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Unformatted text preview: Fall 2009 Truman Bewley Economics 350a
Answers to Homework #1
(Due Thursday, September 17) Problem: 1) Consider the Edgeworth box economy where the endowment of consumer A is (1, O)
and the endowment of consumer 8 is (0, i). For each of the following three cases, find and
sketch the set of Pareto optimal aliocations and the utility possibility set and find the aliocat‘ions
that maximize the sum of the utilities of the two consumers. By “sketch the set of Pareto
optimai allocations,” I mean find the coordinates of a few points on the utility possibility
frontier and fill in the remaining part of the curve. In maximizing the sum of the utilities in
part i), use the symmetry of the problem. In order to see where the sum of the‘ utilities is
maximized, it is important to have a fairly accurate sketch of the utiiity possibility set. {xx andu(x,x)= xx.
12 312 12 h) u (x , x) = x1’5x1’3andu (x , x) = x1’3x1’6.
A 1 2 1 2 B 1 2 1 2 ll a) uA( x1, x2) '2 2
c u x x =xxandu x x =xx.
) A(1’ 2) 12 8(1' 2) 12 Answer: a) Because uA and u are homothetic, the set of Pareto optimal allocations is the
B diagonai of the Edgeworth box, as shown beiow. X
A2 Set of
Pareto Optimal
Allocations mwmm..mm=umammi.ammsmxmsmmmmmmmmwmmw‘mwWesvwmmx‘xwfmmﬁiik mmmaximwmmmwwm mwmxﬁuyrmﬂm‘xlmvwwmww4m\v;\MXVﬁ'ATrwiwmwmmw‘dYWM%%WWNM\W1WQWWAW0“!“me Because uA and uB are téneariy homogeneous, the utiiity possibility frontier is a straight line as sketched below. I I I I I I I I I I I I . Utility Possibility Frontier The allocations that maximize the sum of utility is the entire set of Pareto optimal allocations, namely,{(x,x)x =x ,x =x =1—x }.
A B A1 A2 B1 32 A1 Answer: b) in order to find the set of Pareto optimal altocations, we must sotve the equation au(x ,x )lax ' au(x ,x )lax
A A1 A2 A1 =MRS :MRS = 3 Bi B2 a1_ au(x ,x )lax A B au(x ,x )lax
A A1 A2 A2 13 81 B2 B2 Since the total endowment of each commodity is one, this equation becomes au(x ,x )lax au(1x ,1—x )lax
A A! A2 1;: B A1 A2 1,
au(x ,x )lax au(1—x ,1—x Max
A A1 A2 2 B A1 A2 2 which in turn becomes mumamaura;mama1mmAvMk7»!mmMWNWth>WMvMWWWWNW‘WWWWWW‘VWWWW?“W'W4WHW‘7WWWMWIWlwﬁﬁqwmhwmwﬁww sartgamewwmmmwww :‘ﬁm’nﬁﬂimwﬁ mzmmmwww warm» mew/uwmmwtewﬂ IsmVmymItxmas/tummmmmmwww.” x113 (1—K )11'6
A2 i A2 l—
6 x5”6 3 (1—K )2,3
A1 = A1 , E.
1I6 _ 1/3 t
i XA1 i (1 XA1) E
3 x2’3 6 (1—x )5’5 E
A2 A2 E
which implies that E
x 1 — x E
_A2 = 4 _A2 , E
x 1 —x ‘1
A1 A1
and hence that
x —x x =4): —4x x .
A2 A1 A2 A1 A1 A2 Solving this equatioh for x , we see that
A2 4x
x =______L_. '12 1+3x
A1 The set of Pareto optimal allocations looks approximately as in the drawing below. mmmwwmm.la«”mayrmxxml:MmmmWwvwIﬁmﬁWwWanmeWWWMWWW? Set of
Pareto
Optimai
Ailocations The utility possibility frontier looks approximately as follows. .......... Utility Possibility Frontier  (0.727, 0.727) The point (0.727, 727) on the frontier may be found by letting xA1 = 1/3. Then xA2 = 2/3, so 1 3
that x = 2/3 and x = 1/3. At these values for the allocation, V = v = 2 I a 0.727.
