Economics 386-A1 Practice Assignment 1
S Landon Fall 2003
This assignment will not be graded. Answers will be made available on the Economics 386 web page:
http://www.arts.ualberta.ca/~econweb/landon/E38603.html. The goal of this assignment is to help you review the course material and to expose you to the types of questions that I may ask on exams. The assignment is not intended to be comprehensive (as not all the questions asked on the exams will necessarily relate directly to the questions on the assignments). In particular, the exams may include more questions that ask you to explain concepts or answers in words than does this assignment. The material covered by this assignment generally corresponds to the material in the first two sections of the reading list (Chapters 1-5 of Chiang). I am happy to discuss any of the questions and answers with you.
1. Consider the following set:
{
x y x y M x x M y}
Z = ( , )| + ≤ ,0≤ ≤ < ,0≤ . a) Illustrate set Z. (Be sure to label all intercepts.)
b) Briefly define a closed set (including definitions of all the terms you use). Is Z closed?
Explain.
2. Consider the set S:
[ ] [
1,2 ⊗ 3,4 .=
S
]
a) Set S represents points in which dimension of space (R, R2, R3, etc.)? Explain.
b) Use the method of "description" to provide an alternative way of describing the elements of set S.
c) Illustrate set S.
d) Is S compact? Explain.
3. Consider the set:
4. Consider the following two sets:
S1 = {-1,0,1}
S2 = {-2,2}.
Enumerate the set S = S1⊗S2.
5. Consider the set:
{
x y x y}
C = ( , )|0<α ≤ ,0< β ≤ . a) Illustrate this set.
b) Is this set closed? Explain.
c) Is this set bounded? Explain.
d) Define a convex set.
e) Is this set convex? Explain.
6. Consider the market for beef. The demand and supply functions in this market are:
BD = f-aPB + bPC + cY, BS = -d + ePB,
where BD is beef demand, PB is the price of beef, PC is the exogenous price of chicken, Y is exogenous income, and BS is beef supply. The parameters a, b, c, d, e and f are all positive and exogenous.
Solve for the equilibrium quantity of beef as a function of exogenous parameters and variables only.
7. Consider the following market for the Canadian dollar:
Supply of Canadian dollars: CS = α0 + α1ec+ α2Dr Demand for Canadian dollars: CD = β0 - β1ec - β2Dr
where ec = the price of Canadian dollars (in terms of US dollars), Dr = the difference between US and Canadian interest rates.
The parameters α0, α1, α2, β0, β1 and β2 are all positive and α0<β0.
Find an equation (without using derivatives or matrices) that can be used to determine the sign of the impact of Dr on the equilibrium price of Canadian dollars. What is the effect of a one unit fall in Dr on the equilibrium price of Canadian dollars?
8. Consider the following three equation model:
=0
−
−βx γy α
0
2 1
=
− − β z x
0
2 1
=
− − γ z y
where x, y and z are endogenous and α, β and γ are exogenous.
Find the reduced form solution equation for y.
9. Consider the following model of price (P) and quantity (Q):
Q2 + 5Q - P + 1 = 0, 2Q2 + P - 9 = 0.
Solve for the equilibrium quantity.
10. Consider the two equation model determining Y and C:
11. Consider the following two equation model. The endogenous variables are L and K while the exogenous variables are w and r and α is a parameter:
, w K
Lα−1 1/2 = α
r K
L −1/2 = 2
1 α
, where α∈(0,½).
Find the reduced form equation for K, simplify it as much as possible, and write it in a form so that none of the exponents has a negative value.
12. Consider the following two equation model:
, 0 0
1 − =
− K w
Lα β α
1− = ,
− r
K Lα β β
where 0<α, 0<β, and α+β<1.
The endogenous variables are L and K. All other variables and parameters are assumed to be exogenous and are positive.
Find the solution equation that describes the equilibrium value of L as a function of exogenous variables and parameters only. Show your work and simplify your answer as much as possible.
All the exponents included in your answer should be positive.
13. Consider the following system of three equations:
αxα-1yβ = λPx
βxαyβ-1 = λPy
Pxx + Pyy = I,
where x, y and λ are endogenous.
Find the reduced form equation for y.
14. Let QD = a-bP and QS = -c+dP where P is the per unit price, QD is the quantity demanded, QS is the quantity supplied and a, b, c and d are constant parameters.
a) If (b+d)=0, how many equilibrium prices and quantities exist? Explain.
b) What is the relationship between the demand and supply curves if (b+d)=0? Explain.
15. Consider the following two matrices:
.
⎥⎥
⎥
⎦
⎤
⎢⎢
⎢
⎣
⎡
= 0 3
3 0
2 1
A ⎥
⎦
⎢ ⎤
⎣
=⎡
1 0 1
0 2 B 2
a) Find the matrix products AB.
b) What is the trace of the matrix BA?
16. Consider the following two matrices:
.
⎥⎥
⎥
⎦
⎤
⎢⎢
⎢
⎣
⎡
=
2 0 5
4 3 0
3 2 1 A
⎥⎥
⎥
⎦
⎤
⎢⎢
⎢
⎣
⎡
=
1 1 1
0 6 5
1 4 2 B
a) Find AB.
b) Find BA.
c) Find Trace(BA) and Trace(AB) and show that they are equal.
d) Find B′ and A′. Find the matrix product B′A′.
e) Find (AB)′ and show that it is equal to B′A′.
17. The following vectors must be linearly dependent.
⎥⎦
⎢ ⎤
⎣
=⎡ 2
A 2 ⎥
⎦
⎢ ⎤
⎣
=⎡ 4
B 2 ⎥
⎦
⎢ ⎤
⎣
=⎡ 26 C 16
18. Consider the following matrix:
.
