PART A: For each worker, determine that worker's marginal product of labor.


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1 ECON 3310 Homework #4  Solutions 1: Suppose the following indicates how many units of output y you can produce per hour with different levels of labor input (given your current factory capacity): PART A: For each worker, determine that worker's marginal product of labor. Labor Output Marginal Product PART B: Suppose you can sell each unit of y that you produce for $1, and suppose that you can hire workers at a rate of $5 per hour. How many workers will you hire? How much output will you produce per hour? How much money will you make? You will hire 7 workers. (Or 6 workers; The 7 th worker produces 5 units of goods worth $1 each, and I have to pay him $5. So you will break even on the 7 th worker and will be indifference as to whether or not you hire this 7 th worker.) You will therefore produce output of 75 or 80 units per hour, and you will make a total of $45 (after paying for labor). PART C: Suppose instead that you have to pay workers $12 per hour. How does your answer change? You will hire 5 workers, produce 65 units per hour and make $5. PART D: Suppose instead that you have to pay workers $20 per hour. How does your answer change? At this new wage you would choose to not produce at all.
2 PART E: Use the information above to derive the labor demand curve. wage labor (# workers) Note: the above assumes that labor is perfectly divisible, rather than being only available in the discrete units in the table; If only discrete whole numbers of workers are allowed, then the labor demand will look as shown below: w labor Question 2  PART A: True or False: Average costs can never rise when marginal costs are below average costs. (Explain) TRUE: The only way an average can rise is for the marginal quantity to be above that average. The example given in class is that of grades  the only way the (marginal) final exam grade can raise your average is for it to be higher than your average for the class up to that date. Similarly, the only way the average cost of production can rise when we produce one more unit is if that unit costs more to produce than the previous average.
3 PART B: True or False: Average costs can never rise while marginal costs are falling. (Explain) FALSE: As long as marginal cost (MC) lies above the average cost, it will lift the average cost. The following graphical example is logically possible: MC AC PART C: True or False: A producer would never produce when his profits are zero because, since he is not making anything, he would be better off just doing nothing and enjoying life. (Explain) An economic profit of zero just means that the producer is making as much in this business as he could be making in his or her next best alternative business. If he stopped working, he would be making nothing and would thus be making negative economic profit. Therefore the answer is FALSE. QUESTION 3: Suppose a firm faces the following short run total costs (and suppose the firm can only produce integer units of y): Y TC PART A: Construct a table in which you show for each level of y the fixed cost, the variable cost, the average total cost, the average variable cost, and the marginal cost. Y TC FC VC AC AVC MC
4 PART B: Suppose you are a price taker and face a market price of $6 per unit. How much will you produce? How much profit will you make? You will produce at the point where MC = MR (which is $6). You will therefore produce 7 units. You total profit will be: (Note: Fixed Costs are not economic cost in the short run so we use the following formula to calculate profit) TRVC = Profit = Revenue TC = $13 Therefore you will make a profit of $13, and produce 7 units. PART C: Suppose the price rises to $9. How much will you produce now? At price = $9, you will produce 9 units (or 8). The new profit level will be $36. PART D: Suppose the price falls to $4. How much will you produce now in the short run? At price = $4, you will produce 7 units (or 6) in the short run. Your profit will be as follows: profit = revenue TC = 4*7 29 = $1 Notice that this profit is negative. Nonetheless, you will continue to produce (7 units) in the short run. This is because you are covering your variable costs. By shutting down, you would lose more money ($6 due to the fact that your fixed costs are sunk). You could therefore define their short run profit as being: SR profit= revenue VC = 4*7 23 = $5. Since Revenue Variable Costs > 0, you should continue to produce in the short run. (In long run, assuming nothing changes, you will leave the industry.) QUESTION 4. Suppose the hourly wage for labor (l) is $5 and the price of each unit of capital (K) is $25. The price of output is constant at $50 per unit. The production function is q = l 1/2 K 1/2 so that the marginal product of labor is MP l =1/2 (K/l) 1/2 a. If the current capital stock is fixed in the short run at 1,600 units, how much labor should the firm employ in the short run? How much output is produced? What is the firm s short run profit? MRP L = p*mp L = w 50 * (1/2) (K/L) 1/2 = 5 50 * (1/2) (1600/L) 1/2 = 5 5 * 40 = L 1/2 L= = 40,000
5 Q = L 1/2 K 1/2 Q = (40,000) 1/2 (1600) 1/2 = 200*40 = 8,000 SR Profit = p*q w*l = (50)(8,000) (5)(40,000) = 200,000 QUESTION 5. Consider a firm that specializes in typing papers for professors who do not know how to type. Suppose that the technology for the firm is such that it has two inputs: secretaries and computers. Each secretary needs a computer in order to produce something, and no secretary can work on more than one computer at a time. Thus, computers and secretaries are perfect complements in production. However, secretaries can specialize with some typing text, others typing equations, yet others drawing graphs on the computer. Thus, as the number of secretaries and computers goes up by x%, output of typed papers goes up by more than x%. a. Suppose the firm currently employs one secretary and one computer, and this secretary is therefore able to type 2 papers per day. What is the daily marginal product of this first secretary? Given the technology of the firm (as described above), what is the marginal product of the second secretary? Is the marginal product declining? The daily marginal product of the first secretary is 2 papers. The marginal product of any additional secretaries is zero, as each secretary needs their own computer in order to produce anything. b. Now suppose that both a second secretary and a second computer are hired. As a result of specialization, the two secretaries together can now type 5 papers per day. Does this firm have increasing or decreasing returns to scale in the current range of its production function? The firm has increasing returns to scale. By doubling the inputs, the output is more than doubled. Note that in this example, we have declining marginal products, but increasing returns to scale. This is because marginal product is defined with respect to an individual input, holding other inputs constant. Returns to scale is defined in terms of varying ALL inputs.
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