A Firms CobbDouglas production function takes the following

A Firm\'s Cobb-Douglas production function takes the following form: q(k, l) = 100 k^3/4 l^1/4. Rent (r), the cost of capital, is $200 per unit. Wage (w), the cost of labor, is $250 per unit. Use the LaGrange Multiplier Method to find q*, the maximum possible production, k*, the optimal amount of capital employed, and l*, the optimal amount of labor employed, if the firm\'s budget is $50,000.

Solution

Given,

rent=200 , wage= 250,

Therefore, if q=100k^3/4 l^1/4

   ^{50000x3/4=37500}

^{Labor= 50000x1/4= 12500}

^Therefore,

^{Maximum output q= 100x37500x12500}

^{= 46875000000 quintals}

^{If q=100k3/4l1/4}

^{dq/dl= l 1/4}

   ^{=1/4x250x50000= 3125000}

^dq/dk=k3/4

   ^{3/4x200x50000}

^{= 750000}

^{Hence capital max would be 750000}

^{and labor=3125000}

^By,

^{Nishant Bhatt}