Info Yse
c -50 21. -52 21 neither an extremum nor a saddle point d 0, 0 neither an extremum nor a saddle point 2 3, 2 31 maximum e 0. 0 neither an extremum nor a saddle point 4 3. 4 3 minimum b Coke and hot dogs are complements a , 2y. 3 1 , , pz 7 . b The solution depends on h as shown in the table below
Info Gjt
linear, first-order differential equations in this model. We re-express the model as Using theorem 24.2 the solutions are where p and N are the steady-state price and number of firms respectively. The roots of Ihe characteristic equation are The determinant of the coefficient matrix is The positive determinant indicates that the roots are either both negative or both positive if real valued. The trace of the coefficient matrix is Both roots are negative if and only if ab lt 0. Since a gt 0. the...
Info Sju
25.6 FREE-TERMINAL-TIME PROBLEMS TFREE 1063 to solve this problem for the optimal path of investment, capital accumulation, and labor demand, subject to K l -8K A' 0 K0 lt lt gt where L is labor input. K is the slock of capital, I is investment, p. w. and are the constant prices of output, labor, and investment goods respectively, and cv and fi are positive constants whose sum is less than one. 25.6 Free-Terminal-Time Problems TFree Until now- we have assumed that T is specified exogenously. In...
Info Lct
Figure 25.7 Phase diagram for die exhaustible resource problem drawn in R the resource stock remaining and c consumption space Figure 25.7 Phase diagram for die exhaustible resource problem drawn in R the resource stock remaining and c consumption space point or. if there is a satiation point, that it is at a high enough consumption rate that the resource constraint becomes binding. The relevant second boundary condition then is R T 0. Because we have not specified a functional form for...
Info Uzm
optimal trajectory starts with Aft0 K and ends with T 0 and takes an amount of lime exactly equal to T. v' , p'r ra Xo ,--' a _ g l,-r Tlue-rl j These diffevcwial eqilWlous yield the phase diagram shown here The phase diagram shows that the saddle path is the horizontal line which reaches the steady-state point at R a and p p,. li is clearly not possible to reach a negative resource stock. The optimal trajectory begins at RiQ R . finishes ai R T 0 and also satisfies p T P. This occurs in...
Info Uxl
Figure 24.2 Phase diagram for example 24.12 southwestern sector of the phase plane move in a northeasterly direction. Trajectories in the northwest move southeast and trajectories in the northeast move southwest. Overall, the arrows of motion tell us that no matter what sector of the phase plane we start in. trajectories always move towards the steady state. The phase diagram Ihen provides a good indication that the steady state is globally stable. To get a precise picture of Lhe trajectories...
Ip
These results are illustrated in figure 5.36. Find the point elasticity of demand r- with respect to own price for the demand function v 100 ' Use both the direct approach, using equation 5.6 , and ihe method of first taking logarithms, and then apply equation 5.8 . where, in this case, dy dp -200 and so Alternatively, by first taking logs, we have Noting that e d In y d in p gives We can also define the price elasticity of supply as the ratio of the percentage change in quantity supplied...
I Tdm
4 - 1 3 gt 0 for any x e R Since i -2 lt 0. applying theorem 12.4 shows that the .stationary point 3.33. 2.67 is a maximum. v 4.Vf - JT2 - 2 x'l f 8 6.V1 - x2 - 3.V,2. fz X 2.V ti 8 6.x , n -l, fn 1, fzi 2 This function has stationary points 0, J and 2.5. 1.5 . At the lirst of these
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...
Info Wsp
b There is a single point of nondilTerentiability at 5 10,000. 1. a . 10i., f' L 10, f L 0, so die rate at which output rises wish respect to more input being used does not change. b f L 8L f' L 8 3 2 3, f L - 16 9 lt 0. so the rate at which output rises with respect to more input being used is falling. c f L 3L f L I2L f L 36L1 gt 0. so the rate at which output rises with respect to more input being -used is increasing.