⎢⎢
⎢
⎣
⎡
= 0 5 1 A
⎥⎥
⎥
⎦
⎤ 3 2 1
Let Z=AA′.
Without undertaking any calculations, find the determinant of Z. Explain your answer. Your mark will depend primarily on your explanation.
19. Consider the following matrix:
M = I - X(X′X)-1X′, where X is an n×k matrix, n≠k.
a) Find the trace of M. Show your work.
b) Does M have an inverse? Explain.
c) Suppose Y = Xβ + u where β is a k×1 matrix.
Show that MY = Mu.
d) What does your answer to (c) imply about the relationship between M and X? Explain briefly.
20. Consider the following matrices:
ˆ uˆ X Y= β +
Y X' X) ˆ=(X' -1 β
where Y is n×1 and X is n×k.
Find X'uˆand give its dimension. Show your work.
21. Consider the matrix:
′Ω
′Ω
= X X X − X
W ( ) 1 ,
where X is n×k and Ω is a symmetric n×n matrix.
a) Is W symmetric? Show why or why not.
b) Is W idempotent? Show why or why not.
c) What is the trace of W? Show all the steps of your derivation.
22. Consider the matrix Z:
Z = C′AC
where A and C are both n×n symmetric idempotent matrices (neither of which is the identity matrix).
a) Is Z symmetric? Show why or why not.
b) Is Z idempotent? Show why or why not.
23. Let i be an n×1 matrix of ones (n>1) and let W=ii′.
a) Briefly define a symmetric matrix. Is W symmetric? Show why or why not without having to make reference to the individual elements of W.
b) Define an idempotent matrix. Is W idempotent? Show why or why not.
c) Does W have an inverse? Explain.
24. Let A, B and C be n×n matrices. Prove that (ABC)-1 = C-1B-1A-1.
25. Consider the matrix M=X(X′X)-1X′ where X is an n×k matrix with n≠k.
a) Is k less than or greater than n? Explain.
b) Does M-1 exist? Explain.
c) Let P=I-M where I is an n×n identity matrix. What do we know about the relationship between the rows of P and the columns of M? Explain.
d) Are the rows and columns of M orthogonal? Explain.
e) What is the determinant of M? Explain.
27. Consider the matrix:
.
⎥⎥
⎥
⎦
⎤
⎢⎢
⎢
⎣
⎡ −
=
7 1 3
4 4 4
1 2 1 A
a) Are the columns of A linearly independent? Show how you know.
b) Can the three column vectors that make up A be used to describe every other vector in R3? Explain briefly.
28. Consider the matrix:
⎥.
⎦
⎢ ⎤
⎣
⎡
−
−
−
−
= − −−21 1−− −1 −−−1 )
1 ( )
1 (
) 1 ( )
1 (
α α α
α
α α α
α
α α α
α
α α α
α
Y X Y
X
Y X Y
A X
Does the inverse of A exist?
29. Consider the following linear IS/LM/BP macro model:
y = α0 + α1r + α2e + g, (M/P) = β0 + β1y + β2r,
0 = γ0 + γ1e + γ2(r-r*).
The endogenous variables are e (exchange rate), r (interest rate) and y (output). Solve for the equilibrium value of y (if possible).
30. Consider the following simple linear macro model consisting of an IS, LM and aggregate supply (AS) curve:
IS: y = αy + βg, LM: M = P + γy – δr, AS: y = y*.
The endogenous variables are y, P and r.
a) If possible, solve for the equilibrium value of y as a function of exogenous variables and parameters only using the substitution method. Explain your answer.
c) How does your answer to (a) relate to linear dependence?
d) Find the determinant of the A matrix.
e) What do we know about the rank of A in general? (Do not find the actual value of the rank.)
31. Consider the following macro model:
M = P + β0 + β1y y = γL
w = θ0 + θ1L w = δ0 – δ1L
The endogenous variables are y, L, P and w. All other variables and parameters are exogenous.
The parameters are all positive and δ0>θ0. a) Set up the model in Az=d form.
b) Does the model yield a unique solution for all the endogenous variables? Show why or why not.
c) If possible, solve for the reduced form solution equation for y.
32. Consider the linear macro model:
y = α1y - α2r + g, M = β1y + β2P, W = W, y = γ1(P-W).
The endogenous variables are y (output), r (interest rate), W (nominal wage) and P (price). All other variables are exogenous.
33. Consider the following linear IS/LM/BP macro model:
IS: y = αy + βe, 0<α<1, LM: M = γy-δr,
BP: 0 = λe + µ(r-rf).
Assume that y, e and r are endogenous and M and rf are exogenous, where rf is the foreign rate of interest. The parameters α, β, γ, δ, λ and µ are positive. Note that µ represents a parameter, not a function.
a) Is there a unique solution for the endogenous variables of this model? Explain how you know.
b) Solve for the equilibrium value of e using Cramer's rule.
34. Consider the markets for beef and chicken. The demand and supply functions in these markets are:
BD = -a1PB + a2PC + a3Y, BS = -b1 + b2PB,
CD = c1PB + c3Y, CS = -d1 + d2PC,
where BD is beef demand, PB is the price of beef, PC is the price of chicken, Y is exogenous income, BS is beef supply, CD is the demand for chicken, and CS is the supply of chicken. The parameters a1, a2, a3, b1, b2, c1, c3, d1 and d2 are all positive and exogenous.
a) Is this model complete? Add any equations that are required to complete it.
b) Impose the equilibrium conditions and write the model in Az=d form.
c) Find the determinant of A.
d) What is it possible to say about the rank of A?
e) Solve for the equilibrium price of chickens using Cramer's rule. (You might also try using the substitution method and see if you get the same answer.)
