mathematics of econmics

⊃

"includes"

∈

"is an element of"

⊂

"is included in"

Properties of a Production Set Y

(1) 0 belongs to Y (possibility of inaction) (2) Y ∩ {R(l)+} = {0} (no free lunch) (3) -{R(l)+} is a subset of Y (free disposal) (4) Y ∩ {-Y} = {0} (irreversibility) (5) Y is closed (6) Y is convex

Point Income-Elasticity of Demand

(dQdx/dI)(I/Qdx)

Point Cross Price-Elasticity of Demand

(dQdx/dPw)(Pw/Qdx)

decreasing returns to scale

(diseconomies) an increase in a firms scale of production leads to higher costs per unit produced

increasing returns to scale

(economies of scale) an increase in a firms scale of production leads to lower costs per unit produced

profit

(pie) = TR - TC

null set

(Ø or {}) Empty set, considered a subset of every set

The Power Rule of differentiation

*see notes

Convexity/concavity in multivariate functions

- A function f is convex at x if and only if its Hessian matrix is positive semidefinite at x - A function f is strictly convex at x if and only if its Hessian matrix is positive definite at x - A function f is concave at x if and only if its Hessian matrix is negative semidefinite at x - A function f is strictly concave at x if and only if its Hessian matrix is negative definite at x

[1st Three of Six] Steps of Risk Filtering, Ranking, and Mgmt.

- Identify risk scenarios - Scenario filtering based on scope, temporal domain, and level of decision making - P-I matrix filtering and ranking

[Last Three of Six] Steps of Risk Filtering, Ranking, and Mgmt.

- Multi-criteria evaluation, e.g., multi-objective, sensitivity analysis - Quantitative ranking - Risk management: response planning, report, implementation, review

Equation for MRS

-MUx1/MUx2

Decreasing returns to scale

-Q(λK,λL)<λQ(K,L) -Degree of homogeneity is less than 1 *If input is 2, output will be 1

Constant return to scale

-Q(λK,λL)=λQ(K,L) -Degree of homogeneity is 1 *If input is 2, output will be 2

Increasing returns to scale

-Q(λK,λL)>λQ(K,L) -Degree of homogeneity is greater than 1 *If input is 2, output will be 3

Normal Good: Necessity Value of Ei

0.0 < Ei < 1.0

What is the antiderivative of e^2x?

0.5e^2x

Rules for matrix multiplication

1. (AB)C = A(BC) 2. A(B + C) = AB + AC 3. (A + B)C = AC + BC 4. (αA)B = A(αB) = αAB Also: • AB ≠ BA • AB = 0 does not imply that either A or B = 0 • AB = AC and A ≠ 0 does not imply that B = C

Elements of a function

1. Domain - representing X 2. Target space - representing Y 3. Rule for mapping

Solving Differential Equations

1. Find Homogenous Solution 2. Find Particular Solution (Steady State) 3. Find the General Solution 4. Solve Initial Value problem

Give the steps for finding the max profit for a firm that produces into two different markets

1. Find TR=Tr1+Tr2 2. Find profit = TR-TC 3. Partially derive profit with respect to each good and solve 4. Find second order derivatives and prove it is a max using f'(x1)f'(x2) -f'(x1x2) > 0

Verifying critical points for functions of several variables

1. Find all four second partial derivatives 2. Find the Hessian determinant (D2) 3. Use the second partial derivative with respect to x or respect to y as D1 4. Substitute the critical point(s) into D1 and D2 5. Observe if D2 is greater than 0, if it is then observe D1's value *D1>0: minimum D1<0: maximum

Properties of a function

1. Function has to be defined for every element in the domain 2. Each element in the domain is mapped to only one

What is the substitution method and how can it be used to find maximum output?

