ECON2272 Intermediate Mathematics for Economists

Tutorial 4. Applications
Solve all questions. A selection of questions will be marked.
Question 1
General equilibrium analysis considers the interaction between markets. For example, an
increase in the price of silver may affect the market for lead because silver and lead are often
found in the same geological formations. Consider the following demand and supply functions
for the two metals:
Demand for Sliver:
0 1
d
S
S a a P = +
Supply of sliver:
0 1 2
S
S l
S b b P b P = + +
Demand for Lead:
0 1
d
l l
L P u α α = + +
Supply of lead:
0 1 2
S
S l
L P P β β β = + +
The term
l
u indicates an exogenous shock to the demand for lead.
a) What are the signs of
2
b and
1
β if silver and lead are joint products?
b) In a general equilibrium demand equals supply in each market.
0 1 0 1 2 S S l
a a P b b P b P + = + +
0 1 0 1 2 l l S l
P u P P α α β β β + + = + +
This can be written as:
( )
( ) ( )
1 1 2 0 0
1 1 2 0 0
S l
S l l
a b P b P b a
P P u β α β β α
− − = −
− + − = − −
Use Cramer’s rule to solve this system of equations for the price of silver.
c) Suppose the shift to alternative energy sources causes an exogenous increase in the demand
for batteries that increases the demand for lead ( 0
l
u ∆ > ).What is the effect on the price of
silver?
Question 2
Consider the multivariate linear regression model that is described in Turkington (2007, p.71-
72.)

y = X β + u
The ordinary least squares estimator (OLS) of the vector of coefficients β is
β
ˆ =(X’X) -1 X’y
a) Must the matrix X’X be square? Must X be square?
b) The matrix X’X is invertible if X has full column rank (i.e. the columns of X are linearly
independent). What is the minimum number of rows (observations) if the model includes k
regressors and a constant?
c) For a given X, the predicted values of y are:
ｙˆ
= X β
ˆ = X(X’X) -1 X’y
Show that the matrix N = X(X’X) -1 X’ is symmetric and idempotent. Is N invertible?
d) the prediction errors (residuals) are:
e = y -
ｙˆ
= y - X β
ˆ = y - X(X’X) -1 X’y = y – Ny = (I-N)y = My
Show that the matrix M is symmetric and idempotent.

Tutorial 5. Quadratic Forms and Eigenvalues
Solve all questions. A selection of questions will be marked.
Question 1
Express the following quadratic forms as x’Ax where A is a symmetric matrix.
(a)
2 2 2
1 2 3
5 3 4 x x x + −
(b)
2 2 2
1 2 3 1 2 1 3 2 3
2 9 3 4 6 2 x x x x x x x x x − − + − +
(c)
2 2
1 3 1 3
2 5 x x x x + −
Question 2
Consider the matrix
2 2 1
1
1 2 2
3
2 1 2
A
−  
 
=
 
 
−
 
.
Use the following two methods to show that A is an orthogonal matrix.
a) Check that the columns of A form an orthonormal set of vectors.
b) Check that A’ = A -1 .
Question 3
Consider the multivariate linear regression model that is described in Turkington (2007, p.71-
72.)
y = X β + u
The ordinary least squares estimator (OLS) of the vector of coefficients β is
β
ˆ =(X’X) -1 X’y
The prediction errors (residuals) are:
e = y -
ｙˆ
= y - X β
ˆ = y - X(X’X) -1 X’y = y – Ny = (I-N)y = My
Please represent the sum of squared prediction errors e’e as a quadratic form of the random
errors u.

Tutorial 6. Topics in Matrix Algebra, Comparative Static Analysis in a Linear Model
Solve all questions. Questions with an asterisk (*) will be marked
Question 1
a) Show that the determinant of an orthogonal matrix is either 1 or -1.
b) Is an orthogonal matrix invertible (nonsingular). Why or why not?
Question 2*
a) Use the property of determinants |AB|=|A||B| to show that the determinant of an
idempotent matrix is either 0 or 1.
b) Show that the only idempotent matrix that is invertible (nonsingular) is the identity matrix.
Question 3
a) What does the covariance matrix of a multivariate probability distribution show?
b) Which of the following matrices is possibly a covariance matrix?
5 3
0 7
1 9
A
−  
 
=
 
 
 
5 1 4
1 3 2
4 2 7
B
−  
 
=
 
 
−
 
2 5 4
5 2 3
4 3 0
C
−  
 
= −
 
 
 
