## Constrained Optimization And Lagrange Multipliers

5.12, (I) Use Lagrange multipliers to optimize the following functions subject to the given constraint, and (2) estimate the effect on the value of the objective function from a 1-unit change io the constant of the constraint a) z m 4x* - 2xy + 6y* subject to x + y 72 1) Set the constraint equal to zero, multiply it by A, and add it to the objective function, to obtain Z-4*z-2xy + 6yi+ A(72-x-y) The first-order conditions arc Z,53 & r - 2y - A 0 (5.27) Subtract (5.2S) Irom (5.27) to...

## The M Arginal Rate Of Technical Substitution

An isoquant depicts the different combinations of inputs A' and L that can be used to producc a specific level of output Q. One such isoquant for the output level Q - 2144 is (0) Use implicit differentiation from Section 3.9 to find the slope of the isoquant dKldL which in economics is called the marginal rate of technical substitution (MRTS). (b) Evaluate the marginal rate of technical substitution at K - 256. L 108. a) Take the derivative of each term with respect to L and treat A' as a...

## 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...

## 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...

## 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* -...

## 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...

## Optimizing Multivariable Functions

For cach of the following quadratic function . 1 find the critical points at which the function may be optimized and 2 determine whether at these points the function is maximized, is minimized, is at an inflection point, or is at a saddle point. a z - 3 ' xy ly2 - 4. - ly 12 1 Take the first-order partial derivatives, set them equal to zero, and solve simultaneously, using the methods of Section 1.4. 2 Take the second-order direct partial derivatives from 5 5 and 5.16 . evaluate them at...

## 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...