1. Given Q(L,K) 2. Find PlL + PkK = C Where Pl=price of labour, Pk=price of capital, C = total input cost 3. Express K as a function of L 4. Partially derive with respect to L and prove it is a maximum Same for utility

Give the steps for finding max profit for a monopolist producing 2 goods

1. Identity the constraint 2. Find profit equation 3. Form Lagrangian function (pi - constraint x Llamda) 4. Form Lagrangian equations by deriving with respect to each variable 5. Find Q1, Q2, and Llamda 6. Calculate optimal profit

Obtaining critical points for functions of several variables

1. Take the first partial derivative with respect to x and with respect to y 2. Set each result equal to 0 (FOC) to obtain the critical point(s) *By setting equal to 0, one finds the point(s) where the minimum or maximum is the same in both x and y directions

What are the 6 rules of differentiation for a function with one independent variable such as y= f(x)

1. The Constant Rule 2. The Power Rule 3. The Addition Rule 4. The Product Rule 5. The Quotient Rule 6. The Chain Rule (Function of a Function Rule

What are two concepts that the derivative can represent?

1. The slope of a function 2. The "marginal" function of a "total" function

The derivative of log base b of x

1/ (lnb*x)

The derivative of ln(x)

1/x

Number of subsets of a set with n elements

2^n

Opt. Profit (60%) Leisure (20%) Health (20%) Utility A 8 2 3 B 5 5 5 C 5 6 2 What're the Values and Which one Should He choose?

5.6 4.2 5.4

Risk Loving 𝑢(𝑝𝑥+(1−𝑝)𝑦) O? 𝑝𝑢(𝑥)+(1−𝑝)𝑢(𝑦)

<

Risk Neutral 𝑢(𝑝𝑥+(1−𝑝)𝑦) O? 𝑝𝑢(𝑥)+(1−𝑝)𝑢(𝑦)

=

Elasticity of supply or demand

= (dQ/dP) * (P/Q)

Marginal product of capital

=dQ/dK

Marginal product of labour

=dQ/dL

Risk Averse 𝑢(𝑝𝑥+(1−𝑝)𝑦) O? 𝑝𝑢(𝑥)+(1−𝑝)𝑢(𝑦)

>

Risk Premium if Risk Averse? RP>0, RP<0, or RP=0

>

Set (finite and infinite) (disjoint)

A collection of distinct objects. Membership of an element in a set is indicated by the symbol € (epsilon). { (is contained in) and ) } (includes) the symbol relates a subset to a set

Parameter

A constant value that is undetermined and, therefore, represented as a variable (usually lowercase a, b, c)

Homothetic function

A function is homothetic if they are indifferent between x and y whilst maintaining indifferent between tx and ty. f(x) = f(y) --> f(tx) = f(ty)

Homogeneous function

A function with multiplicative scaling behavior *if all its arguments are multiplied by a factor, then its value is multiplied by some power of this factor

Hessian determinant

A matrix composed of the four second partial derivatives used to find D2 First row: f(subscript xx) f(subscript xy) Second row: f(subscript yx) f(subscript yy) Multiply f(subscript xx) by f(subscript yy) and subtract Multiply f(subscript xy) by f(subscript yx) from it

Rational Number

A number that can be written as a fraction, including all integers and fractions. All rational numbers can be written as either terminating or repeating decimals.

Irrational Number

A number which cannot be represented as a fraction and whose decimal form never repeats or terminates (i.e. pi)

Matrix

A rectangular array of numbers considered as one mathematical object.

Behavioral Equation

A relationship that specifies the manner in which a variable behaves in response to changes in other variables

Bound and unbound sets

A set is bounded if the whole set is contained within a sufficiently large n-dimensional ball with a finite radius A set is unbounded if no ball with a finite radius can enclose it

Compact sets

A set is compact if it is closed and bounded.

Rule for convex set

A set is convex if any weighted average of any two points lies on the set. xa = ax1 + (1-a)x2 is in domain

Open and closed sets

A set is open if it consists only of interior points A set is closed if it contains all of its boundary points

Endogenous Variable

A value that depends on other variables in a model and is thus explained within the model

Exogenous Variable

A value that is wholly causally independent from other variables in the system (usually symbolized with a "₀" as in "P₀" instead of "P")

average cost

AC = TC/Q or FC/Q + VC

average variable cost

AVC = TVC/q Rising cost intersects average variable cost at the minimum point of AVC when MC < AVC, AVC is declining when MC > AVC, AVC is increasing

Real Numbers

All rational and irrational numbers (commonly symbolized as R)

