###### 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 ...

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) = ...

(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. ...

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, ...

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. ...

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). ...

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). ...

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 (∗) ...

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) ...

(a) Because $(x − 2)^{2} ≥ 0$ for all...