Question

Consider the function: Y = X_A^1/4 * X_B^3/4

a) Find the value of the function at the point (81,16).

b) Find the first order partial derivatives and evaluate them at the point (81,16). Interpret your results.

c) Suppose that xA decreases 3 units and xB increases 4 units. Using the small increments formula, calculate the impact of these changes in the value of the function y.

d) Suppose that we don’t want to change the value of y, but we want to substitute xA for xB. What is the rate we can substitute xA for xB (meaning: how many units of xB we need to substitute 1 unit of xA)?

Answer 1

We are given the function: Y = X_A^1/4 X_B^3/4

a) The value of the function at the point (81,16) is given by

Y(81, 16) = (81)^(1/4) x (16)^(3/4) = 3 x 8 = 24.

Hence the value of Y at (81, 16) is 24

b) The first order partial derivatives are

dY/dXA = (1/4)XA^(-3/4) * XB^(3/4) = 0.25(XB/XA)^0.75

dY/dXB =  (3/4)XA^(1/4) * XB^(-1/4) = 0.75(XA/XB)^0.25

Find the value of these at the point (81,16)

dY/dXA = 0.25*(16/81)^0.75 = 0.0741

dY/dXB = 0.75*(81/16)^0.25 = 1.125

The value of the function when XA is increased by 1 unit (from XA = 81 and XB = 16) is 0.0741 while the value of the function when XB is increased by 1 unit (from XA = 81 and XB = 16) is 1.125. This happens because the weightage/exponential value of XB is greater at 0.75. Hence the effect of any change in XB on Y is greater than that of XA.  

c) Suppose that xA decreases 3 units and xB increases 4 units.

Since XA decreases by 3 units, Y decreases by 3 x 0.0741. When XB increases by 4 units, Y increases by 1.125 x 4. Net increase in Y = 1.125 x 4 - 3 x 0.0741 = 4.2777

d) Suppose that we don’t want to change the value of y, but we want to substitute xA for xB. The rate we can substitute xA for xB is given by (dY/dXB) / (dY/dXA) = [0.75(XA/XB)^0.25]/[0.25(XB/XA)^0.75] = 3XA/XB.

The units of xB we get when we substitute 1 unit of xA is 1/3.