Economics I Lecture: Anna Della Valle TA Andrea Venegoni. Tutorial 4 Production theory, theory of the firm

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1 Economics I Lecture: Anna Della Valle TA Andrea Venegoni Tutorial 4 Production theory, theory of the firm PROBLEM 1 Consider the following investment financed with equity and debt. Calculate the expected rate of return for stockholders and bondholders. Assumptions: Investment: $400,000 Financing: $200,000 equity $200,000 debt (backed in full by collateral value of capital equipment or plant) Interest rate on debt: 10% Prob. Of Success: 50% Prob. Of Failure: 50% Outcome if success: $700,000 Outcome if failure: The investment is liquidated for $200,000 (resale value of the capital equipment or plant and paid back to debt holders.) Important Note: Debt is always senior to equity. That is, if the investment runs into trouble, the debt holders get paid first, i.e. they have first claim on any proceeds associated with the investment. From those proceeds, debtholders first get back their principal (i.e. the original amount loaned), then interest if there is money left over after the principal is reimbursed). Equity holders get paid only if there is money left from the investment proceeds over after debtholders are paid. SOLUTIONS: Payoff to Debtholders: Success: $600,000 ($500, *$500,000 = principal + interest) Failure: $500,000 (get $500,000 principal back) Expected payoff: $550,000 (.5*$600,000+.5*$500,000) Initial investment: $500,000 Expected Net payoff: 10% ($50,000/$500,000) Payoff to Stockholders: Success: $1,400,000 ($2,000,000 $600,000 paid to bondholders) Failure: $0 Expected payoff: $700,000 (.5*$1,400,000+.5*$0) Initial investment: $500,000 Expected Net payoff: 40% ($200,000/$500,000)

2 PROBLEM 2 A firm s total cost function is given by the equation: TC = Q + 40Q 2 Based on the TC function above, write the value or the expression (formula) for each of the following cost measures: Average total cost (ATC) Fixed cost (FC) Average fixed cost (AFC) Variable cost (VC) Marginal cost (MC) Please explain how to determine the quantity that minimizes the average total costs and calculate it. If marginal cost is > average total cost, what will be the behavior of the average total cost? In such a situation, we will have economies or diseconomies of scale? SOLUTION Average total cost (ATC) = ( Q +40Q 2 ) /Q Fixed cost (FC) = Average fixed cost (AFC) = 16000/Q Variable cost (VC) = 20Q + 40Q 2 Marginal Cost (MC) = Q b. ATC is minimized where MC = ATC: Equating MC to ATC yields: ( Q +40Q 2 ) /Q = Q Q +40Q 2 = 20Q + 80Q = 40Q = Q 2 Q = 20 So, ATC is minimized at 20 units of output. When MC > ATC ATC is rising. That is due by the fact that the cost of producing an additional unit of the good will be higher than the average total cost per unit of output already realized. This makes the average total cost increase. In this case we can say that we are observing diseconomies of scale, because the increase of units of output makes my average total costs increase (The ATC curve is upward sloping)

3 PROBLEM 3 1. The cost function for a firm is C(Q) = Q where C is the total cost (in $) per year and Q is the total output per year. a. What is the firm s annual fixed cost? b. At a level of production Q = 10, what is the firm s average cost (ATC) and marginal cost (MC)? c. Does this firm s cost function exhibit economies of scale, and, if so, over which level of production? d. At what level of output Q does this firm s marginal cost intersect its average cost. (Do not answer by simply saying at its minimum. ) SOLUTION a. Annual fixed cost = $600 b. At Q = 10, the firm s average cost = C(Q)/Q [ (10)]/10= $75 The firm s marginal cost = C(Q)/ Q = 15. (Note: the marginal cost for this cost function is constant. It does not vary with the level of output.) c. A firm exhibits economies of scale if its average cost is declining. Average cost for this firm = ( Q)/Q = 600/Q + 15 From observation of the average cost function above, we see that average cost will decline indefinitely as Q increases. (It will decline approaching but never reaching 15.) Therefore, yes, this firm s cost function exhibits economies of scale and these scale economies are present over its entire production process or level of output. d. At no level of output, because the average cost is everywhere declining and therefore the marginal cost will always be < average cost and the marginal cost curve (which is really a horizontal line at $15) will never intersect the average cost curve. In fact MC = 15 and AC = 600/Q + 15 which is always > 15. Therefore, MC is always less than AC. PROBLEM 4 2. Suppose the cost function of a company is given by the following equation: TC(Q) = Q 2 a. What is the company s fixed cost? b. What is the company s variable cost? c. At production level Q = 10what is the average fixed cost AFC(Q), the average variable cost AVC(Q), the average total cost ATC(Q) and the marginal cost MC(Q)? d. Does this cost function exhibit economies of scale? If so, over what range of Q? e. At what level of Q does the average cost = marginal cost? Answer: a. Fixed cost = 400. b. The variable cost = 0.5Q 2. c. At Q = 10, AFC(Q) = 400/10= $40 The AVC(Q) = 0.5Q 2 /Q = 0.5Q At Q =10, AVC(Q) = 0.5(10) 2 /10 = 5 ATC(Q) = AFC(Q) + AVC(Q) At Q = 10, AFC(Q) = 40 and AVC(Q) = 5 ATC(Q) = = 45 Alternatively, we can find ATC(Q) = TC(Q)/Q

