Maria's Umbrellas has a production function given by Q = L0.5K0.5. The wage (W) is $80 per day and the rental per unit of capital (R) is $5 per day. In the long run, how many units of capital will Maria want to buy for each unit of labor?

Suppose that a firm has the following production function: q = 2L + K. The wage for a $10 per hour and the rental rate of capital is r = worker is w $10 per hour. In the short run the firm amount of capital is fixed at K = 20 units. If the firm's goal is to minimize their cost of producing 100 units, then in the short run they will hire units of labor and in the long run they will hire units of labor. (Round answers to two decimals if necessary.)

15. Apple uses labour and machines to produce iPhones according to the production function y = LM, where L is the number of units of labour used and M is the number of machines. The cost of labour is w₁ = 4 per unit and the cost of using a machine is wM = 16. Apple faces a fixed cost of F = 25 for renting its factory. Given these input prices, Apple chooses how much labour and machines to employ in order to minimize the cost of producing any given amount of iPhones y. Apple is therefore solving a cost-minimization problem. (a) If Apple is cost-minimizing, how many workers will it use for each machine? That is, find the number of workers per machine that Apple will employ. (Hint: setup the tangency condition.) Continuing to assume that Apple produces any given number of iPhones y in the cheapest way possible, use the equation defining the number of workers per machine from part (a) and Apple's production function to find expressions for the cost-minimizing choices of labour and machines…

A firm has the production function Q = 20K0.2 10.8, where Q measures output, K represents machine hours, and L measures labor hours. If rental rate of capital is R=$15, the wage rate is W = $10, and the firm wants to produce 40,000 units of output, what is the cost- minimizing bundle of capital and labor? Hints: 60.8 4.19