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After reading Chapter 10 and working the problems




Learning Objectives;After reading Chapter 10 and working the problems for Chapter 10 in the textbook and in;this Student Workbook, you should be able to;Specify and estimate a short-run production function using a cubic specification of;the production function.;Specify and estimate a short-run cost function using a cubic specification.;In order to accomplish these goals, Chapter 10 shows you how to;Estimate the parameters of a cubic short-run production function by using the;technique of regression through the origin.;Find the region of diminishing returns.;Estimate the output level at which AVC reaches its minimum value.;Estimate the parameters of a cubic short-run total variable cost equation along with;the associated average variable cost and marginal cost equations.;Essential Concepts;1.;The cubic empirical specification for a short-run production function is derived;from a long-run cubic production function. The cubic form of the long-run;production function is expressed as;Q = aK 3 L3 + bK 2 L2;2.;The properties of a short-run cubic production function (Q = AL3 + BL2) are;a.;Holding capital constant at K units, the short-run cubic production function is;derived as follows;Q = aK 3 L3 + bK 2 L2;= AL3 + BL2;where A = aK 3and B = bK 2;b.;The average and marginal products of labor are respectively;AP = Q / L = AL2 + BL and MP = Q/ L = 3AL2 + 2 BL;Chapter 10: Production and Cost Estimation;205;c.;Marginal product of labor begins to diminish beyond Lm units of labor and;average product of labor begins to diminish beyond La units of labor, where;Lm =;d.;B;B;and La =;3A;2A;In order to have the necessary properties of a production function, the;parameters must satisfy the following restrictions;A 0;3.;To estimate a cubic short-run production function using linear regression analysis;you must first transform the cubic equation into linear form;Q = AX + BW;where X = L3 and W = L2. In order to correctly estimate the cubic equation, the;estimated regression line must pass through the origin, that is, when L = 0, Q = 0.;Regression through the origin simply requires the analyst to specify in the;computer routine that the origin be suppressed.;4.;Short-run cost functions should be estimated using data for which the level of;usage of one or more of the inputs is fixed. Usually time-series data for a specific;firm are used to estimate short-run cost functions.;5.;Collecting data may be complicated by the fact that accounting data are based on;expenditures and may not include the firms opportunity cost of using the various;inputs. In particular, capital costs should reflect not only acquisition cost but also;the rental income forgone by using (rather than renting) the capital, the;depreciation, and any capital gain or loss.;6.;The effects of inflation on cost data must be eliminated. To adjust nominal cost;figures for inflation, divide each observation by the appropriate price index for that;time period.;7.;The properties of a short-run cubic cost function (TVC = aQ + bQ 2 + cQ3) are;a.;The average variable cost and marginal cost functions are, respectively;AVC = a + bQ + cQ 2 and SMC = a + 2bQ + 3cQ 2;b.;Average variable cost reaches its minimum value at Qm = b / 2c.;c.;To conform to the theoretical properties of a cost function, the parameters;must satisfy the following restrictions;a > 0, b 0;d.;The cubic specification produces an S-shaped TVC curve and -shaped AVC;and SMC curves.;e.;Because all three cost curves (TVC, AVC, and SMC) employ the same parameters, it is only necessary to estimate any one of these functions in order to;obtain estimates of all three curves.;f.;In the short-run cubic specification, input prices are assumed to be constant;and are not explicitly included in the cost equation.;Chapter 10: Production and Cost Estimation;206;Summary of Short-Run Empirical Production and Cost Functions;Short-run cubic production;equations;3;2;Total product;Q = AL + BL;Average product of labor;AP = AL + BL;Marginal product of labor;MP = 3AL + 