4 At Q = 10, ATC(Q) = [ (10) 2 ]/10= $45 MC(Q) = TC(Q)/ Q = Q = 1(10) = $10 To compute MC(Q), you have to obtain the first derivative of the cost function with respect to Q. d. For this cost function, the average cost ATC(Q) = [ Q 2 ]/Q = 400/Q + 0.5Q To determine whether the ATC function declines as Q increases is necessary to recall the relationship between the AC and the MC to establish whether the ATC reaches a minimum. That is, AC(Q) reach a minimum where ATC(Q) = MC(Q). Economically, it is easy to understand that until the marginal costs (MC) are lower than the average total costs (ATC), to produce a unit more of the good makes the average total costs decline. Such a situation is defined as economies of scale, and we have for every quantity that makes MC<ATC. e. ATC(Q) = 400/Q + 0.5Q and MC(Q) = Q Set ATC(Q) = MC(Q): 400/Q + 0.5Q = Q 400/Q = 0.5Q 0.5Q 2 = 400 Q 2 = 400/0.5 Q 2 = 800 Q = 800 Q = Thus ATC(Q) = MC(Q) at Q = That is, the ATC and MC curves intersect at Q = It is also the point at which the ATC curve reaches its minimum. Knowing that a firm exhibits economies of scale when its marginal costs (MC) are lower than its average total costs, it is immediate to conclude that a firm with such an ATC function exhibits economies of scale for Q = 0 to Q =

5 PROBLEM 5 The McGrady Enterprise that produces cutlery wants to assess which is the right level of production that will allow it to minimize its average production costs in order to make its business more efficient. The use of the plant and the machineries costs $ while for each unit produced the cost to be substained is 50Q + 100Q 2 Which is the cost function for the McGrady Enterprise? Calculate fixed costs, average fixed costs, variable costs and marginal costs for the production function of McGrady enterprise. How can we determine the quantity to be produced in order to minimize costs? d)calculate the quantity that minimizes the average total costs. SOLUTION a) The cost function for McGrady Enterprise is: Q + 100Q 2 b) Fixed Costs = Variable Costs = 50Q + 100Q 2 Average Total Costs = ( Q + 100Q 2 )/Q Marginal Costs= Q c) To determine the quantity to be produced in order to minimize the average total costs we have to calculate the quantity that makes marginal costs equal average total costs. ( Q +100Q 2 ) /Q = Q Q +100Q 2 = 50Q + 200Q = 100Q = Q 2 Q = 50 So, ATC is minimized at 50 units of output.

