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Advanced microeconomics: solutions of the TD
exam
Nicolas Gravel & Jo„o Varandas, Aix-Marseille Univ.
November 30th 2016, Marseille
Question 1 (4 points) Iara has preferences for champagne (good 1) and Paneer (good 2) deÖned by:
(x 1 ; x 2 ) (z 1 ; z 2 ) , min(x 1 ; x 2 ) min(z 1 ; z 2 ) and max(x 1 ; x 2 ) max(z 1 ; z 2 )
Are Iaraís preferences complete? Reáexive? Transitive? Locally Non- Satiable? Convex? Justify with care and draw the No-Worse-Than, No- Better-Than and the Indi§erence Set of a representative bundle.
Solution:
champagne
paneer
O
x
x
O
No worse than
45°
no better than (x1,x2)
(x1,x2)
x
x O
The "no-better than" and "no worse than" sets associated to an arbitrary bundle (x 1 ; x 2 ) are depicted in blue and red-pink (respectively), with the blue color continuing indeÖnitely to the north-east and the pink color continuing up to
the origin (I was lazy). The (boundary of the) red half rectangle corresponds to bundles whose maximal quantity of one of the two goods is the same than that of (x 1 ; x 2 ) while the (boundary of the) blue half rectangle describes the bundles for which the smallest of the two quantities of the two goods is the same than that of (x 1 ; x 2 ). The white zone describes the bundles that are not comparable to (x 1 ; x 2 ). As this zone is non-empty, this means that the binary relation is not complete. The indi§erence curve associated to (x 1 ; x 2 ) is made of (x 1 ; x 2 ) itself and its ordered permutation. These two bundles are represented by red circles on the pictures. Since (x 1 ; x 2 ) is comparable to itself. the binary relation is reáexive. It is also transitive. Indeed assume that (x 1 ; x 2 ) (z 1 ; z 2 ) and (z 1 ; z 2 ) (w 1 ; w 2 ). By deÖnition of , one has:
min(x 1 ; x 2 ) min(z 1 ; z 2 ) and max(x 1 ; x 2 ) max(z 1 ; z 2 ) (1)
and min(z 1 ; z 2 ) min(w 1 ; w 2 ) and max(z 1 ; z 2 ) max(w 1 ; w 2 ) (2)
combining (1) and (1) yields:
min(x 1 ; x 2 ) min(w 1 ; w 2 ) and max(x 1 ; x 2 ) max(w 1 ; w 2 )
and, therefore, (x 1 ; x 2 ) (w 1 ; w 2 ), as required by transitivity. Iaraís prefer- ences are locally non-satiable because, for any bundle (x 1 ; x 2 ), and for any " 2 R++, one can Önd a bundle (z 1 ; z 2 ) such that jzj xj j < " for j = 1; 2 and (z 1 ; z 2 ) (x 1 ; x 2 ). Indeed, just pick a strictly positive real number < " and deÖne (z 1 ; z 2 ) by
z 1 = x 1 + z 2 = x 2 +
Iaraís preferences are not convex because (x 2 ; x 1 ) % (x 1 ; x 2 ) and (x 1 ; x 2 ) % (x 1 ; x 2 ) but the bundle ((x 1 + x 2 )= 2 ; (x 1 + x 2 )=2) (represented by the red circle on the picture) is not weakly preferred to (x 1 + x 2 ).
Question 2 (4 points) A household consumes three goods that are sold on perfectly competitive markets. One has observed the behavior of this households under three di§erent circumstances. The data that describes this householdís be- havior are described in the following table (where xj and pj denote respectively the quantity and price of good j for j = 1; 2 ; 3 )
observation p 1 p 2 p 3 x 1 x 2 x 3 1 1 2 3 3 2 1 2 3 1 2 2 1 2 3 3 4 8 2 3 1 (a) Is the consumption behavior described by these data satisfying the Weak Axiom of Revealed Preference (WARP)? Justify with care (2 points).
where xi denote the (non-negative) quantity of good i (for i = 1; 2 ). We can say that.
