- Ultimate test of any theory?
Ability to predict real world events.
- All economic models incorporate three common elements. What are they?
- the ceteris paribus assumption
- supposition that economic decision makers seek to optimize something.
- careful distinction between “positive” and “normative” questions.
- Inputs into economic models are ______ variables. Basic examples?
Exogenous variables.
Households: Prices of goods.
Firms: Prices of inputs and output.
- Output of economic models are _________ variables. Examples?
Endogenous variables.
Households: Quantities bought.
Firms: Output produced, inputs hired.
- Formula for elasticity?
– 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
- Necessary and sufficient conditions for a maximum.
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).
- What is the importance of “own” 2nd partial derivative [f(ii)]?
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.
- What is Young’s Theorem?
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.
- What is the implicit function and what is an example of the implicit function?
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).
- With respect to the Production Possility Fronter (PPF), what does dx(2)/dx(1) = -f(1)/f(2) demonstrate?
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.
- Define the envelope theorem.
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).
- Envelope shortcut –> one variable case.
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
- Envelope theorem –> many variable case.
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.
- Solving constrained maximization with the lagrange multiplier (LM) results in what two properties?
- x’s obey the constraint ‘cuz ⍺L/⍺λ imposes that condition.
- 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).
- Interpret the Lagrange Multiplier LM.
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.
- With constrained maximization, what is the optimal value for all x’s?
When the MB/MC of each x are equal, i.e.
λ1 = λ2 = λ3 = λ4
- Define duality and give an example of duality.
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.
- What are the Kuhn-Tucker conditions?
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.
- Describe steps necessary to solve inequality constraints.
- 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]
- Use Lagrangian method to solve optimization problem. Include partials for all slack variables and all constraints (w/slack variables).
- Kuhn-Tucker conditions apply.
- How optimize functions w/ two variables (unconstrained maximazation)?
- FOC: Partial derivatives w.r.t. to each x = 0.
- 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.
- SOC for optimization w/2 variables is also a condition for what?
Concavity. Functions that obey the condition f(11)*f(22) - f(12)^2 > 0 are concave functions (resemble inverted tea cup).
- Define concavity (Nicholson, p. 54-55)
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.
- Define convexity
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.
- What is a quasi-concave function. (Nicholson, p. 54-55)
- 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.