2BL;Diminishing marginal returns;beginning at Lm =;Restrictions on parameters;A 0;Short-run cubic cost equations;Total variable cost;TVC = aQ + bQ2 + cQ3;Average variable cost;AVC = a + bQ + cQ2;Marginal cost;SMC = a + 2bQ + 3cQ2;Average variable cost reaches minimum at;Restrictions on parameters;Chapter 10: Production and Cost Estimation;207;b;2c;a > 0, b 0;Qm =;Matching Definitions;empirical production function;long-run production function;short-run production function;cubic production function;short-run cubic production function;regression through the origin;nominal cost data;deflating;user cost of capital;1.;2.;3.;The exact mathematical form of the equation to be;estimated.;Production function in which all inputs are considered;variable.;Production function in which at least one input is fixed.;4.;Production function of the form Q = aK 3 L3 + bK 2 L2.;5.;Production function of the form Q = AL3 + BL2.;6.;7.;8.;9.;A regression with the intercept parameter suppressed.;Data that have not been corrected for the effects of;inflation.;The process of correcting for inflation by dividing;nominal data by a price index.;The firms opportunity cost of using capital.;Study Problems;1.;Name the following empirical specifications of production and cost functions;TVC = aQ + bQ 2 + cQ 3;a.;b.;SMC = a + 2bQ + 3cQ 2;c.;Q = aK L + bK L;d.;AVC = a + bQ + cQ;e.;Q = AL + BL;33;3;22;2;2;2.;What restrictions must be placed on the parameters in the empirical production and;cost functions in question 1 above?;3.;A firm estimates its long-run production function to be;Q = 0.008 K 3 L3 + 10 K 2 L2;Suppose the firm employs 15 units of capital.;a.;The equations for the product curves in the short run are;TP =;AP =;MP =;Chapter 10: Production and Cost Estimation;208;b.;c.;d.;4.;At _________ units of labor, marginal product of labor begins to diminish.;At _________ units of labor, average product of labor begins to diminish.;Calculate the marginal product and average product of labor when 20 units of;labor are employed.;MPL = 20 =;APL = 2 0;=;A firm estimates its cubic production function of the following form;Q = AL3 + BL2;and obtains the following estimation results;Q;RSQUARE;FRATIO;PVALUE ON F;62;0.7032;142.175;0.0001;PARAMETER;ESTIMATE;STANDARD;ERROR;TRATIO;PVALUE;L3;-0.050;0.013;3.85;0.0003;L2;0.600;0.250;2.40;0.0195;DEPENDENT VARIABLE;OBSERVATIONS;VARIABLE;INTERCEPT;The firm pays $36 per unit for labor services.;a.;b.;c.;d.;e.;f.;5.;The estimated total, average, and marginal product functions are;Q =;AP =;MP =;Are the parameters of the correct sign and are they significant? Discuss the pvalues.;Average product reaches its maximum value at _________ units of labor.;Average product reaches its maximum value at _________ units of output.;At the output level for part d, AVC = $ __________ and SMC = $ ________.;When labor usage is 7 units, AVC = $ ___________ and SMC = $ ________.;Consider a firm that estimates the following average variable cost function;AVC = a + bQ + cQ 2;The computer printout for the regression analysis is;Chapter 10: Production and Cost Estimation;209;AVC;RSQUARE;FRATIO;PVALUE ON F;16;0.9000;58.50;0.0001;VARIABLE;PARAMETER;ESTIMATE;STANDARD;ERROR;TRATIO;PVALUE;INTERCEPT;75.00;25.00;3.00;0.0102;Q;2.40;0.40;6.00;0.0001;Q2;0.06;0.20;3.00;0.0102;DEPENDENT VARIABLE;OBSERVATIONS;a.;b.;c.;d.;e.;Determine whether the estimate values of the coefficients indicate a;shaped AVC curve at the 5 percent level of significance.;The marginal cost function associated with this AVC function is;SMC = _____________________________.;The total variable cost function associated with this function is;TVC = _____________________________.;AVC reaches its minimum value at Qm = __________.;Minimum AVC = $_________.;-;Computer Problem;Mercantile Metalworks, Inc. manufactures wire carts for grocery stores. The production;manager at Mercantile wishes to estimate an empirical production function for the;assembly of carts using the following time-series data for the last 22 days of assembly;operations. L is the daily number of assembly workers employed, and Q is the number of;carts