 2a) Using the result of problem 1, x + ’x =6,
A1 A1 so that XM = 4. Hence feasibility implies that x81 = 6 — 4 = 2. Because person B’s utility function is CobbDougtas, we see immediately that x82 = 6/3 = 2. Hence feasibility implies that xA2 = 6 — 2 = 4. The budget equation of person A is x +px =6,
A1 2A2 that is, 4 + 4p2 = 6, and hence p2 = 112. The equilibrium is (XA’ xB. p) = ((4. 4). (2, 2), (1, 112)).
2b) The marginal utility of wealth of consumer A, AA, satisﬁes the equation __1.. = A p
X A 2
A2
That is,
1_ = 32.
4 2
so that AA = i. The marginal utility of wealth of consumer B, AB, satisfies the equation
2
1— = A p
X B 2
82
That is,
1_ = E,
2 2 so that AB = 1. The weights are proportional to (A: 22‘) = (2, 1). Since the weights must add
to 1, they are (aA, a3) = (2/3, 113). Problem: 3) State Walras’ Law and state conditions under which it is valid. Answer: Walras' Law says that for an economy, if x. 6 l;( p), for all i, and y. e n_(p), for all j,
I l I l  .J
then p. XXXH31) — 2y] 2 0, for all price vectors p, whether it be an equilibrium price
i=1 ' )«1 1 vector or not. This law applies if, for all E, the utility functions u_, are iocally nonsatiated. Problem: 4) Give an example of a Robinson Crusoe economy (u,_‘e, Y) with no optimal allocation. Answer: A Robinson Crusoe economy has an optimal allocation if the utility function is
continuous and the set of feasibie allocations is nonempty and is closed and bounded. Violation
of any of these conditions can give an exampie. For instance, suppose that e = (1, 0), Wmmmm’t‘m‘ﬁ‘ﬂWW—Wm l?
i
g u(x1, x2) = x2, andY= {(y , y2) E R2} y1 s 0 and y2 < —y1}. In this example, the inputoutput
1 possibility set is not closed, so that the set of feasibie allocations is not closed. The consumption
allocation (0, 1 — t/n) is feasible, for all positive integers n, and the utiiity of this allocation is 1 — iln. if x is any feasible consumption allocation, then u(x) < 1 and so
u(x) < u(0, 1 — 1m) = 1 —— 1/n, for some n. However, the consumption allocation (0, t) is not feasible, and so there is no optimal allocation. The visual picture is as foilows. Problem: 5) An economy ((u_, e)! , (Y)J ) is in equilibrium at (( x , _y), E), and all the l E l=1 j j=1
u_tility functions are iooally nonsatiated. Firm 1 produces nothing in the equilibrium. That is,
y1 = 0. A government obliges firm 1 to produce 116 Y , where y at 0. The economy then 1 ‘— reaches a new equilibrium (( x , y), E) . Could the new equilibrium allocation (3:, y ) Pareto dominate the original equilibrium aiiocation ( x , y )? Explain why or why not. Answer: Begausg the utility functions are locally nonsatiated, the original equilibrium
allocation ( x , y ) is Pareto optimal and therefore cannot be Pareto dominated by the allocation ( x , y ). Question: 6) Two consumer workers A and B can produce goods 1 and 2. The two consumers have
access to the same technology, but have different types of labor. Each consumer worker is
endowed with one unit of their own labor. With her or his labor, consumer worker A can
produce 3 units of good 1 per unit of labor time but oniy 1 unit of good 2 per unit of labor time.
That is, if consumer worker A devotes Li units of labor time to the production of good 1 and L2 units of labor time to the production of good 2, then she or he produces 3i.1 units of good 1 and
L2 units of good 2. Consumer B can produce 1 unit of good 1 per unit of labor time and 2 units of good 2. Both consumers have utility function u (x1, x ) = In x1 + In x2. Find a competitive 2
equilibrium for the economy consisting of the two consumer workers and in which the price of
good 1 is 1. (Hint: Draw the production possibility frontier for goods 1 and 2 and use the fact
that both consumers have the same Cobb—Douglas utility function.) Answer: The ‘output possibility sets, YA and YB, for persons A and B are as pictured below. The total inputoutput possibility set, YA + YB, is as in the next diagram. mmw«wmumrvmww nasmmnﬁmwmywm (1, 1/2) The vector (1, 3) is perpendicular to the upper face and the vector (1, 1/2) is perpendicular to the righthand face. Let x and x be the totai consumptions of goods 1 and 2, respectivety,
. 1 2 and let w be the total of the incomes of consumers A and B. Because the two consumers have the
same CobbDouglas utility function, we know that There are three possibilities, p2 = 3, p2 = 1 ori < p2< 3. If p2 = 3, then 2 2
y1 = x1: 3X2 = 3y2, which means that (y1, y2) lies to the right of the upper face of YA + Y3 and this is inconsistent with equilibrium. ff p = l, theny = x = ix = ly , which means that
2 2 1 1 2 2 2 2 (y1, V2) lies to the left of the righthand face of YA + Y3 and this is inconsistent with equilibrium. Therefore 1_ < p2< 3, which means that consumer A maximizes profit or
2 revenue by producing (ym, yAz) = (3, 0), and consumer B maximizes profit or revenue by producing (ym, y ) = (0, 2), andso(x1, x2) = (ym + yB1,yA2 + yBE) = (3,2). Hence 82 w‘a w 1. ”‘0
ll
x 1..
II
M Because consumer A’s production process has constant returns to scale, she or he earns no
profits and the wage for consumer A's labor is :E
E
g}?
E
E
§
3, it Mwwsxiivlﬁmwwmamw which is the same as consumer A’s weaith, since A has 1 unit of labor. Similarly the wage for
consumer B’s labor and B’s wealth are «mmammwmw;rmqquwmmcwMmmnummw m: Wm?» A1 2p 2 wwmnwwwmuw w ‘ mm Tugw v wmm‘mwrwmamwmmmw mwcmmrmw W
8:1. p
2 x=l
B22 In summary, the equilibrium is (xA, x3, yA, yB, WA, WE, (p1, p2)) = ((3/2, 1), (3/2, 1), (3, 0), (0, 2), 3, 3, (1, 3/2)). , ...
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