1. A firm has the following production function: = 150 0.25 0.75 where Q = total output per day, K = machine hours per day and L = worker hours per day. The price of labour (w) is 20 while the price of capital (r) is 5. The firm wants to minimise the costs of producing 1500 units of output. a. Using the Lagrange method, find the cost minimising quantities of labour and capital the firm should employ to produce 1500 units of output per day. (You must show all working and clearly explain the economic reasoning). b. Using the total cost function, calculate the total cost of producing 1500 units of output using the cost minimising method? (You must show all working and clearly explain the economic reasoning). c. Using the Lagrange multiplier, calculate the marginal cost of producing an additional unit of output at the cost minimising point? (You must show all working and clearly explain the economic reasoning). d. Assume the price of labour (w) decreases from 20 to 10 while the price of capital (r) remains unchanged. Holding total cost constant, calculate both (i) the cost minimising method of production and (ii) the output level, at the new lower price of labour. You must use the Lagrange method to calculate your answers. (You must show all working and clearly explain the economic reasoning.) e. Assume now that total output remains unchanged instead of total cost. Using the input demand function for a Cobb-Douglas production function (holding output constant) calculate the demand for labour and capital at this new lower price of labour. What is the total cost of using this new method production? (You must show all working and clearly explain the economic reasoning). f. Illustrate your answers to all the previous parts to this question on a clearly labelled isoquant/isocost diagram. In particular, illustrate and explain. i. The intercepts and slope of the initial isocost. ii. The initial cost minimising method of producing 1500 units of output. iii. The intercepts and slope of the isocost after the change in the price of labour, holding total output constant at 1500 units. iv. The new cost minimising method of producing 1500 units after the change in the price of labour. v. The intercepts and slope of the isocost after the change in the price of labour, holding total cost constant. vi. The new cost minimising method of output after the change in the price of labour, holding total cost constant. vii. The scale and substitution effects of the fall in the price of labour g. Derive a labour demand curve from your isoquant/isocost diagram, holding output constant at 1500 units. h. Explain why total costs are unlikely to remain constant at the firm's profit maximising response to the fall in the wage rate. Illustrate the implications for the isoquant/isocost diagram carefully explaining any assumptions you have made.