Essential Mathematics for Economic Analysis 5th. Edition by Knut Sydsaeter; Peter Hammond; Arne Strom; Andrés Carvajal | 9781292074658 | 1292074655

Engineering Economy, Mathematics

Essential Mathematics for Economic Analysis

366 SOLVED PROBLEMS

Question: 12.11.1

Consider the following system of two linear equations in four variables: 5u + 5v = 2x − 3y 2u + 4v = 3x − 2y It has two degrees of freedom. In fact, it defines u and v as functions of x and y. Differentiate the system and then find the differentials du and dv expressed in terms of dx and dy. Derive ...

Verified Answer:

For both equations, take the differential of each ...

Question: 12.11.2

Consider the system of two nonlinear equations: u² + v = xy; uv = −x² + y² (a) What has the counting rule to say about this system? (b) Find the differentials of u and v expressed in terms of dx and dy. What are the partial derivatives of u and v w.r.t. x and y? (c) The point P = (x, y, u, v) = ...

Verified Answer:

(a) There are four variables and two equations, so...

Question: 12.10.2

Consider the alternative macroeconomic model (i) Y = C + I + G (ii) C = f (Y − T) (iii) G = G whose variables have the same interpretations as in the previous example. Here the level of public expenditure is a constant,G. Determine the number of degrees of freedom in the model. ...

Verified Answer:

There are now three equations in the five variable...

Question: 11.8.2

The demand D1 for potatoes in the USA, for the period 1927 to 1941, was estimated to be D1 = Ap^−0.28m^0.34, where p is the price of potatoes and m is mean income. The demand for apples was estimated to be D2 = Bq^−1.27m^1.32, where q is the price of apples. Find the price elasticities of demand, ...

Verified Answer:

According to part (a) of Example 11.8.1,

El...

Question: 11.8.3

Suppose D = Ax^a11 x^a22· · · x^ann is defined for all x1 > 0, x2 > 0, . . . , xn > 0, where A > 0 and a1, a2, . . . , an are constants. Find the elasticity of D w.r.t. xi, for i = 1, . . . , n. ...

Verified Answer:

Because all the factors except

x^{a_{i}}_{i...

Question: 11.2.5

If f (x, y) = x³e^y², find the first- and second-order partial derivatives at the point (x, y) = (1, 0). ...

Verified Answer:

To find

f^{\prime}_{1}(x, y),

we di...

Question: 16.7.2

Find the inverse of A =(1 3 3 1 3 4 1 4 3). ...

Verified Answer:

First, write down the 3 ×6 matrix

(A:I)=\le...

Question: 14.1.3

A consumer who has Cobb–Douglas utility function u(x, y) = Ax^ay^b faces the budget constraint px + qy = m, where A, a, b, p, q, and m are all positive constants. Find the only solution candidate to the consumer demand problem max Ax^ay^b s.t. px + qy = m (∗) ...

Verified Answer:

The Lagrangian is

{\mathcal{L}}(x,y)=A x^{a...

Question: 8.3.3

A monopolist is faced with the inverse demand function P(Q) denoting the price when output is Q. The monopolist has a constant average cost k per unit produced. (a) Find the profit function π(Q), and prove that the first-order condition for maximal profit at Q^∗> 0 is P(Q^∗) + Q^∗P'(Q^∗) = k (∗) (b) ...

Verified Answer:

(a) The profit function is π(Q) = QP(Q) − kQ, so [...

Question: 8.1.1

Find possible maximum and minimum points for: (a) f (x) = 3 − (x − 2)² (b) g(x) =√x − 5 − 100, for x ≥ 5 ...

Verified Answer:

(a) Because

(x − 2)^{2} ≥ 0

for all...