Calculate a firms utility function

Question

A firm has the following utility function this is the function u(x,y) = K - aln(x) -bln(y) where a equals 3.95, b equals 7.47 and k equals 137.5. x and y are the number of units of two different items under the following budget constraint: mx+ny = c, where m = 5, n = 14, and c= 43

Determine the maximum utility under these contidions

Answer 1

u(x,y) = 137.5 - 3.95lnx - 7.47lny and 5x + 14y = 43; x = 8.6 - 2.8y;

u(x) = 137.5 - 3.95ln(8.6 - 2.8y) - 7.47lny; u'() = -3.95(- 2.8)/(8.6 - 2.8y) - 7.47/y = 11.06/(8.6 - 2.8y) - 7.47/y = 0; 11.06y - 7.47(8.6 - 2.8y) = 0; 11.06y + 20.916y - 64.242 = 0; 31.976y = 64.242; y = 2.00907 = 2.

You can use 1st derivative test to check that at y = 2 and x = 8.6 - 5.6 = 3 u(x, y) = 137.5 - 3.95ln3 - 7.47ln2 =

127.98 = 128 maximum