## Info

Figure 4.11 Marginal product of an input under a capacity constraint Revenue Function, Cost Function, and Profit Function for a Perfectly Competitive Firm In the model of perfect competition it is assumed that each lirm treats the market price as given. The lirm does not believe its own choice of output level will influence the market price, and so it treats this value as fixed. This assumption is usually only made to describe markets in which a large number of producers each produces a small...

## Economic Applications of Continuous and Discontinuous Functions

There arc many natural examples of discontinuities from economics, In fact economists often adopt continuous functions to represent economic relationships when the use of discontinuous functions would be a more literal interpretation of reality. It is important to know when the simplifying assumption of continuity can be safely made for the sake of convenience and when it is likely to distort the true relationship between economic variables too much. Our first example illustrates a class of...

## 2 I I I oo39

Which is clearly a divergent series, This means that if one is willing to accept the idea that individuals value risky outcomes according to the expected value of the monetary outcome, then this gamble is preferred to a gain of 10 million for certain or any other finite amount, no matter how large. Bernoulli believed this was a ludicrous conclusion and offered a way out of the apparent paradox. An intuitive expUuiation of his argument is as follows. The vulue to an individual of different...

## Solutions by Substitution and Elimination

Perhaps the simplest analytical (rather than diagrammatic) ways ol solving these simple linear systems is by the substitution and elimination methods. Consider again the two equations represented in figure 7.1. These are The analytical solution by substitution is obtained simply by solving one of these equations for either x or v and substituting the result in the other equation. For example, we can solve equation (7.8) for .r to lind a- y + 1 and then substitute this value for x in equation...

## Chapter Review 697

The price of meal is 10 per pound and that of potatoes is I per pound. She has an income of 80. In addition to her budget constraint, she has a subsistence-calorie constraint she must consume at least 1,000 calories. One pound of potatoes yields 20 calories, one pound of meat yields 60 calories. Find her optimal consumption bundle. Now suppose that the pricc of potatoes rises to 1.60 per pound. Find the new optimal bundle. Explain your results, and discuss their...

## C fx i to Ai

Applying the envelope theorem directly gives Thus the Lagrange multiplier measures the rate at which the value function changes when the corresponding constraint is lightened or relaxed slightly. This interpretation of the Lagrange multiplier is of fundamental importance in economic applications of methods of constrained optimization. One implication is immediate if a constraint is nonbinding at the optimum, so that a small lightening or relaxing of it has no effect on the solution, then the...

## Rip x pim p 100 r

Jrip Rip - C p 100, - p - 2.500 - 25, I25p- p2 2.500 Then maximizing with respect to gt gives 7t' p 125 - 2p' 0 giving p 62.50, just as before. A monopolist has inverse demand function p 50 - 2x. The total-cost function is C 20 2.x 0.5x2. What are the profit-maximizing price and output TT .x 50x - 2x - 20 2x 0.5x2 48.r - 2.5.v2 - 20 x' 9.6 p 50 - 2 9.6 30.80 The level of profit at the maximum is then rtix 48 9.6 - 2.5 9.6 f - 20 210.40 See figure 6.10. A monopolist has inverse demand function p...

## Mathematics For Economic Analysis

Part I Introduction and Fundamentais 1.1 What Is an Economic Model 3 2.3 Some Properties of Point Sets in R 33 2.5 Proof, Necessary and Sufficient Conditions 60 Chapter 3 3.1 Definition of a Sequence 67 3.3 Present-Value Calculations 75 Part It Univariate Calculus and Optimization 4.1 Continuity of a Function of One Variable 115 4.2 Economic Applications of Continuous and Discontinuous Functions 125 4.3 Intermediate-Value Theorem 143 The Derivative and Differential for Functions of One Variable...

## Exercises

For the function u x . a2 5.V 3.t2 a Find the total differential. b Draw the level curve for u 120. c Use the pair of points 12. 20 and 18, 10. lo illustrate thai the MRS 5 3 and derive this result from the total differential in part a . 2. For the. function u xi. .r2 ax gt x2. a Find the total differential. b Draw a representative level curve for u. c Use the expression for the total differential to illustrate that the MRS a b. 3. Use the total differential lo hnd the MRTS for the...