6 PROBLEM 6 Define what is meant by economies of scope and economies of scale. Calculate the degree of economies of scope for DELTA enterprise obtained by producing jointly 1000 Belts and 1000 Shoes given that their joint production function is: C(B,S)= B + 20S Derive the individual production function. SOLUTION By economies of scope we mean the cost reduction obtained by the joint production of two goods. If producing the goods together yields a reduction in costs compared to the separate production of the same goods we can say we have an economy of scope. By economies of scale we mean the average total cost decreases as the quantity produced increase. C(B)= B C(S)= S C(B,S)= * *1000 = C(B) + C(S) = B S = * *1000 = = = The degree of economies of scope is given by: [C(B)+C(S) C(B,S)]/C(B,S) = ( )/ = 51,6% SAMPLE PROBLEMS 1. The cost function for a firm is C(Q) = 190, Q where C is the total cost (in $) per year and Q is the total output per year. a. What is the firm s annual fixed cost? b. At a level of production Q = 10,000, what is the firm s average cost (AC) and marginal cost (MC)? c. Does this firm s cost function exhibit economies of scale, and, if so, over which level of production? d. At what level of output Q does this firm s marginal cost intersect its average cost. (Do not answer by simply saying at its minimum. ) Answer: a. Annual fixed cost = $190,000 b. At Q = 10,000, the firm s average cost = C(Q)/Q [190, (10,000)]/10,000 = $72 The firm s marginal cost = C(Q)/ Q = 53. (Note: the marginal cost for this cost function is constant. It does not vary with the level of output.) c. A firm exhibits economies of scale if its average cost is declining.

7 Average cost for this firm = (190, Q)/Q = 190,000/Q + 53 From observation of the average cost function above, we see that average cost will decline indefinitely as Q increases. (It will decline approaching but never reaching 53.) Therefore, yes, this firm s cost function exhibits economies of scale and these scale economies are present over its entire production process or level of output. d. At no level of output, because the average cost is everywhere declining and therefore the marginal cost will always be < average cost and the marginal cost curve (which is really a horizontal line at $53) will never intersect the average cost curve. In fact MC = 53 and AC = 190,000/Q + 53 which is always > 53. Therefore, MC is always less than AC. 2. Suppose the cost function of a company is given by the following equation: TC(Q) = 200, Q 2 where Q is measured in lbs. and C(Q) is measured in $. a. What is the company s fixed cost? b. What is the company s variable cost? c. At production level Q = 10,000 lbs, what is the average fixed cost AFC(Q), the average variable cost AVC(Q), the average total cost ATC(Q) and the marginal cost MC(Q)? d. Does this cost function exhibit economies of scale? If so, over what range of Q? e. At what level of Q does the average cost = marginal cost? Answer: a. Fixed cost = $200,000. b. The variable cost =.50Q 2. c. At Q = 10,000, AFC(Q) = 200,000/10,000 = $20 The AVC(Q) =.50Q 2 /Q = $0.50Q Note: with this functional form of the variable component of the cost function, the AVC is not the same for all levels of output Q but varies with Q. At Q =10,000, AVC(Q) =.50(10,000) 2 /10,000 = $5,000 ATC(Q) = AFC(Q) + AVC(Q) At Q = 10,000, AFC(Q) = $20 and AVC(Q) = $5000 ATC(Q) = $20 + $5000 = $5020 Alternatively, we can find ATC(Q) = TC(Q)/Q At Q = 10,000, ATC(Q) = [200, (10,000) 2 ]/10,000 = $5020 MC(Q) = TC(Q)/ Q = 1Q = 1(10,000) = $10,000 Note: the MC(Q) is the first derivative of the cost function with respect to Q. d. For this cost function, the average cost ATC(Q) = [200, Q 2 ]/Q = 200,000/Q Q At first glance it is difficult to determine whether the ATC function declines as Q increases since the first part of the function (200,000/Q) declines but the second part (0.50Q) increases as Q increases. So, one way of answering the question is by using the relationship between the AC and the MC to establish whether the ATC reaches a minimum. That is, AC(Q) reach a minimum where ATC(Q) = MC(Q). Then, we can establish that for levels of Q less than the Q that solves the above equality, the ATC function exhibits economies of scale. See answer to part (e) below. Note: Some students may attempt to evaluate the ATC function and/or plot the curve to determine whether and over what range it exhibits economies of scale. This is an OK approach, although it is a bit cumbersome to do (unless they are using a graphing calculator) but is fine. e. ATC(Q) = 200,000/Q Q and MC(Q) = Q Set ATC(Q) = MC(Q): 200,000/Q Q = 1.00Q

8 200,000/Q = 0.50Q 0.50Q 2 = 200,000 Q 2 = 200,000/0.50 Q 2 = 400,000 Q = 400, 000 Q = Thus ATC(Q) = MC(Q) at Q = That is, the ATC and MC curves intersect at Q = It is also the point at which the ATC curve reaches its minimum. Thus, the ATC function exhibits economies of scale for Q = 0 to Q =