Mathematical model

An approximate description or representation of a real-life situation expressed in terms of mathematical symbols (constants, variables, equations, inequalities, etc)

Law of diminishing marginal productivity

An economic principle that states that while increasing one input and keeping other inputs at the same level may initially increase output, further increases in that input will have a limited effect, and eventually no effect or a negative effect, on output

Conditional Equation

An equation that states a requirement to be satisfied

Definitional Equation

An identity between two expressions that have exactly the same meaning (Profit ≡ Revenue - Cost)

Labor Supply Decision

As in output markets, households face constrained choices in input markets. They must decide: 1. Whether to work 2. How much to work 3. What kind of job to take

appendix to chapter 9

Assumption: The new firms that enter the market are pretty much identical to the old firms

Inada conditions

Assumptions about the shape of a neoclassical production function that guarantee the stability of economic growth in a neoclassical growth model

Let P be a simplex and z:P->R(l) be a continuous function s.t. p*z(p) <= 0 for all p in P. Then there exists a p' in P s.t. z(p') <= 0

Baby Gale-Nikaido-Debreu

Marginal Cost

C'(x). Rate of change of cost as "x" changes

Cost function

C(x)

Cost Function

C(x). The cost of making "x" units.

Average cost

C(x)/x

Average Cost

C(x)/x. Cost per unit when "x" units are made.

Give the equation for consumption

C=aY + b where a= marginal prosperity to consume and b=autonomous consumption

Small increments formula

Change in z = Partial derivative of x . change in x + partial derivative of y . change in y

Define marginal cost

Cost of producing just one more item / the rate of change of cost with respect to x units produced

Define average cost

Cost per unit when x units are produced

Three types of equation

Definitional equations, behavioral equations, and conditional equations

Partial derivative of f(x) with respect to y

Derive in terms of y holding all other functions constant

Partial-Equilibrium Supply & Demand Model

Describes the behavior of a single good in the market as opposed to a general equilibrium model which describes all goods in the market and their inter-relationships

Implicit differentiation

Differentiate both sides of the equation with repect to x considering y as a function of x; then solve for y'.

Effective Annual Interest Rate (EAR) Formula

EAR = (1 + (i/m))^(m) -1

Relationship between Two Goods: Compliments

Ecp < 0.0 or negative

(Ecp) cross-price elasticity of demand formula

Ecp = % change in demand of Good x / % change in change in price of Good y

Relationship between Two Goods: Substitutes

Ecp > 0.0 or positive

Point Price-Elasticity of Demand

Ed= (dQdx/dPx)(Px/Qdx)

Point Price-Elasticity of Supply

Ed= (dQsx/dPx)(Px/Qsx)

Inferior Good Value of Ei

Ei < 0.0 or negative

Income Elasticity of Demand formula

Ei = % change in demand / % change in income

Normal Good: Luxery Value of Ei

Ei > 1.0

How do you decide if a good is inelastic or elastic?

Ep < 0 = inelastic Ep > 0 = elastic

Function for number of individuals between a and b (N)

F(∞)= n∫f(r)dr (from a to b)

Function for total income in a group (M)

F(∞)= n∫rf(r)dr (from a to b)

Cumulative distribution function

F(∞)=∫f(r)dr (from 0 to infinity)

Total Costs (TC)

FC + ( VC×Q)

First and second order conditions

FOC: f'(x)=0 SOC: looking at the sign of f''(x) (if <0, local maximum, if >0, local minimum - when f'(x) = 0 that is)

Characteristic of the Difference Equation

First Order, Linear, and Autonomous

Concave (in relation to derivatives)

For X in domain I; increasing in I, f''(x) </= 0

Convex (in relation to derivatives)

For X in domain I; increasing in I, f''(x) >/= 0

Young's theorem

For any two partials that involve differentiating with respect to each variable the same number of times, they are equal (cross-partials equal).

multivariate function

Function with more than one independent variable. The value of the dependent variable is determined by the values two variables.