## Ifffl 0 ihI 0Ii ll j o

In this case dry lt 0 and so . ' is concave. Moreover, if is concave this set of conditions must hold. The following examples illustrate how to use the results in theorem 11.9 to determine the concavity convexity properties of a function. Example 11.28 Use theorem 11.9 to determine the convexity concavity property of the function y jci , x2 x x2 l2 defined on x 6 R . The second-order partial derivatives are Since all oi these are negative, we check first for strict concavity of Note...

## Review Of Fundamentals

A set X C R is convex if for every pair of points .v, x' e X, and any k e fO, I j. the point 2.3 SOME PROPERTIES OF POINT SETS IN R 41 In words, a set is convex if every point on the line segment between every pair of points in the set is also in the set. In a of figure 2.18 we show some convex sets in K2, and in b some nonconvex sets. Although the idea of convexity is a very simple one geometrically, it is extremely important. An interior point of a set X C K is a point A'o e X for which there...

## Marginal Rate Of Technical Substitution And Monotonic Transformation

Addilively separable function bordered Hessian cross-partial derivatives elasticity of substitution Euler's theorem first-order total differential gradient vector Hessian matrix homogeneous function homothetic function implicit differentiation implicit function theorem indifference curves marginal rate of substitution MRS marginal rate of technical substitution positive monotonic transformation remainder formula second-order total differential Taylor series Young's theorem 2. Why is...

## Bakery Advertises Its Bagels By Noting The Price Per Dozen

Find the slope of each of the following production functions, y f L . Graph he functions and their derivative functions. Give the economic significance of the sign of the slope of the derivative functions i.e whether the derivative is increasing or decreasing in L . a y 10L lt b v 8JLl 3 c y 3LA 2. Find the slope of each of the following production functions, y ' L . Graph the functions and their derivative functions. Give the economic significance of the sign of the slope of the derivative...

## Concavity Convexity Quasiconcavity Quasiconvexity

In our description of some specific functions we used the terms convexity and concavity. Visually the meaning should be clear, but we now present a formal definition. Figure 2.29 shows how we proceed in the case of a concave function. First we must assume that the domain of the function is a convex set. because we want convex combinations of points in the domain to be in the domain. Take any two points x' and x i n the domain of the function and the corresponding function values f x' and f x ....

## Z crfp p

Where p is the input price, and a, b. u. p gt 0. Find the profit-maximizing price and quantity of the input the monopsonist will choose, and compare the analysis to that of the profil-maximizing monopoly. 5. A firm has the production function x f L , where x is output and L is labor input. The linn buys the input in a competitive market. a Assuming the linn sells its output in a competitive market, show that setting output where price equals marginal cost is equivalent to setting labor input...

## By 05x70fx5V2

X. a Given the strictly quasiconcave function y f xt.x2 . sketch a typical level set in each of the following cases i The function is increasing in x and decreasing in xj. ii The function is decreasing in X and increasing in x . iii The function is decreasing in both variables Him First determine which way the curve of the level set must slope, then identify the area that gives the better set, and then find how the curvature must look to make the better set convex. fb Repeat part a , assuming...

## Rule 2 Derivative of a Linear Function fx mx b

If fix nix b, with m and b constants, then fix in. Figure 5.20 Linear demand has a constant slope example 5.5 Figure 5.20 Linear demand has a constant slope example 5.5 This result follows because Ay fix Ax - J' x - mix A.v b- m x b m A.v. Then Ay Ax m. and so lima, o Ay Ax in. For example, the derivative of the function v 3.v 5 is fix 3, The important implication of this result is that for a linear function the rate at which the variahle y changes with respect to a change in x is the same at...

## Unconstrained Maxima and Minima

Given some function i.e., y . , we optimize it by finding a value of x at which it takes on a maximum or minimum value. Such values are called extreme values of the function. If the set of v-values from which we can choose is the entire real line, the problem of finding an extreme value is unconstrained, while if the set of.r-values is restricted to be a proper subset of the real line, the problem is constrained. To begin with we consider only unconstrained problems. We also assume that the...

## Rectangular Hyperbola

A rectangular hyperbola may be written for some positive constant a. The name stems from the fact that every rectangle drawn to the curve has the same area a. Note that the graph of the function in figure 2.25 has two parts, one entirely in the positive quadrant and the other entirely in the negative quadrant. In economics we often restrict x to so only Figure 2.25 Rectangular the upper curve is relevant. As x tends to zero, the curve approaches the y-axis hyperbola asymptotically, and as x...