Download Microeconomics 1 / M1: Final exam, Fall 2013

Microeconomics 1 / M1: Final exam, Fall 2013
No class notes, no calculators are allowed. Write brief answers. If you get stuck somewhere,
move on to another question. Good luck!
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Production (4 points)
p
A …rm’s production funtion is y = x. Prices are p for the output y, and w for the input x.
1. Are returns increasing, decreasing or constant?
2. Find the unconditional input demand function, the output supply function, and the
pro…t function.
2
Revealed Preferences (3 points)
Suppose that n = 2: You observe that a consumer whose income is y = 12 consumes x = (2; 5)
when prices are given by the vector p0 = (1; 2) and that he consumes x = (5; 2) when the
price vector is p00 = (2; 1) : Could these choices be generated by maximization of some strictly
quasi-concave utility function on the budget set? In other words, can there be a strictly
quasi-concave utility function u on R2+ such that such that these choices equal the Marshallian
demand (obtained from this utility function) at (p0 ; y) and (p00 ; y)? If your answer is positive,
provide an example of such a utility function. Otherwise, explain why it is impossible.
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General Equilibrium (6 points)
Consider an economy with two goods, two types of consumers, and no production. Let px
denote the price of the …rst good and py the price of the second. The consumers of type i (
i = 1; 2 ) have the following utility functions and endowments:
U1 = x1 y1 ;
U2 = x2 y22 ;
e1 = (ex1 ; ey1 );
e2 = (ex2 ; ey2 ):
1. Write the budget constraints for both consumers and derive their demand functions.
2. Assume that e1 = (15; 30) and e2 = (45; 30). Find the Walrasian Equilibrium of the
economy and the corresponding Walrasian Equilibrium allocations. (Hint: Set p = px =py .)
3. Are e1 = (15; 30) and e2 = (45; 30) the only the initial endowments that generate the
Walrasian equilibrium you found above? Provide a precise geometric description of the set
of all initial endowments that generate the same Walrasian Equilibrium. (Hint: Consider a
generic initial endowment e1 = (ex1 ; ey1 ) and e2 = (ex2 ; ey2 ). Then use the the equilibrium price
vector and the equilibrium allocations found in question 2 and combine them with budget
constraints.)
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4
Quasi-linear Preferences (9 points)
Let U (x1 ; :::; xn ) be a utility function de…ned on Rn+ such that there exists a continuously
di¤erentiable function u de…ned on Rn+ 1 ; such that
U (x1 ; x2 ; :::; xn ) = x1 + u (x2 ; :::; xn ) :
Preferences that can be represented by such a function are called quasi-linear with respect to
good 1: You can think of good 1 as being “money”.
In addition, assume that the preferences are strictly monotonic and strictly convex.
[For example, it could be U (x1 ; :::; xn ) = x1 + ( 2 x2 + ::: + n xn ) where k > 0 for all
k 2; and 0 < < 1 but you should not assume this particular functional form.]
The price of good 1 is normalized to 1; so that p = (1; p2 ; :::; pn ) : For any price p and any
income level y; the consumer problem at (p; y) is
max U (x) such that p x
x2Rn
+
y:
Note that y can be considered either as an income or as an endowment vector of the form
e = (y; 0; :::; 0) :
1. Show that if x and x0 are two consumption bundles such that U (x) = U (x0 ) ; then for
any positive real number ; we have
U (x1 +
; x2 ; :::; xn ) = U (x01 +
; x02 ; :::; x0n ) :
2. For this question, suppose that n = 2: What does the property established in question 1
imply for the map of indi¤erence curves of the utility function U in R2+ ? You may explain this
by drawing two distinct indi¤erence curves of the utility function U:
Let x (p; y) be the Marshallian demand function. For all questions except question 10,
always suppose that the income y is high enough for the inequality x1 (p; y) > 0 to
hold:
3. Show that for any two (high enough) income levels y and y 0 ; with y 6= y 0 ; and all k 2;
we have
xk (p2 ; y 0 ) = xk (p2 ; y 00 ) :
This means that for price vector p; the consumer’s consumption of goods 2; :::; n does not
depend on his precise income.level, as long as this level y is high enough:
4. Hold the price p at some …xed level: How does the income expansion path (or Engle
curve) looks like for high incomes? Assuming n = 2; draw a picture with x1 and x2 on the
axes, including two indi¤erence curves and the income expansion path. Both indi¤erence curves
should intersect the Engle curve at bundles corresponding to high incomes:
2
k
5. What can you say about the income e¤ect @x
(p; y) for each k
1 at a high income
@y
level y? In particular, do not forget the case k = 1. Is good 1 normal or inferior? Are the other
goods normal or inferior?
6. Can any of the goods be a Gi¤en good at a high income level?
7. For this question and question 8, suppose that n = 2; so that p = (1; p2 ). Consider two
prices p02 and p002 such that p02 < p002 and a high income level y: Let u0 = U (x ((1; p02 ) ; y)) and
u00 = U (x ((1; p002 ) ; y)) : Draw in the same diagram with p2 on the vertical axis and x2 on the
horizontal axis the graph of the Marshallian demand function p2 7! x2 ((1; p2 ) ; y 0 ) and of the
Hicksian demand functions p2 7! xh2 ((1; p2 ) ; u0 ) and p2 7! xh2 ((1; p2 ) ; u00 )on [p02 ; p002 ] :
8. Suppose that price p2 increases to price p02 ; while the (high) income is held …xed at y:
Compare EV (p2 ; p02 ; y) and CV (p2 ; p02 ; y) : What is the sign of these numbers? On a graph
similar to the one you have drawn for question 7; provide a geometric representation of these
numbers.
9. At (p; y) such that y > y (p) ; assuming that n 3; show that
@x2
@x3
(p; y) =
(p; y) :
@p3
@p2
10. How does the answer to questions 3 to 9 change if the income y is not assumed to be
high, so that x1 (p; y) = 0 may hold?
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