Baby Brouwer

Given continuous f:[0,1] -> [0,1], there exists x' in [0,1] s.t. f(x') = x' (prove it)

Interpreting λ at maximum/minimum point

How optimal value of f changes with c (constant) • Function needs to satisfy FOC (as in, be at an optimum) ○ At a corner solution, this will not work • Needs to satisfy budget constraint

Give the equation for investment

I=I*

Brouwer's Theorem

If S (subset of R(l)) is a nonempty, convex, and compact set and f:S -> S is continuous, then there exists an x' in S s.t. f(x') = x'

Extreme value theorem (multivariate)

In a continuous function through a nonempty, compact domain D; there exists both a global minimum and a global maximum in D.

Union

In set theory, the union (U) of a collection of sets is the set of all elements in the collection.

Mean value theorem

In the interval [a, b] for a function connecting a to b, there will be at least one point on this function that has the same gradient as the linear line connecting a to b.

Let S be a non-empty, convex, compact set. additionally, let φ:S -> S be a upper semi-continuous, convex-valued correspondence. Then there exists a x' in S s.t. x' is in φ(x')

Kakutani's Theorem

The transpose

Let A be a m × n matrix. The transpose of A, denoted by A' or sometimes by AT, is defined as the n × m matrix whose first column is the first row of A, whose second column is the second row of A, and so on.

Gale-Nikaido-Debreu Lemma

Let P be a simplex in R(l) and Z a compact convex subset of R(l). Let ζ:P -> Z be a USC and convex valued correspondence s.t. p*ζ(p) <= 0 for all p in P. Then there exists a p' in P s.t. ζ(p') ∩ -{R(l)+} ≠ ∅

Profit maximizing theorem

Let Y subset of R(l) be convex. Let y' in Y be technologically efficient. Then there exists a p in R(l) s.t. p > 0 and py' >= py for all y in Y.

Intermediate value theorem

Let f be a function which is continuous in the interval [a, b]: 1. If f(a) and f(b) have different signs, then there is at least one c in (a, b) such that f(c) = 0 2. If f(a) ≠ f(b), then for every intermediate value y in the open interval between f(a) and f(b) there is at least one c in (a, b) such that f(c) = y

Mean income

M/N

Marginal cost

MC / C'(x)

Short run: supply curve for an individual firm

MC curve of a perfectly competitive profit maximizing firm

Average Product Trends

MP > AP, AP increases MP < AP, AP decreases

MRTS (marginal rate of technical substitution)

MPL/MPk

Equation for marginal rate of technical substitution

MPl/MPk

Short run: how much output will maximize profit?

MR = MC = p in perfect competition

marginal revenue formula

MR=(1+(1/Ed))

application to labor markets and land

MRPa = Pa (a = acres/land)

Multivariate functions

Map vectors to vectors

What does the second derivative represent?

Marginal changes of a marginal function.

Matrix operations

Matrix addition: if A = (aij)m x n and B = (bij)m x n are two matrices of the same order, we define: A + B = (aij)m x n + (bij)m x n = (aij + bij)m x n If α is a real number, we define: αA = α(aij)m x n = (αaij)m x n

Function

Method for mapping elements from one set (X) to elements of another set (Y) such that each element of X goes to one element of Y.

Assumptions for Cobweb Model

No Inventory Adaptive Expectations

Utility function

Numerical representation of ranking one's preferences over alternatives (this is descriptive and ordinal).

Interval of Compounding formula

P(1 + (i/m))^(mt) i = nominal interest rate m = number of times a bank pays you a year

profit maximizing conditions for inputs

PL = MRPL = (MPL x Px) for L, K, and A

Give the equation for PS

PQ - [Integral between Q and 0] P(q) dq

Give the equation for Total surplus

PS + CS

Marginal utility of a good

Partial derivative

Marginal product of capital MPk

Partial derivative of Q with respect to K

Equation for cross price elasticity of demand

Partial derivative of Q with respect to alternative Pa x Pa/Q

Equation for income elasticity of demand

Partial derivative of Q with respect to income x Y/Q

Equation for MPl

Partial derivative of Q with respect to labour

Equation for price elasticity of demand

Partial derivative of Q with respect to price x P/Q

How do you decide if a good is a normal or inferior good?

Positive - normal Negative - inferior

How do you decide if a good is a substitute or a complement good?