Info Ajo
2. Show that a profit-maximizing monopolist's output is unaffected by a proportional profit tax, but is reduced by a tax of per unit of output. Explain these results, 3. Find the supply curve of a competitive firm with the total-cost function 4. The demand function facing a monopolist is What range of values must b lie in for a solution to the profit-maximization problem to exist 5. A monopolist faces a linear demand function. Show that if it maximizes sales revenue, it sets an output exactly...
Rule 4 Derivative of the Constant Multiple of a Function
ffg x ef x . then g' x cf' x , This rule is a very straightforwajd one to implement Example 5.7 Find the derivative of the function g L 5L2. Solution This function can be written as 5f L where f L L and so the derivativem g L is g' L 5 ' . 5 2L 10L. I This general rule is easy to prove. Note that
Exercises Kpt
1. Find the partial derivatives of the function y 3 j. i 5xz using definition III see example 1 I.I . 2. Find the partial derivatives of the function where a and b are any constants, using definition 11.1 see example 11.1 . 3. For the revenue function of example 11.1. R x , x2 p x Pz i- find the partial derivative dR X ,xi dxi by using definition 1 I.I. Give an intuitive explanation of your result. 4. Discuss why it is the case that the partial derivatives in questions 1. 2. and 3 are constant...
Exercises Unw
1. 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...
Equation For Bhn
Although the diagrammatic approach is suggestive, we now study die problem algebraically. Consider again the constraint ,t xi. x2 0, and assume that it can be solved to give, say. x2 as a function of jti This is. in fact, simply the function which gives the curve G in figure 13.1. We know from the presentation of implicit differentiation theorem 11.2 that the derivative of this function is Now, if we substitute y .i for a 2 in the function . ', we are left with the unconstrained problem in one...
Ifffl 0 ihI 0Ii ll j o
for x e R. 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...
Chapter 5 The Derivative And Differential For Functions Of One Variable
8. For the same production function as in question 7. q ciL1', show that the cost function is convex concave if the production function is concave convex . Relate your answer to the answer in question 7. 9. Let C y y3 - 12v2 50y 20, v gt 0 be a firm's cost function. Find the interval over which it is concave and the interval over which it is convex. Use this information and a table such as that of example 5.20 to draw this function. 10. For the following functions, find the Taylor series...
Exercises Cls
1. In the simple Keynesian model of income determination, assume that c 0.8, that I - 1,000 initially, and then that I increases to 1,200, Draw the counterparts of figure 14.2 and solve for the two levels of national income. What is the value of the multiplier in this case 2. In the linear market model, take the functions with v 10 initially and then y 20. Draw the counterpart of figure 14.4 and solve for the equilibrium prices. 3. In the model of a profit-maximizing monopoly with tax, take the...
Cramers Rule
In this section we present Cramer's rule, a method for solving for 11 unknown variables in a system of n equations that is an alternative to the inverse-matrix method. The system of equations may he written as where A is a square matrix of order n that has an inverse A '. x is an array of order x 1 of n unknowns, and b is an array of order n x 1 of known elements. Pre-multiplying both sides of equation 9.3 by ,41 yields the solution for x. given by In other words, to solve for the n unknowns,...
Chapter 2 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...
Review Pvp
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...
A Bakery Advertises Its Bagels By Noting The Price Per Dozen
1. 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 0
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...
Review Qpr
constrained extrema extreme values first-order conditions global maximum minimum interior solution local maximum minimum necessary condition nonbindins constraint Hth derivative test points of inflection second-order conditions shadow price stationary value sufficient condition unconstrained extrema 1. Distinguish between local and global optima. 2. What is the first-order condition for a maximum 3. Whai is the first-order condition for a minimum 4. What is a sufficient condition for a maximum...
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...
Exercises Mwf
1. Form the Cartesian products of the following sets a 11. 2, 3. 4, 5, 6 and 7, 8.9 c The set of even elements of Z and the set of odd elements of Z., Illustrate these product sets in R2. 2. The consumption set of a consumer is C .v. y e jc gt x' gt 0, y gt y' gt 0 Illustrate this set. Is it closed bounded convex How would you interpret x' and where p , p gt 0 are prices and m gt 0 is income. Illustrate this set. Is il closed bounded convex Consider the set X B HC where C is defined in exercise...
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...