assembled (completely) for that day. Mercantile pays its assembly workers $160 per;day in wages and benefits.;Day;Number of;carts;assembled;Q;Day;Number of;workers;L;1;2;3;4;5;6;7;8;9;10;11;1.;Number of;workers;L;15;21;24;32;36;38;18;18;41;36;44;75;897;1,280;1,251;1,315;2,837;590;129;1,572;2,005;1,024;12;13;14;15;16;17;18;19;20;21;22;40;21;27;20;15;36;14;24;25;32;21;Number of;carts;assembled;Q;2,165;1,534;835;906;102;1,424;111;868;916;1,341;806;Use a computer regression package or Excel to estimate the following short-run;cubic production function;Q = AL3 + BL2;Chapter 10: Production and Cost Estimation;210;Do the parameter estimates have the appropriate algebraic signs? Are they;statistically significant at the 1 percent level of statistical significance? How well;did the empirical model do in explaining the variation in the number of carts;assembled each day?;2.;What are the estimated total, average, and marginal product functions from your;regression results in Part 1?;3.;At what level of labor usage does average product reach its maximum value? In a;day, how many carts per worker are assembled when average product is;maximized? What is average variable cost when average product is maximized?;4.;What is short-run marginal cost when average product is maximized?;5.;Beyond what level of labor employment does the law of diminishing returns set in?;Beyond what level of output?;Multiple Choice / True-False;1.;Empirical production and cost functions;a.;can be obtained using regression analysis.;b.;require data from actual production operations.;c.;can be used in making profit-maximizing decisions.;d.;are curvilinear functions that can be estimated using regression analysis.;e.;all of the above.;2.;Time-series data for a specific firm are often used to estimate short-run cost;functions because;a.;over the chosen period of time, a firm will not be able to vary the usage of;one or more inputs.;b.;cross-section data would probably include firms with different levels of;capital usage.;c.;time-series data are best suited for investment decisions.;d.;both a and b.;e.;both b and c.;3.;A cubic specification for a short-run cost function is appropriate when the scatter;diagram indicates;a.;an S-shaped short-run marginal cost curve.;b.;total cost increases at an increasing rate throughout the range of output.;c.;an S-shaped short-run total variable cost curve.;d.;an S-shaped short-run average total cost curve.;e.;a -shaped short-run total cost curve.;4.;The user cost of capital includes;a.;acquisition cost.;b.;depreciation from the use of capital.;c.;capital gains or losses.;d.;revenue foregone by using rather than renting the capital.;e.;all of the above.;Chapter 10: Production and Cost Estimation;211;5.;To adjust cost data for the effects of inflation;a.;throw out the observations that occur in years with high inflation rates.;b.;deflate cost figures by dividing by an appropriate price index.;c.;inflate cost figures by multiplying by an appropriate price index.;d.;adjust cost data by dividing by the percentage rate of inflation.;6.;An estimated short-run cost function;a.;would be used to make price and output decisions.;b.;holds the capital stock constant.;c.;can be estimated using time-series data.;d.;all of the above.;7.;For the short-run cost function AVC = a + bQ + cQ2;a.;the AVC curve is -shaped when a 0, and c 0, b 0.;c.;the corresponding SMC function is SMC = aQ + 2bQ2 + 3cQ3.;d.;both a and c.;e.;all of the above.;8.;A potential problem with cross-section cost data is that;a.;nominal cost data include the effect of inflation.;b.;different firms face different input prices.;c.;at least one input is fixed over time.;d.;both a and b.;e.;none of the above.;The next six questions refer to the following;A firm estimated its short-run costs using an average variable cost function of the form;AVC = a + bQ + cQ 2;and obtained the following results. Total fixed cost is $1,000.;AVC;RSQUARE;FRATIO;PVALUE ON F;35;0.8713;108.3;0.0001;VARIABLE;PARAMETER;ESTIMATE;STANDARD;ERROR;TRATIO;PVALUE;INTERCEPT;43.40;13.80;3.14;0.0036;Q;2.80;0.90;3.11;0.0039;Q2;0.20;0.05;4.00;0.0004;DEPENDENT