Positive = substitute Negative = complement

Profit, revenue, total costs

Profit = total revenue - total cost Economic Profit = total revenue - total economic cost

Equilibrium of supply and demand curve

Q1 = Q2 (intersection x coordinate)

Mathematical solution for equilibrium

Qd = Qs

Marginal revenue

R'(x)

Marginal Revenue

R'(x). Rate of change in revenue as "x" changes.

Profit function

R(x) - C(x)

Revenue function

R(x) = p(x) • x

Profit

R(x)-C(x). The revenue minus the cost.

Revenue Function

R(x). Total income from selling "x" units. R(x)=p(x)*x

Lagrange Multiplier

Rate at which the optimal value of the objective function changes with respect to changes in the constraint constant

Define marginal revenue

Rate of change of revenue with respect to the number of units sold

Kuhn Tucker Method (steps)

STEP 1: Associate Lagrange multipliers with the m constraints, then write down Lagrangian. STEP 2: Find the critical points of the Lagrangian by solving the system of equations (derive with respect to x, not λ) STEP 3: Impose the complementary slackness conditions: λ≥0, g_j (x)≤c_j, λ(g_j (x)-c_j) = 0 STEP 4: Find all vectors x that, together with associated λ's, satisfy conditions 2-3.

How do we use the substitution method for finding cost minimisation?

Same method but prove that stationary point is a minimum

Returns to scale

Scale determining whether or not output is multiplied by the same factor if all inputs are multiplied by that same factor *Used in production functions with labor and capital

How do you prove that a stationary point is a maximum?

Second order partial derivatives should be greater than 0 Derive one of the first oder partial derivatives in terms of the other variable Use equation: f"(x1).f"(x2) - f"(x1x2)^2 > 0

Partial Derivative

Shows the change in the dependent variable resulting from a small change in one independent while holding constant the value of other independent variables.

Equality of Cross Partial Derivatives

Smooth Functions

Sum of Geometric Sequence

Sn = (a(1-r^n))/(1-r)

Sum of Arithmetic Sequence

Sn = (n/2)(2a+(n-1)d)

Singular Matrix

Square-matrix that is not invertible

Introduction for Lagrangian Multiplier

State the Extreme Value Theorem: As the constraint is a closed and a bounded set as both x and y are non-negative, by Extreme Value Theorem, there will exist a minimum (maximum).

What does Particular Solution Mean?

Steady State, dy/dx = 0

Sequence

String of objects, like numbers, that follow a particular pattern (a 'set' does not need to follow such a pattern).

Matrix multiplication

Suppose A = (aij)m x n and B = (bij)n x p. Then the product C = AB is the m x p matric C = (cij)m x p whose element in the i-th row and the j-th column is the inner product.

Extreme value theorem

Suppose that f is a continuous function over a closed and bounded interval [a, b]. Then there exists a point d in [a, b] where f has a minimum, and a point c in [a, b] where f has a maximum. That is, one has f(d) ≤ f(x) ≤ f(c) for all x in [a, b].

Convex/concave theorem

Suppose that f is a function defined in an interval I and that c is a critical point for f in the interior of I... 1. If f is concave, then c is a maximum point for f in I. 2. If f is convex, then c is a minimum point for f in I.

Average Total Cost Formula

TC/Q

Average Fixed Cost

TFC/Q

total revenue

TR = P x Q

Total revenue quadratic formula

TR = aQ^2 +bQ

Total Revenue Test: When price is lowered when demand is elastic...

TR will rise

Total Variable Cost (TVC)

TVC = VC × Q

Average Variable Cost (AVC) Formula

TVC/Q

Long run

That period of time for which there are no fixed factors of production: firms can increase or decrease the scale of operation, and new firms can enter and existing firms can exit the industry

Elastic

The absolute value of Ed is greater that 1.0. Large response to a change. Located on the highest region of the demand curve.

Combinatorial mathematics

The branch of mathematics concerned with developing counting rules for given situations

What is the second derivative?

The derivative of the first derivative

Third Inada condition

The function is concave in K and L (law of diminishing marginal productivity) -Proven through the use of second partial derivatives of K and L *The second partial derivative of K must be less than 0 The second partial derivative of L must be less than 0

Second Inada condition

The function is increasing in K and L -Proven through use of first partial derivatives of K and L *The first partial derivative of K must be greater than 0 The first partial derivative of L must be greater than 0

Identity matrix

The identity matrix of order n, denoted by In or just I, is the n x n matric whose entries are 1 along the main diagonal and 0 elsewhere.

