Question

The total revenue function for two goods is given by the equation TR = 36 x-3x²+ 56y- 4y².find the number of unit of each goods which must be sold. if profit is to be maximize when the firm subject to a budget constraint 5 x + 10 y = 80in Lagrange multiplayer method

Answer 1

Total revenue function:

$TR=36x-3x^2+56y-4y^2$

Budget constraint:

$5x+10y=80$

$x+2y=16$

$x=16-2y$

Substituting the value of x in the total revenue equation gives:

$TR=36(16-2y)+3(16-2y)^2+56y-4y^2.$

Differentiating the above equation w.r.t y:

$\frac{\delta TR}{\delta y}=-72y+ 6(16-2y)+56+8y$

$=-104y+248=0$

$\implies$ $y=2.39$

$x=16-2y$

$=16-2(2.385)=11.23$

$x=11.23$

Thus, the quantities of goods x and y to be sold are 11.23 and 2.39 respectively.