## Concave Programming And Inequality Constraints

In the classical method for constrained optimization seen thus far. the constraints have always been strict equalities Some economic problems call for weak inequality constraints, however, as when individuals want to maximize utility subject to spending not more than x dollars or business seeks to minimize costs subject to producing no less than x units of output. Concave programming, so called because the objective and constraint functions are all assumed to be concave, is a form of nonlinear...

## Graphs In Thk Income Determination Model

C -50+ 0-8K and 50. (< r) Graph Ihc consumption function. ( ) Graph the aggregate demand function. < 4 . (c) Find the equilibrium level of income from the graph d) Since consumption is a function ol income. it is graphed on the sertieal axis income is graphed n the horizontal See Rg, 2-7 When other cimiponents of aggregate demand such as Ci. and X arc added to the model, they are also graphed on the sertieal aus It is easily determined from the linear form ol the...

## Homogeneous Production Functions

A production function is said to be homogeneous if when each input factor is multiplied by a positive real constant ky the constant can be completely factored out. If the exponent of the factor is 1, ihe function is homogeneous of degree 1 if the exponent of the factor is greater than 1. the function is homogeneous of degree greater than 1 and if the exponent of the factor is less than I. the function is homogeneous of degree less than 1 Mathematically, a function z (x. y) is homogeneous of...

## Comparative Statics With More Than One Endogenous Variable

In a model with more than one endogenous variable, comparative statics requires that there be a unique equilibrium condition for each of the endogenous variables. A system of n endogenous variables must have n equilibrium conditions. Measuring the effect of a particular exogenous variable on any or all of the endogenous variables involves taking the total derivative of each of the equilibrium conditions with respcct to the particular exogenous variable and solving simultaneously for each of the...

## Iiffi o p P

The substitution cffcct in the first term is unambiguously negative. The income cffcct in the second term will depend on the nature of the good. For a normal good, dtiJdY > 0 and the mcomc cffcct above will be negative, nuking da dp. < 0. For a weakly inferior good, dalOY 0. and Ai dpm< 0. For a strictly inferior good, dd dY < 0 and the sign o Aldp, will depend on the relative magnitude of the different effects. If the income effect overwhelms the substitution cffcct, as m the ease of a...

## Firstorder Partial Derivatives

Find the first-order partial derivatives lor each of the following functions a) z + 14.xy + Sy2 b) z - 4** + 2x*y - 7y* Z. - 16ur 4 14y j, - I2r 4xy *r - 14jr 4 lOy z, 2*' - 35 c) 2 6k-3 4 4w* 4 a 2 - 7jr> - -8 d) z 2w* + fiwxy - x2 + zf 4h- + 6r - ly zt 8wy - Ix 5.2. Use the product rule from (5.2) lo find the first-order partials for each of the following functions and jy M7)4< 5x4 7> -)(0) 21jt Z, (9x-4y)(l2)4(12x + 2y)(9) lO& r - 48.v + I08x - 18y - 216x - 30y and z, - (9* -...

## L

Are improper integrals because 33 is not a number and cannot be substituted for x in F(x). They can. however, be defined as the limits of other integrals as shown below. f(x)dx lim j'f(x)dx and J f(x)dx - lim J f(x)dx If the limit in either case exists, the improper integral is said to converge. 'Ihe integral has a definite value, and the arc-a under the curve can be evaluated. If the limit docs not exist, the improper intcRral diverges and is meaningless. See Example 5 and Problems 15.19 to...

## Determinants And Nonsingularity

The determinant A of a 2 X 2 matnx. called a second-order determinant, is derived by taking the product of the two elements on the principal diagonal and subtracting from it the product of the two elements off the principal diagonal. Given a general 2x2 matrix I'Ik determinant is a single number or scalar and is found only for square matrices. If the determinant < > ( a matrix is equal to zero, the determinant is said to vanish and the matrix is termed menhir singular matrix is one in which...

## Yo2 vcv

C-100 gt 0.6 V c - 100 0.6 Yd T -50 The first system of equations presents no problems the second requires that C lirst Ik converted from a function of Yd to a function of Y 100 06V 40 - 100 4- 0.6 Vd 40 140 0.6 V - T 140 0.6V - 140 06 lt V-50 - 110 ObV A lumpsum tax ha a negative effect on the vertical intercept of the aggregate demand unction equal to MPC lt Here - lt MK 0 - - 30. Ihe sli pc is not affected note the parallel linc for the two graph in Fig. 2-'f . Income falls from 350 to 275...

## Index

Ihe letter p full im m a gsagc number refer to a Problem Additkm ol matrices. 200-201. 208-209 Antidiffcrcr'.i in n rr Integration between cunev 345-346. 354- undet a curse. 342-343 of resolution 4 V Argument of lundion. 5 Arrows of Motion. V 8 Associative lau 204-20 . 219-222 Auhmomous l-'qu.ition. 42H Auxiliary aquation. 4 fft 4iW. 457p Average concepts. 63-64, 72-74p relationship to total and marginal concept . 63-61. 2 74p. 80-81 Ascrch-Johnson effect. 323 325p Bordered Hesst m...

## J V a i a jt x x jr x TJ

C CO - J X Of Rule 3 r ' yH yKJijf Uj' lt m Hyyyy - Since i i i and from Rule I exponents of a commno base arc added in multiplication, the exponent of V jr. when added to itself. must equal 1. With I - J 1. the exponent of t i J, Thus. 4i. 14 '' gt c 4 ' -2 ' -8. equally valid. 4 - HT' M '7 - M - 2H Ruk 0

## Constrained Optimization In Economics

628. a What combination of goods x and y should a linn produce to minimize costs when the joint cost unction is c 6jt 10y ' - xy 30 and the firm has a production quota of x y 34 b Estimate the effect on costs if the production quota is reduced by 1 unit. a Form a new function by setting the constraint equal to zero, multiplying it by A. and adding it to the original or objective function. Thus. C - 6t2 10 - xy 30 A 34 - x - y C, 12 - y - A 0 Cr - 20v - x - A 0 CA - 34-x- v - 0 Solving...