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The short-run production

Question 2 and 3 The short-run production function of a competitive firm is given by = 6!!where L is the amount of labor it uses. (For those who do not know calculus—iftotal output is aLb , where a and b are constants, and where L is the amout of somefactor of production, then the marginal product of L is given by the formula abLb-1.)The cost per unit of labor is w=6 and the price per unit of output is p=3.(a) Plot a few points on the graph of this firm’s production function and sketch thegraph of the production function, using blue ink. Use black ink to draw the EC 301 Madraisoprofit line that passes through the point (0, 12), the isoprofit line that passesthrough (0,8), and the isoprofit line that passes through the point (0, 4). What isthe slope of each of the isoprofit lines? How many points on the isoprofit linethrough (0, 12) consist of input-output points that are actually possible? Markthe part of the isoprofit line through (0,4) that consists of outputs that areactually possible.(b) How many units of labor will the firm hire? How much output will it produce? Ifthe firm has no other costs, how much will its total profits be?(c) Suppose that the wage of labor falls to 4, and the price of output remains at p.On the graph, use red ink to draw the new isoprofit line for the firm that passesthrough its old choice of input and output. Will the firm increas its output atthe new price? Explain why referring to your diagramA firm uses labor and machines to produce output according to theproduction function , = 4!!!!where L is the number of units of labor used and M is the number of machines. Thecost of labor is $40 per unit and the cost of using a machine is $10. (Also note that themarginal product of labor is equal to 2 !!!!!!and the marginal product of machinery isequal to 2 !!!!!!)(a) On a graph, draw an isocost line for this firm, showing combinations ofmachines and labor that cost $400 and another isocost line showingcombinations that cost $200. What is the slope of these isocost lines?(b) Suppose that the firm wants to produce its output in the cheapest possible way.Find the number of machines it would use per worker. (Hint: The firm willproduce at a point where the slope of the production isoquant equals the slopeof the isocost line.)(c) On the graph, sketch the production isoquant corresponding to an output of 40.Calculate the amount of labor and the number of machines that are used toproduce 40 units of output in the cheapest possible way, given the above factorprices. Calculate the cost of producing 40 units at these factor prices: C(40, 10,40)=?(d) How many units of labor and how many machines would the firm use toproduce y units in the cheapest possible way? How much would this cost?(Hint: Notice that there are constant returns to scale.)

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