Fifth Inada condition

The limit of the first derivative is 0 as K and L approach infinity "Increasing input infinitely gives no change in output" *The limit of the first partial derivative of K as K approaches infinity must equal 0 The limit of the first partial derivative of L as L approaches infinity must equal 0 **Remember that a number divided by infinity is equal to 0

Fourth Inada condition

The limit of the first derivative is infinity as K and L approach 0 "Decreasing input gives an infinitely changing output" *The limit of the first partial derivative of K as K approaches 0 must equal infinity The limit of the first partial derivative of L as L approaches 0 must equal infinity **Remember that a positive number divided by a positive zero is equal to positive infinity

Budget Constraint

The limits imposed on household choices by income, wealth, and product prices PxX + PyY = I

Saddle point

The minimum of the function is in one direction and the maximum of the function is in another direction *Indicated by D2<0 If D2<0, the function has no minimum or maximum If D2=0, the function is inconclusive

Law of Diminishing Marginal Utility

The more of any one good consumed in a given period, the less satisfaction (utility) generated by consuming each additional (marginal) unit of the same good

Elements

The objects in a set

Short run

The period of time for which two conditions hold: the firm is operating under a fixed scale or production and firms can neither enter nor exit an industry

Define demand / price function

The price per unit that the company can charge if it sells x number of units

What does a derivative measure?

The rate of change of one variable in response to change in another.

Utility

The satisfaction a product yields

Substitution effect of price change

The shift in a purchasing pattern when a fall in the price of product X causes a household to buy more of X than its substitutes

Microeconomics

The study of the behavior of individual decision-making and how individuals or firms allocate scarce resources to make themselves and society as well off as possibe.

Define revenue function

The total amount of money taken in by selling x units

First Inada condition

The value of the function at (0,0) is 0

Geometric Sequence

Un = ar^(n−1)

Arithmetic Sequence

Un=a+(n-1)d

Three principal operations on sets

Union, intersection, and complement of sets

Gradient vector

Vector of all first-order partial derivatives. Can be used to draw an arrow.

Interval of Compounding

When banks pays interest more than once for example quarterly or monthly

external economies

When industry growth results in a decrease in long-run average costs

external diseconomies

When industry growth results in an increase in long-run average costs

Compound interest

When interest is NOT withdrawn, and yet for a period of time

Semidefinite

When the determinants and the diagonals are not strictly positive nor strictly negative but maybe equal to 0, they are positive (negative) semidefinite.

Simple interest

When you just keep your initial investment in the bank and withdrawn interest repayments

Total derivative

When z = f (x, y) with x = g(t) and y = h(t), then dz/dt = (∂f/∂x)(dx/dt) + (∂f/∂y)(dy/dt)

Level set

Where you let a function of multiple variables equal a constant and then plot various combinations of variables to equal that constant (e.g. indifference curve).

System of equations matricies

X=A^-1 * B

Give the equation for equilibrium

Y=C+I

Compound interest formula

Yn = P(1 + i)^n

Saving Formula

Yn = P(1+i)^t

Percentage Change in Quantity Demanded Formula

[(Qnew-Qold)/(Qnew+Qold/2)]/[(Pnew-Pold)/(Pnew+Pold/2)]

Give the equation for CS

[Integral between Qo and 0] of P(Q) dQ - PQ

() [] in denoting domain

[inclusive], (not inclusive)

Production function

a numerical or mathematical expression of a relationship between input and outputs. It shows units of total product as a function of units of inputs

uncertainty

a situation in which no reasonable probability can be assigned to potential outcomes. (the inability to determine the true state of affairs of a system)

What does Homogenous Solution Mean?

a=0

Product linear equation

aQ+b

The derivative of a^x

a^x * lna

Cost of Capital

add normal rate of return to capital as part of economic costs

Rule for concave functions

af(x1) + (1-a)f(x2) </= f[ax1 + (1-a)x2]

