Arne Strøm

Elasticities. Elasticities of substitution

• When f is a utility function, and x and y are goods, R yx is called the marginal rate of substitution (abbreviated MRS).

Probability and statistics

f is the discrete/continuous probability density function for the random (or stochastic) variable X. F is the cumulative discrete/continuous distribution function. In the continuous case, P(X = x) = 0. Expectation of a random variable X with discrete/continuous probability density function f. μ = E[X] is called the mean. Expectation of a function g of a random variable X with discrete/continuous probability density function f.

Method of least squares

The method of least squares with k explanatory variables. X is often called the design matrix. Definition of the coefficient of determination and the multiple correlation coefficient. 100R2 is percentage of explained variation in y.

Topics from trade theory

Standard neoclassical trade model (2 × 2 factor model ). Two factors of production, K and L, that are mobile between two output producing sectors A and B. Production functions are neoclassical (i.e. the production set is closed, convex, contains zero, has free disposal, and its intersection with the positive orthant is empty) and exhibit constant returns to scale. The economy has incomplete specialization when both goods are produced.

Equations. Functions of one variable. Complex numbers

Let k be the number of changes of sign in the sequence of coefficients a n , an −1, … , a1, a0 in (2.8). The number of positive real roots of P(x) = 0, counting the multiplicities of the roots, is k or k minus a positive even number. If k = 1, the equation has exactly one positive real root.

Set Theory. Relations. Functions

Let R be a relation from A to B and S a relation from B to C. Then we define the composition S ○ R of R and S as the set of all (a, c) in A × C such that there is an element b in B with aRb and bSc. S ○ R is a relation from A to C.

Series. Taylor’s formula

If a series is absolutely convergent, then the sum is independent of the order in which terms are summed. A conditionally convergent series can be made to converge to any number (or even diverge) by suitable rearranging the order of the terms.

Eigenvalues. Quadratic forms

Eigenvalues and eigenvectors are also called characteristic rootsand characteristic vectors. λ and c may be complex even if A is real. The eigenvalue polynomial (the characteristic polynomial) of A = (a ij ) n×n . I is the unit matrix of order n.

Vectors in ℝ n . Abstract spaces

Definition of a linear combination of vectors. Definition of linear dependence and independence. A characterization of linear independence for m vectors in ℝ n . (See (19.23) for the definition of rank.) A characterization of linear independence for n vectors in ℝ n . (A special case of (18.4).)

Limits. Continuity. Differentiation (one variable)

Any function built from continuous functions by additions, subtractions, multiplications, divisions, and compositions, is continuous where defined.