● You have 1 hour to complete the following 2 questions. Each question is worth 50 marks.

● We suggest you handwrite the answers to the test, then photograph your answers and then upload to a single document into Tabula.

● Accepted submissions: pdf or word document.

Q1.  Suppose the consumption set X is R0  and the consumer’s prefer- ences are defined by x < y if

m;n {x1 , x2 } - m;n {y1 , y2 } and

m;n {x1 , x2 } = m;n {y1 , y2 } =→ max {x1 , x2 } o max {y1 , y2 }

Note that a description of these preferences would be that they are similar to perfect complements but with the caveat that the consumer prefers not to have strictly more of one good than the other.  Consider the following 3 bundles:

a = ((, (〉    b = ((, ≥〉    c = (≥ , (〉

a) Argue that a ~ b and derive the strict preference relation and indiffer- ence relation over the set of bundles {a, b, c}. (10 marks)

b) Draw a diagram of the consumption set and partition it into 3 sets:

i) the set of bundles strictly worse than b =  ((, ≥〉, ii) the set of bundles indifferent to b = ((, ≥〉, iii) the set of bundles strictly better than b = ((, ≥〉. (10 marks)

c) Explain whether this preference relation satisfies i) strong monotonic- ity, ii) monotonicity, iii) local nonsatiation. (12 marks)

d) Are preferences continuous? Justify your answer. (12 marks)

e) Show that these preferences satisfy transitivity. (6 marks)

Solution:

a) Note that m;n {a1 , a2 } = m;n {b1 , b2 } and max {a1 , a2 } < max {b1 , b2 }. So by definition of the preference relation we have a < b but not b < a and so a  ~ b.  Similarl logic also gives a  ~ c and b > c.  Also j  > j for all j ∈ {a, b, c}.

b) Diagram looks very similar to that for perfect complements except for other bundles at which m;n {x1 , x2 } = (. Bundles along the m;n {x1 , x2 } = ( set which are closer to ((, (〉than b are preferred to b.  Bundles that are further away from ((, (〉than b are inferior to b.

2

A

Better

than b

Indifferent

   to b

Worse

than b

1

c) Strong monotonicity is violated - for example we don’t have (≥ , (〉~ ((, (〉- in fact the opposite.  Monotonicity is satisfied since if x1  and x2  are both higher then m;n {x1 , x2 } must be higher too.   Since monotonicity is

satisfied, LNS is also satisfied.

d) Preferences are not continuous. Students could obtain up to 8 marks

out of 10 by drawing a diagram and arguing that fixing some allocation x and moving in a straight line between two other allocations w and y where w is better and y is worse than x, we don’t need to pass through a point of indifference. For the full 10 marks, students should apply the mathematical definition and give an example of sequences that give a counterexample to the continuity definition. An example is

xn  = ╱( ↓  , ≥、         lnx = ((, ≥〉

yn  = ((, (〉       ln y = ((, (〉

For all points along the sequence xn  ~ yn . However in the limit we have the opposite - y ~ x.

e) To show this we assume

x = (x1 , x2〉< y = (y1 , y2〉       y = (y1 , y2〉< z = (z1 , z2〉

and we need to show x < z .  Students can get 2 marks for knowing what they need to show. Now for the proof: by assumption

m;n {x1 , x2 } - m;n {y1 , y2 } - m;n {z1 , z2 }

If at least one of the two above inequalities is strict then m;n {x1 , x2 }  > m;n {z1 , z2 } and so we have x ~ z . If we have

m;n {x1 , x2 } = m;n {y1 , y2 } = m;n {z1 , z2 }

then by assumption we also have

max {x1 , x2 } o max {y1 , y2 } o max {z1 , z2 }

and so in total we have

m;n {x1 , x2 } = m;n {z1 , z2 } and max {x1 , x2 } o max {z1 , z2 }

This is what is required for x < z .

Q2.  Consider a pure exchange economy with 2 goods and 2 consumers. Andy has initial endowment eA   =  (≥ , ≥〉and Bob has initial endowment eB   = (4, ≥〉.  Preferences of Andy and Bob respectively are represented by

utility functions uA , uB  : R0  - R where

uA (xA1, xA2〉= xA1 ↓ xA2           uB (xB1, xB2〉= ^xB1xB2

a) Calculate each person’s Marginal Rate of Substitution at the initial endowment. (10 marks)

b) Draw an Edgeworth box showing the initial endowment and indiffer- ence curves of both person going through the initial endowment. (10 marks)

c) Shade on your Edgeworth box a set of allocations that Pareto dominate the initial endowment.  Identify a specific allocation in the set and explain why it Pareto dominates the initial endowment. (12 marks)

d) Find the Pareto Set. Justify your answer. (12 marks)

e) Can a wasteful allocation (an allocation with excess supply of at least 1

good) ever Pareto dominate a non-wasteful allocation? Justify your answer. (6 marks)

Solution:

a) MRS for Andy is MRSA  = o( for any bundle.  For Bob the MRS is

found by:

MRSB  = o  = o 

at xB  = (4, 2〉    MRSB  = o

b) See below:  key points are curvature of indifference curves, slope of indifference curves through initial endowment and location of initial endow- ment in the Edgeowrth box.

A2

B

A

B2

c) See diagram above for lens of allocations Pareto dominating initial endowment.  Without doing extra calculations it is not clear whether the lens is enclosed just by the two indifference curves (as pictured above) or whether it is enclosed by the indifference curves and the southern edge of the Edgeworth box - so mark both as correct.  To find a specific allocation

that Pareto dominates the initial endowment, note that from diagram, if we

draw a straight line of slope ∈ ╱ o  , 1、going through the initial endowment then there will be some points on that line to the south east of the initial

endowment that Pareto dominate the initial endowment. Such an allocation

can be found as

xA  = (2 + 4e, 2 o 3e)        xB  = (4 o 4e, 2 + 3e)    e > 0 but small In this example e = 0.1 is small enough and we find

xA  = (2, 2)    xB  = (4, 2) =→ (uA , uB ) =╱4, ^8、

xA  = (2.4, 1.7)    xB  = (3.6, 2.3) =→ (uA , uB ) =╱4.1, ^8.28、

To get full marks students need to explain why they expect their allocation

to Pareto dominate the chosen allocation or calculate utilities to show Pareto

dominance as done here.

d) First note that only non-wasteful allocations can be Pareto efficient

and so we are looking for points in the Edgeworth box where both consume at the same point.   The Pareto efficient alocations in the interior of the Edgeworth box satisfy

B2

Students should justify why this is - perhaps by drawing Edgeworth box and showing indifference curves at points of tangency or arguing that preferences of both consumers are convex. Thus we have a diagonal line of points from

0B  downwards to xA  = (2, 0)    xB  = (4, 4). To continue to make Bob better off from this latter allocation, we need to give Bob more good 1 and so we trace out points between this and 0A .  To confirm this students could draw indifference curves at such allocations, show that - Andy’s are still of slope -1 but Bob’s are shallower and so there are no allocations in the upper level sets for both agents. So in total the Pareto Set is