2 4 5
9 7 2
3 0 3
D
 
 
= −
 
 
−
 
Hint: Use R to calculate the eigenvalues where needed.
c) Are all covariance matrices nonsingular (invertible)? Explain.
Question 4*
Theorem: Cholesky Factorization
Any symmetric positive definite matrix A can be decomposed uniquely as
A = LL’
Where L is a lower triangular matrix with positive entries on the main diagonal.
Find the Choleski factorization for
8 2
2 5
A
−  
= 

−
 
Question 5
The basic Keynesian macroeconomic model for a closed economy consists of the national
income identity which determines aggregate demand and a linear consumption function.
Y C I G
C Y α β
= + +
= +

Using matrix notation:
Ax b =
1 1
1 β
−  
 
−
 
Y
C
 
 
 
=
I G
α
+  
 
 
a) Show that the model has a unique solution.
b) Solve the model for vector of endogenous variables 𝑥𝑥, using matrix inversion.
c) Comparative static analysis: Suppose there is an investment boom in the mining sector that
increases 𝐼𝐼. Determine the response in the endogenous variables Y and C.
Question 6*
a)For the symmetric matrix
A =








２　３
６　２
Find the orthogonal matrix Q such that Q’AQ is diagonal.
Hint: The columns of the orthogonal matrix Q are the eigenvectors of A.
b) What is the definiteness of A and the sign of x’Ax for all vectors x.
Question 7
Consider the following symmetric matrix A:
A =










　０　　２　－２
　２　－５　　２
－４　　２　　０
Using the leading principal minors of A show that A is negative definite. What is the sign of
x’Ax for all x other than the null vector?

Tutorial 7. Differentiation
Solve all questions. Questions with an asterisk (*) will be marked.
Question 1
In each graph determine which function is the derivative of the other.
(A) (B)
Question 2
Verify the power rule of differentiation for y = x 2 .
Question 3*
Compute the slope of the following functions at the point x = 2. Apply the chain rule where
applicable.
(i) y =
2
5
x
+7 (ii) y = (x 3 - 7) 5
(iii) y = ln(x 2 - 2) (iv) y =
3
x
e
Question 4*
a) Compute the gradient vector for the following functions:
(i)
2 3 2 5
( , ) 2 5 f x y x y xy y = + −
g(x)
f(x)
g(x)
f(x)
Business School
10
(ii)
3
2 4
( , )
xy
f x y e
−
=
(iii)
2
( , ) ln( 3 5) f x y x xy = − +
b) Compute the Hessian matrix for (i).

Tutorial 8. Implicit Function Theorem, Comparative Static Analysis in a Nonlinear
Model
Solve all questions. Questions with an asterisk (*) will be marked.
Question 1*
The generalized Cobb-Douglas production function is:
Y=AK α L β 0 < α < 1, 0 < β < 1
a) Determine the marginal product of labor and its rate of change. Comment.
b) Is the Cobb-Douglas production function homogeneous? If so, of what degree?
c) Is the marginal product of labor a homogeneous function? Comment.
d) Is it possible that a production function has diminishing marginal products and at the
same time constant or even increasing returns to scale?
Question 2
Consider again the Cobb-Douglas production function given in question 2.
Show that the parameter β is the elasticity of output with respect to labor input. Derive the
elasticity both with and without logarithmic differentiation.
Question 3*
a) Does the nonlinear equation
x 2 + 2y 3 - 9z + 7 = 0
define an implicit function y = f(x, z) around the point y = 1, x = 3, z = 2？If so, find ∂y/∂x and
∂y/∂z and calculate these partial derivatives at the given point.
b) In what sense is the implicit function theorem about “compensatory “changes?
Question 4 (Chiang 2005,pp.205-207) *
Suppose that the demand for some good depends on its price and household income and the
supply depends only on price. Both functions may be nonlinear.
Demand： S d = D(P, I)
Supply： S s = S(P)
The price is the endogenous variable and income is exogenous, i.e. the price adjusts until the
market for the good is in equilibrium.
a) Market equilibrium requires:
F(P, I) = D(P, I) - S(P) = 0
What does the function F show? Under what condition does the implicit function P = f(I)
exist？Provide the necessary and sufficient condition.
b) Determine the effect of an exogenous increase in income on the price of the good. Assume

the good is normal. Illustrate with a graph.
Hint: Use the implicit function theorem to determine the comparative-static derivative
dP/dI.
Question 5
Derive the rate of technical substitution (RTS) for the Cobb-Douglas production.
Hint: Set the total differential equal to zero and calculate dK/dL.
Question 6
Are the following functions homogeneous? If so, of what degree?
(i)
2
( , ) 5 f x y x y =
(ii)
3
2 2
( , )
2 5
x
f x y
x y xy
=
+
(iii)
2
( , ) 7 f x y xy = +