Bt B2 A B 31I2 The concavity of the utitity possibility frontier is a consequence of the concavity of the utility
functions. The symmetry of the utility possibility set with respect to the two axes foliows from
the symmetry of the problem. If the allocation ((X,X),(X,X))=((X ,X),(X ,X)) A1 A2 B1 82 A1 A2 B1 B2 is feasible and achieves utility levels (VA, VB) = ( v A, 73)’ then the allocation ((XA1axA2)J(XBIX)):((X :X ),(X ,X )) f 132 32 BE A2 A1 is feasible and achieves the utility levels (VA, VB) = (7, 7). From the symmetry and
B convexity of the utility possibility set, it foliows that the point on the frontier where the sum of E
E
i
E
g.
i WWW—w w 1 3 .
the utilities is maximized is the point where vA = v = 2 I . The corresponding allocation is
B 1m
3 ((x ,x ),(x ,x )) =((1/3,2/3),(2l3,1/3)). A1 A2 B1 BE Answer: c) The utility functions of part b are the sixth power of the utility functions of part a.
Since the sixth power is an increasing function, the utility functions of part c are monotone
transformations of the corresponding utility functionsoi part b. Therefore the indifference
curves and hence the set of Pareto optima are the same in the two parts. The utility possibiiity
frontier, however, differs, because it depends on the shape of the utility functions, not just on
the indifference curves. ln attempting to plot the utiiity possibility frontier, notice that the
points (VA, VB) = (t, 0) and (VA, VB) = (0, 1) belong to the frontier. If we calculate the utility levels at the allocation ((XA1,XA2),(X , )) =((1l3,2/3),(2/3,1/3)), we find that
B x
1 82
VA=VB=4/9. Therefore the utility possibility frontier is not concave and the utility
possibility set is not convex. if we calculate more points, we see that the utility possibility frontier is convex. This is so because the utility functions are convex. The utility possibility
set looks approximately as follows. I : I Z I I I I I Utility Possibility Frontier ........................ E
i;
g
g;
E E
E
E
E
E
E
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E?
E
E .
E
E
E
E
E
ii
i
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i E
E
E
i
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i
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E
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g. The allocations that maximize the sum of the utilities are the ones corresponding to the points
(1, 0) and (0, 1) on the utility possibiiity frontier. That is, they are the allocations ((X IX ),(X ,X )) =((1,1),(0,0)) A1 A2 B1 32 and ((X ,x ),(X :X )) =((0,0).(1,1)') A1 A2 B1 B2 Problem: 2) Find the optimum allocation and draw the feasible set for each of the following
three Robinson Crusoe economies, where L is the input of labor, l! is the consumption of ieisure, x is the consumption of food, and y is the production of food. Indicate Crusoe's indifference curves and the optimum on the drawing. in a commodity vector, the first
component is laborleisure time and the second is food. a) y = f(L) = 2L, e = (1,0), u(l?, x) = ifmxm.
b) y = f(L) = L, e = (1, O), u(t’, x) = min(ﬂ, 2x). my=nu=3ﬁﬂe=m,m,m£m=£+2x Answer: a) The first order condition for an optimum is au(x, I)
m _dﬂU
au(x, I) — dL '
6X which implies that 2 x213
:9, 31"3 2 £13
3 )(1:3 =2, so that . i=2.
t' Feasibility implies that ﬂ=x=2L=a1—a=2—m View»;maxummmimmmwswwmwxwmmmmmwmmw «mum‘s any“ 3
a
3 and hence The corresponding diagram is as follows. The feasibility set is indicated by the letter 3. x,y Optimum 0 1 LI Answer: b) The diagram looks as follows. The optimum is at the point (1/3, 2/3), as is evident
from the diagram. The indifference curves are the each made up of two half lines meeting at
right angles at their endpoints. E
i;
ii
i
3;;
it
E
i
g
i
g
E
:E.
g wwwwtwwmmsewm uﬁnﬁﬂWwWM‘A «vywmMmedurum:vaWWWMWmimiwmm‘mmmmmﬂm:mmﬁrwm e» Answer: c) If we apply the equation au(x, f)
653 = am.) ,
au(x, f) dE—
‘ax we find that 1 i 1 2 '— r N
'— which is infeasible. If L = 1, so that if = O and x = y = 3, then df(_) =_3_>l= au(0,3)/alf’
dl. 2 2 au(0,3)/ax so that we have a corner solution to the maximization problem. The optimum occurs at this
point. The diagram is as follows. The parallel straight lines are indifference curves. , u A“mummy;kwtwmmwwwmmwmwmmwm‘ “Wmmmwww: gsgzﬂéxﬁéiiﬁaiﬁﬁxEggésﬁiiﬁi ...
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 DonaldBrown

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