Rule for convex functions

af(x1) + (1-a)f(x2) >/= f[ax1 + (1-a)x2]

constant returns to scale

an increase in a firms scale of production has no effect on costs per unit

Sensitivity

changes in system's performance index (or output) to possible variations in the decision variables, constraint levels, and uncontrolled parameters (model coefficients)

Define cost function

cost of producing x units of a certain product

First order conditions provide

critical points that might be maxima or minima *First derivative equals 0

Derivative of a Constant

d/dx (c)=0

Power Rule

d/dx(x*)-n^(x*-1)

The Power Rule

d/dx[x^n]=[nx^(n-1)]

Give the equation for the investment multiplier

dY/dI* = 1/(1-a)

Give the equation for the marginal prospenity to consume multiplier

dY/da = (b+I*)/(1-a)^2

What is the law of diminishing marginal utility?

d^2U/dx1^2 < 0 and d^2U/dx1^2 < 0

Second Order Condition

d^2y/dx^2 < 0

derived demand

demand for input is dependent on demand for outputs those inputs are used to make

demand for inputs

demand up until the MC > MR

Envelope theorem (multivariate)

df*(r)/dr = df(x*(r), r)/dr Often you would need to use the chain rule in such a circumstance, however as we have x*(r), we no longer have to assume x is a function of r. Instead, we can treat both x and r as independent exogenous parameters.

Give the equation for the autonomous consumption multiplier

dv/da = 1-/(1-a)

First Order Condition

dy/dx = 0

Chain rule

dy/dx = dy/du.du/dx

Product rule

dy/dx = v du/dx + u dv/dx

The derivative of e^x

e^x

Utility Maximizing Rule

equating the ratio of the marginal utility of a good to its price for all goods

Homogenous function

f (tx1,tx2,...,txn) = t^k f (x1, x2,..., xn) HOD k

The Chain Rule

f'(g(x)) * g'(x)

Higher-order derivatives

f'(x) = slope of f(x) =dx/dy f"(x) = slope of f'(x) =(d^2y)/(dx^2) f"'(x) = slope of f"(x) = (d^3y)/(dx^3)

The Product Rule

f'(x)g(x)+g'(x)f(x)

Mathematical expression of homogenous function

f(ax, ay)=(a^k)(f(x,y)) a=constant number k=degree of homogeneity

Optimization is

finding the maximum or minimum of a function *Done through use of first order conditions and second order conditions

Separating Hyperplane Theorem

if A and B are two non-empty, disjoint, convex subsets of R(l), then there exists a p vector in R(l) and a scalar c within R s.t. px <= c for all x in A and px >= c for all x in B

Short run: should the firm shut down to minimize losses?

if TR > TC, P = MC; operate in short run; expand in long run and firms can enter if TR > TVC, P = MC; operate at a loss; contract in long run and firms exit if TR < TVC, P = MC; shut down in short run; contract in long run and firms exit

Intermediate Value Theorem

if f:[a,b] -> R is continuous, a<b, f(a)>0 and f(b)<0, then there exists x' in [a,b] s.t. f(x') = 0

increasing cost industries

industry encounters external diseconomies - AC increases as industry grows. Positive slope

decreasing cost industries

industry realizes external economies - AC decreases ad industry grows. Negative slope

constant cost industries

industry shows no diseconomies or economies of scale as industry grows. Flat curve

Definition of Derivative

lim h->0 [ f(x+h)-f(x)/h ]

CRRA when r=1

log(x)

Hessian matrix

n x n matrix of second order partials.

disjoint

not containing any of the same elements

Demand / Price function

p(x)

Demand Function

p(x). The price per unit that the company can charge at "x" units

Technologically Efficient

production plan y' in Y is technologically efficient if there does not exist y in Y s.t. y > y'

shifting the demand for inputs

shifts are caused by change in demand for output, change in quantity and substitutable inputs available, the prices of other inputs, and changes in technology

long-run average cost

shows the way per-unit costs change with output in the long run decrease in LRAC leads to external economies increase in LRAC leads to external diseconomies

minimum efficient scale

smallest size at which the LRAC curve is at its minimum

short run industry supply curve

sum of the MC curves (above AVC) of all the firms in an industry

Marginal Product

the additional output that can be produced by adding one more unit of a specific input