(a) Mihir will always choose to spend all his wealth on good 2. (b) Mihir will always choose to consume strictly positive quantities of the two goods no matter what are the positive prices of the two goods if the individual has a strictly positive wealth. (c) Mihir will spend all of his wealth on good 2 if the price of good 2 is lower than that of good 2. (d) If the prices of good 1 and 2 are respectively 1 and 4 and Mihir has a wealth of 2, then Mihirís Marshallian demand for good 1 will be insensitive to small change in his wealth (e) None of the above.
Question 4: A Örm produces one output out of two inputs. It does so with a technology described by the production function f (x 1 ; x 2 ) = (x^11 = 2 + x^12 = 2 )^1 =^4. We can conclude that:
(a) This Örm may be unable to maximize its proÖt at given prices. (b) ProÖt maximization at given prices is always well-deÖned but the proÖt function resulting from this program may sometimes be discontinuous.
(c) The proÖt function of the Örm is (py ; p 1 ; p 2 ) =
p^3 y=^2 3(p 1 +p 2 )^1 =^2. (d) The proÖt function of the Örm is (py ; p 1 ; p 2 ) = p^2 y 3(p 1 +p 2 ). (e) None of the above.
Question 5 Which of the following statements is true? (a) The function (py ; p 1 ; p 2 ) = 15 p^3 y=^2 (p 1 +p 2 )^1 =^2 is a proÖt function of a Örm.. (b) The function V (p 1 ; p 2 ; W ) = W^
1 = 2 p^11 = 2 +p^12 =^2 is the indirect utility function of a household. (c) The function C(p 1 ; p 2 ; y) = p^11 = 2 p^12 = 2 y^2 p 2 is the cost function of a Örm. (d) The function C(p 1 ; p 2 ; y) = p^11 = 2 p^12 = 2 y^2 + p^32 = 4 is the cost function of a Örm. (e) None of the above.
Question 6 If a Örm who produces one output with two inputs has a tech- nology represented by the production function f (x 1 ; x 2 ) = x^11 = 2 + x^12 = 2 , then, we can say that:
(a) ProÖt maximization may not be possible at certain prices. (b) The proÖt function of the Örm is (py ; p 1 ; p 2 ) =
p^2 y 4 [^
1 p 1 +^
1 p 2 ] (c) The proÖt function of the Örm is (py ; p 1 ; p 2 ) = p^3 y 4 p 1 p 2. (d) The proÖt function of the Örm is (py ; p 1 ; p 2 ) = p^2 y 4 p^11 = 2 p^12 =^2 (e) None of the above.
Question 7 Which of the following statements is true? (a) A reáexive, complete and transitive preference on a convex and closed subset of Rl + can always be numerically measured by a utility function. (b) If a technology is represented by a production set that satisÖes closed- ness, irreversibility, impossibility of free production, possibility of inaction, monotonicity and convexity, then proÖt maximization at given prices always has a solution. (c) The function C(p 1 ; p 2 ; y) = p^11 = 2 p^12 = 2 y^2 + p 2 is the cost function of a Örm. (d) The function C(p 1 ; p 2 ; y) = p^11 = 2 p^12 = 2 y^2 + p^32 = 4 is the cost function of a Örm. (e) None of the above.
Question 8 Which of the following statements is true? (a) If returns to scale are non-increasing and the technology satisÖes the possibility of inaction, then the production set can not be convex. (b) If a production set is convex, it satisÖes the property of irreversibility. (c) If a Örm produces only one output out of, possibly, l 1 inputs and has input requirement sets V (z) that are convex for all output quantity z, then the technology has a convex production set. (d) When the wealth of the household is the value, at market prices, of the householdís initial endowments of the goods and of the shares of the Örmís proÖts, it is possible that the Marshallian demand of a normal good be increasing with respect to that good price if the considered good is in net supply by the household. (e) None of the above.