Tutorial 9. Taylor Approximation, Convexity, Optimization
Solve all questions. Questions with an asterisk (*) will be marked.
Question 1*
At x 0 = 3, the value of the function y = x 4 is 81. Now increase x by 0.1 and estimate the function
value, using Taylor’s approximation. How big is the error for a 1 st order and 2 st order
approximation? Illustrate with a graph.
Question 2
a) Show that the difference in the logarithm of a variable is approximately equal to the
difference measured in percent.
Hint: Use a first order Taylor’s approximation of the logarithmic function.
b) Show that In(1 + r) ≈ r for small r. Illustrate with a graph.
Hint: Calculate a first order Taylor’s approximation around the point 1 on the x axis.
This approximation is often used in financial economics. For an interest rate of 3% the
logarithm of the gross return (1 + r) is approximately equal to the interest rate r: ln(1 +
0.03) ≈ 0.03.
Question 3*
a) The function f(x 1 , …, x n ) is strictly convex (concave) subset X of E n if and only if the Hessian
matrix of f is positive(negative) definite at every point in X. Use this theorem to check out
the strict convexity or strict concavity of the following function:
f(x, y, z) = 3e x + 5y 4 - lnz
b) For the following function find the critical points and classify them:
f(x, y) = y 3 - x 2 + 6x - 12y + 5

Tutorial 10. Optimization Problems
Solve all questions. Questions with an asterisk (*) will be marked.
Question 1*
Consider a consumer who buys x 1 kilos of meat and x 2 liters of petrol per month. The
consumer’s utility function is:
u(x 1 , x 2 ) = ln(x 1 )+ ln(x 2 )
The domain of this function is the positive quadrant,
2
+
R .
Suppose the price of petrol is $1.40 per liter and the price of meat is $28 per kilo. The total
amount that the consumer can spend on both goods is $560.
a) Formulate the consumer’s choice as a constrained optimization problem.
b) What are the first order conditions for this problem?
c) Show that the second order condition holds
d) How much petrol and meat will the consumer buy?
e) Show that the maximum you have obtained is unique (global maximum).
Question 2*
Suppose a firm that produces one good under perfect competition has a production function
Q(L, K) = L 1/2 + K 1/4
p is the price of output, w is the wage rate and r the cost of capital (rental).
a) Find an expression for total profit as a function of L and K.
b) Find the critical point of this function.
c) Show that this critical point is a unique global maximum.
Hint: Remember that the domain of the production function is defined for the first
quadrant
2
+
R , i.e. L and K must be greater than zero.
d) How does the demand for labor change with a small change in w?

Tutorial 11. Comparative Static Analysis in Constrained Optimization Problems,
Envelope Theorem
Solve all questions. Questions with an asterisk (*) will be marked.
Question 1*
Use the utility function u(x 1 , x 2 ) = logx 1 + logx 2 and the budget constraint p 1 x 1 + p 2 x 2 = I.
a) Find the demand functions. Also solve for the optimal Lagrangian multiplier, λ * .
b) Derive the indirect utility function (maximum value function).Differentiate the indirect
utility function with respect to income. Comment on result.
c) Use the envelope theorem to derive the result that you got in question (b).
d) Use the envelope theorem to derive the effect of an increase in p 1 on consumer utility.
Business School

Tutorial 12. Integration
Solve all questions. Questions will not be marked. A selection of questions will be discussed in
tutorial in Week 12.
Question 1
Find the indefinite integrals for:
i) ∫(7𝑥𝑥 3 + 5𝑥𝑥 − 4)d𝑥𝑥 ii) ∫𝑥𝑥 3 e 2𝑥𝑥
4 d𝑥𝑥
Question 2
Evaluate the following definite integrals:
i) ∫ (2𝑥𝑥 2
3
1
− 4𝑥𝑥 + 5)d𝑥𝑥 ii) ∫ ( 5
𝑥𝑥 )d𝑥𝑥
2
1
Question 3
a) Use integration by substitution:
i) ∫4(𝑥𝑥 4 + 2) 5 𝑥𝑥 3 d𝑥𝑥 ii) ∫3𝑥𝑥 2 e 𝑥𝑥
3 +5 d𝑥𝑥
b) Use integration by parts:
i) ∫𝑥𝑥 2 ln𝑥𝑥d𝑥𝑥 ii) ∫ln𝑥𝑥d𝑥𝑥
Question 4
Evaluate the following expression at z = 5:
d
d𝑧𝑧
� (𝑥𝑥 2 +3)d𝑧𝑧
z
2
What does this expression show? Illustrate with a graph.