By Percus J.K.
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L=-=Pt P2 and h1 Pt h2 = P2 The sufficiency condition requires that h 11 h 12 -p 1 h12 -pl h22 -p2 -p2 0 <0. , since p. h22 h2 hi h2 >0. h22 -h2 -hi -h2 0 after multiplying the third row and third column by - 1. Comparing this with the determinant in the previous example, we see that the two problems are essentially the same, or one is the dual of the other. We notice that p. is the reciprocal of A of the previous example. In fact, the total differential of is dR = Pidqi +P2dq2 =A(hidqi + h2dq2) by substituting p 1 =Ah 1 and p 2=Ah2 from the last example.
N(x) 2 x. In actual practice, we frequently shift the graph so that the origin. falls on the point (x,j). Then, instead of the line y =ax+ b, we consider the line y=a(x-x)+b. - n(x) z(x-x). This line is called the regression line of y on x. In the above formulation we considered the vertical deviations of the points on the line from the ,observed points. We assumed that the variable x could be measured quite accurately whereas y was subject to random errors. If, instead, we treat x as a function of y and consider the horizontal deviations, we would be interested in a line of the form x = a'(y- j) + b' and we would get a corresponding regression line of x on y.
Functions of Two or More Variables (with Constraint) where r 1 and r 2 are the prices of X 1 and X2 and a is the price of any fixed inputs also needed for the production of Q. For a given C=c0 = r 1x 1 + r2 x 2 +a, the entrepreneur wishes to maximize q. 2 = '2 which means that the ratio of the marginal productivities of X 1 and X2 must be equal to the ratio of their prices. 2 rl r2, which states that A is equal to the contribution of the money spent on each unit of input to output. Finally, dC=r 1dx 1 + r2dx 2 I = xU~dxl + f2dx2).