Basics

Question 0

2. Ultimate test of any theory?

Answer 0

Ability to predict real world events.

Question 1

1. All economic models incorporate three common elements. What are they?

Answer 1

1. the ceteris paribus assumption
2. supposition that economic decision makers seek to optimize something.
3. careful distinction between “positive” and “normative” questions.

Question 2

3. Inputs into economic models are ______ variables. Basic examples?

Answer 2

Exogenous variables.
Households: Prices of goods.
Firms: Prices of inputs and output.

Question 3

4. Output of economic models are _________ variables. Examples?

Answer 3

Endogenous variables.
Households: Quantities bought.
Firms: Output produced, inputs hired.

Question 4

5. Formula for elasticity?

Answer 4

– e(y,x) = (dy/dx)(x/y)
e(y,x) = elasticity of y given x.
– If y is linear function of x: y = a + bx, then e(y,x) = b * (x/y)
– If functional relationship between x and y is exponential: y = a*x^b, then elasticity is a constant: b

Question 5

6. Necessary and sufficient conditions for a maximum.

Answer 5

Necessary (first order condition): f’(q) = 0 (slope = 0)
f’(q) is df/dq. All partial derivatives = 0.

Sufficient condition (second-order condition): f''(q) < 0.
f''(q) is d^2f/dq^2 (second order partial derivative).

Question 6

7. What is the importance of “own” 2nd partial derivative [f(ii)]?

Answer 6

Shows how marginal influence of x(i) on y changes as the value of x(i) increases.
A negative value for f(ii) indicates mathematically the idea of diminishing effectiveness. Provides info about curvature of the function.

Question 7

8. What is Young’s Theorem?

Answer 7

The order in which 2nd order partial derivation is conducted doesn’t matter: f(ij) = f(ji).
Visual: Gain hiker experiences on a mountain depends on the direction and distance, but not on the order in which these occur.

Question 8

9. What is the implicit function and what is an example of the implicit function?

Answer 8

Value of function is held constant to examine the implicit relationship among independent variables. An example is the “envelope function.”

e.g. of implicit function: y = f(x1, x2) –> hold y constant yields x2 = g(x1). Derivative of g(x1) is related to partial derivative of original funtion f. Derives explicit expression for trade-offs between x1 and x2.

OC (of PPF) = dx(2)/dx(1) = -f(1)/f(2) –> Change in x(2) resultant from a change in x(1) is equal to the negative of the partial deriv. w.r.t. x(1) divided by the partial deriv. w.r.t. x(2).

Question 9

10. With respect to the Production Possility Fronter (PPF), what does dx(2)/dx(1) = -f(1)/f(2) demonstrate?

Answer 9

Concavity of PPF –> As output of x(1), i.e. “x”, increases have to give up increasingly more units of x(2), i.e. “y.”

Resources better suited for x are used up and resources better suited for y are marginal for x so, need more y resources in the propduction of x.

Question 10

11. Define the envelope theorem.

Answer 10

It is a mathematical result. The change in max. value of a function brought about by a change in a parameter can be found by partially differentiating the function w.r.t. the parameter in question (when all other variables take on their optimal values).

Question 11

12. Envelope shortcut –> one variable case.

Answer 11

If y = -x^2 + ax , where a = constant.
Optimal value of x for any a = dy/dx –> x* = a/2

dy/da = ∂ y/∂ a [at x = x(a)] = ∂(-x^2 + ax) / ∂ (a)
= x*(a) = a/2

Question 12

13. Envelope theorem –> many variable case.

Answer 12

y = f(x1, x2, … , x(n), a) , where a is a parameter
Assuming SOC are met, implicit function theorem applies and ensures solution of each x(i) as a function of parameter a, e.g. x(1) = x(1) (a).
y = f[x1(a), x2(a), … , x(n)(a), a] –> Differentiate w.r.t. a yields:
dy/da = (⍺f/⍺x1 * dx1/da) + (⍺f/⍺x2 * dx2/da) + … + ⍺f/⍺a
…but because of FOC, all of the above terms except the last are zero.
–> dy*/da = ⍺f/⍺a at optimal values for all x.

Question 13

14. Solving constrained maximization with the lagrange multiplier (LM) results in what two properties?

Answer 13

1. x’s obey the constraint ‘cuz ⍺L/⍺λ imposes that condition.
2. Among all values of x’s that satisfy the constraint, the partial differentiation equations will make L (and hence f) as large as possible (assuming SOC are met).

Question 14

15. Interpret the Lagrange Multiplier LM.

Answer 14

λ provides an implicit value or “shadow price” to the constraint, i.e. provides measure of how an overall relaxation of constraint will impact value of y.
λ = [MB of x(i)] / [MC of x(i)].
– High λ indicates y could be increased by relaxing the constraint (high MB/MC ratio).
– Low λ indicates not much gained by relaxing constraint.
– λ = 0 indicates constraint isn’t binding.