Marginal revenue product

the additional revenue a firms earns by employing 1 additional unit MRPL = MPL x Px

marginal revenue for price taking firms

the additional revenue that a firm takes in when it increases output by one additional unit. In perfect competition, MR = price = MC

Marginal Utility

the additional satisfaction gained by the consumption or use of one more unit of a good or service

productivity of input

the amount of output that can be produced from 1 unit of input

Average Product

the average amount produced by each unit of a variable factor of production. average product of labor = total product/total units of labor

Income effect of price change

the change in consumption of X due to an improvement in well-being

marginal cost

the derivative of the total cost function

Marginal Cost

the increase in total cost that results from producing 1 more unit of output. Marginal costs reflect changes in variable costs. **MC increases with output in the short run

Risk Neutral (r) is at the point where

the interval crosses 0; from negative to positive

total revenue for price taking firms

the total amount that a firm takes in from the sale of its product total revenue = price x quantity

Total fixed cost

the total of all costs that do not change with output even if output is zero

Total variable cost

the total of all costs that vary with output in the short run

Total average cost

total cost divided by the number of units of output; a per-unit measure of total costs. ATC = TC/q If MC is below ATC, ATC will decline toward MC. If MC is above ATC, ATC will increase. MC intersects ATC at ATC's minimum point for the same reason that it intersects the AVC curve at its minimum point

average fixed cost

total fixed cost divided by the number of units of output; a per-unit measure of fixed costs. AFC = TFC/q as output increases, AFC declines

long run industry supply curve

traces out price and total output over time as an industry expands

Give the equation for integration by parts

uv-[integral] du/dx . v dx

Second order conditions provide

verification of critical points as either maxima or minima *Second derivative>0: minimum Second derivative<0: maximum

The Quotient Rule

vu'-uv'/v^2

Law of Diminishing Returns

when additional units of a variable input are added to fixed inputs, after a certain point, the marginal produce of the variable input declines.

Exponential rule

y = e^x dy/dx = e^x

log rule

y = lnx dy/dx = 1/x

Quotient rule

y = u/v dy/dx = (v . du/dx - u . dv/dx)/ v^2

Rules for transposition

• (A')' = A • (A + B)' = A' + B' • (αA)' = αA' • (AB) = B'A' • Combining rules: AB = (B'A')'

For point c which is an interior point...

• If Hessian matrix of f at c is positive semidefinite, then c is a local minimum point • If Hessian matrix of f at c is negative semidefinite, then c is a local maximum point • If Hessian matrix of f at c is indefinite, then c is a saddle point

Interior and boundary points

• Interior point: a point of set S is an interior point if there exists a n-dimensional ball centred at that point such that all points strictly inside the ball lie in the set S • Boundary point: every n-dimensional ball centred at the point contains points of S as well as points outside S

Roy's identity

• Interpretation: the marginal disutility of a price increase is the marginal utility of income (λ) multiplied by the quantity demanded (xi*) • Intuitively, for a small price change, the loss of real income is approximately equal to the change in price multiplied by the quantity demanded

Pos/neg (semi)definite with quadratic form

• f is a positive definite quadratic form if f(x) > 0 for all non-zero vector x • f is a positive semidefinite quadratic form if f(x) ≥ 0 for every vector x • f is a negative definite quadratic form if f(x) < 0 for all non-zero vector x • f is a negative semidefinite quadratic form if f(x) ≤ 0 for every vector x

Absolute Risk Aversion (ARA) Formula

𝐴(𝑥)= −(𝑢"(𝑥) )/(𝑢′(𝑥) )

Relative Risk Aversion Formula (RRA)

𝑅(𝑥)=−(𝑥𝑢"(𝑥) )/(𝑢^′ (𝑥) )=𝑥𝐴(𝑥)

CRRA Formula

𝑢(𝑥) = (𝑥^(1−𝑟)) − 1 / (1−𝑟) ; 𝑅(𝑥)=𝑟, if 𝑟≠1

CARA Formula

𝑢(𝑥)=1−𝑒^(−𝑎𝑥), 𝐴(𝑥)=𝑎.