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Math Help - Second-order total differential

  1. #1
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    Second-order total differential

    Consider the production function:
    y(x1, x2) = x2/1 + x2/2

    * i put the "/" signs just to show the first number above the 2nd, its not actually divided*
    can someone try to explain how I go about finding the second-order total differential to find if it is concave or convex?
    I'd appreciate it if you can explain as simply as possible(in words not math)

    thanks a lot in advance!
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  2. #2
    MHF Contributor

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    Do you mean y(x_1, y_1)= x_1^2+ x_2^2? If so, a better notation would be y(x_1, x_2)= x_1^2+ x_2^2.

    The total differential is dy= 2x_1 dx_1+ 2x_2 dx_2 and the "second order" total differential is 0 because there is no term "mixing" the two variables.

    If the problem had been y(x_1, x_2)= x_1^2+ 3x_1x_2+ x_2^2, then the total differential would have been dy= (2x_1+ 3x_2)dx_1+ (3x_1+ 2x_2)dx_2 and the second order differential would have been 3 dx_1dx_2
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