Question 5 A manufacturing company produces the quantity q of a product that depends on L units of labour and a capital amount K as given by the equation 1 q = 6 (L)²² (K) ² Challong Labour costs are $10 per labour unit and capital costs are $20 per unit of capital and the total cost budget is $3,000. (a) Find the optimal solution using Lagrange Multipliers. (b) Demonstrate the following economic principle for the optimal solution found in (a) that The marginal productivity of labour / The marginal productivity of capital = (29) / (32) = Cost per unit of labour / Cost per unit of capital (c) Recompute the optimal values for L and K when the budget is increased by $1 and check that this allows for the production of an extra λ units where A is the Lagrangian multiplier.

.A company can sell its only product and in a perfectly competitive market at a price of 15. To produce it, it must purchase two production inputs: labor (x1) and machinery (x2). Initially, the company only has 1 machine and it is not possible to change this in the short term. Inputs are purchased in competitive markets with respective prices (w1, w2) = (5,10). The production function is as follows: 3 2/3 1/3 f(x,x2) a) Present the problem of maximization of the company. b) Find the first order condition that must be solved to find the optimal level of supplies. c) Find the optimal input level x1 that the company must contract in the above problem to maximize your profits.

The daily output at a factory is Q(K, L) = 144KL units, where K is the capital investment measured in units of $1000 and L is the size of the labor force measured in worker-hours. (a) Compute the daily output if the capital investment is $139,000 and the size of the labor force is 1025 worker-hours. (b) What will the output be if both the level of capital investment and the size of the labor force are cut in half?