Question 15

16. With constrained maximization, what is the optimal value for all x’s?

Answer 15

When the MB/MC of each x are equal, i.e.

λ1 = λ2 = λ3 = λ4

Question 16

17. Define duality and give an example of duality.

Answer 16

Any constrained maximization problem has an associated dual problem in constrained minimization.

e.g. Primal problem: Utility maximization subject to budget constraint.
Dual problem: Minimize expenditure needed to achieve certain level of utility.

Question 17

18. What are the Kuhn-Tucker conditions?

Answer 17

Conditions for solving inequality constraints. Additional λs indicate customary optimality conditions may not hold (as w/equality constraints).

– If slack variables introduced to the solution = 0, then the constraints hold exactly. e.g. x1 >= 0 becomes x1 = 0. Additionaly, λ indicates its relative importance to the function.

– If slack variable is not equal to 0, then λ = 0. This shows the availability of some slackness in the constraint; implies λ’s value to the objective is 0. e.g. If person does not spend all their income, an increase of income will not increase well-being. If λ = 0, then x1 >= 0 becomes x1 > 0.

Question 18

19. Describe steps necessary to solve inequality constraints.

Answer 18

1. Introduce slack variables to convert inequality constraints into equalities. e.g. x1 >= 0 becomes x1 - b^2 = 0. [b^2 to ensure values are positive]
2. Use Lagrangian method to solve optimization problem. Include partials for all slack variables and all constraints (w/slack variables).
3. Kuhn-Tucker conditions apply.

Question 19

20. How optimize functions w/ two variables (unconstrained maximazation)?

Answer 19

1. FOC: Partial derivatives w.r.t. to each x = 0.
2. SOC: Complex because optimal point not solely in the x1 or x2 directions but can be some combination of the two. In order for second partials to be unambiguously negative require own second partials to be negative and end up w/ the following SOC for a maximum: f(11)*f(22) - f(12)^2 > 0 (i.e. f(11) = own second partial, f(12) = cross second partial. Both own second partials have to be sufficiently negative for expression to be > 0.

Question 20

21. SOC for optimization w/2 variables is also a condition for what?

Answer 20

Concavity. Functions that obey the condition f(11)*f(22) - f(12)^2 > 0 are concave functions (resemble inverted tea cup).

Question 21

22. Define concavity (Nicholson, p. 54-55)

Answer 21

Concave: all line segments connecting any two points on the curve lie everywhere on or under the surface of the function. (Function looks like an inverted tea cup.) All concave function are quasiconcave, but not all quasiconcave functions are concave.

PPF is concave w.r.t. the origin. Concavity reflects diminishing returns.

Question 22

23. Define convexity

Answer 22

Line segment between any two points on the graph of the function lies above the graph. Examples of functions that are convex: f(x) = x^2 and f(x) = e^x.

Indifference curves are convex.

Question 23

24. What is a quasi-concave function. (Nicholson, p. 54-55)

Answer 23

24. Functions that have property that the set of all points for which a function takes on a value > any specific constant is a convex set (i.e. any two points in the set can be joined by a line contained completely within the set). Quasi-concave functions reflect increasing returns over certain areas of the function. All concave functions are quasi-concave but the reverse is not true.

Question 24

26. What is an homogeneous function and give example.

Answer 24

A function f(x1, x2, … x(n)) is said to be homogeneous of degree k if f[tx(1), tx(2), … , tx(n)] = t^k f(x1, x2, … x(n))

k = 1 –> Function is homogeneous of degree one when a doubling of all its arguments double the values of the function itself.

k = 0 –> Function is homogeneous of degree zero when a doubling of all its arguments leaves the value of the function unchanged.

Question 25

25. How optimize functions w/ two variables (constrained maximazation)?

Answer 25

1. FOC: Set up Lagrange expression and take all partial derivatives.
2. SOC: f(11)f(2)^2 - 2f(12)f(1)f(2) + f(22)f(1)^2 < 0
   
   • -> negative is required for quasi-concavity.

Question 26

27. What is a monotonic transformation?

Answer 26

Transformation that preserves the order of the relationship between the arguments of a function and the value of that function.

Monotonic transformtns are NOT expected to preserve an exact mathematical relationship such as those embodied in homogeneous functions.

Question 27

What is a homothetic function?

Answer 27

Function formed by taking a monotonic transformation of a homogeneous function. Except in special cases, homothetic functions do not possess the homogeneity properties of their underlying functions. HOWEVER, homothetic functions do preserve one important feature of homogenous functions: The implicit trade-offs implied by the function depend only on the ratio of the two variables being traded, NOT on their absolute levels.