VARIABLE;OBSERVATIONS;9.;The estimated marginal cost function is;SMC = 43.4Q 1.4Q 2 + 0.07Q 3;a.;b.;SMC = 43.4 1.4Q + 0.07Q 2;c.;SMC = 43.4Q 5.6Q 2 + 0.6Q 3;d.;SMC = 43.4 5.6Q + 0.6Q 2;Chapter 10: Production and Cost Estimation;212;10.;If the firm produces 20 units of output, what is estimated AVC?;a.;$19.40;b.;$67.40;c.;$171.40;d.;$179.40;11.;If the firm produces 20 units of output, what is estimated total cost?;a.;$1,348;b.;$1,388;c.;$2,348;d.;$4,428;12.;If the firm produces 12 units of output, what is estimated SMC?;a.;$38.60;b.;$62.60;c.;$105.80;d.;$197.00;13.;At what level of output is AVC minimum?;a.;0.14;b.;4.67;c.;7;d.;28;14.;What is the minimum value of AVC?;a.;$ 24.50;b.;$ 33.60;c.;$ 72.80;d.;$121.80;15.;A cubic specification for a short-run production function is appropriate when the;scatter diagram indicates;a.;an S-shaped total product curve.;b.;marginal product of labor falls throughout the range of labor usage.;c.;total product is decreasing throughout the range of labor usage.;d.;an S-shaped marginal product of labor curve.;e.;a -shaped marginal product of labor curve.;16.;T;F;With cross-section data it is not necessary to correct for inflation.;17.;T;F;Estimation of a cubic short-run cost function requires that the intercept;term be suppressed.;18.;T;F;Input prices are commonly omitted in short-run cost estimation;because the span of the time-series data set is generally short enough;that real input prices do not change much.;19.;T;F;Once one of the three cost curves TVC, AVC, or SMC has been estimated, the other two functions cannot be estimated without drawing a;new sample of data.;Chapter 10: Production and Cost Estimation;213;20.;T;F;Short-run cost functions are used by firms to make investment;decisions while long-run cost functions provide information for output;and pricing decisions.;Answers;MATCHING DEFINITIONS;1.;2.;3.;4.;5.;6.;7.;8.;9.;empirical production function;long-run production function;short-run production function;cubic production function;short-run cubic production function;regression through the origin;nominal cost data;deflating;user cost of capital;STUDY PROBLEMS;1.;a.;b.;c.;d.;e.;short-run cubic cost function;short-run cubic marginal cost function;long-run cubic production function;short-run cubic average variable cost function;short-run cubic production function;2.;a.;b.;a > 0, b 0;same as part a;c.;A = aK 3 0;d.;e.;same as part a;A 0;a.;TP = 0.008(15)3L3 + 10(15)2L2 = 27L3 + 2,250L2;AP = 27L2 + 2,250L;MP = 3(27)L2 + 2(2,250)L = 81L2 + 4,500L;Lm = B/3A = 2,250/3(27) = 27.78 units of labor;La = B/2A = 2,250/2(27) = 41.67 units of labor;MPL=20 = 81(20)2 + 4,500(20) = 57,600;APL=20 = 27(20)2 + 2,250(20) = 34,200;3.;b.;c.;d.;4.;a.;b.;c.;d.;Q = 0.05L3 + 0.6L2;AP = 0.05L2 + 0.6L;MP = 3(0.05)L2 + 2(0.6)L = 0.15L2 + 1.2L;The signs of both parameters are correct: A is negative, B is positive. The p-values;indicate significance at better than the 2 percent level for both parameter estimates.;La = B/2A = 0.6/0.1 = 6;AP reaches its maximum value when 6 units of labor are employed.;Q = 0.05(6)3 + 0.6(6)2 = 10.8;At 10.8 units of output, AP reaches its maximum value.;Chapter 10: Production and Cost Estimation;214;e.;f.;5.;a.;b.;c.;d.;f.;APmax = 0.05(6)2 + 0.6(6) = 1.8 (or APmax = Q/L = 10.8/6 = 1.8);So, AVC = w/AP = 36/1.8 = $20;Since AP = MP when AP is at its maximum value, AVC = SMC = $20 at L = 6 and Q;= 10.8.;When L = 7, AP = 1.75 and MP = 1.05. Thus, AVC = 36/1.75 = $20.57 and SMC =;36/1.05 = $34.29.;The parameter restrictions are: a > 0, b 0. In each case, the absolute;value of the t-ratio is greater than the critical value of 2.160.;SMC = 75 4.8Q + 0.18Q2;TVC = 75Q 2.4Q2 + 0.06Q3;Qm = b/2c = 2.4/0.12 = 20;AVCmin = 75 2.4(20) + 0.06(20)2 = 51;COMPUTER PROBLEM;1.;Yes, A 0. Both A and B are statistically significant at better than the 1 percent;level. The estimated model explained only about 62 percent of the variation in output. The;computer printout looks like this;Q;R-SQUARE;F-RATIO;P-VALUE ON F;22;0.6198;83.72;0.0000;VARIABLE;PARAMETER;ESTIMATE;STD.;ERROR;T-RATIO;P-VALUE;L3;0.04249;0.01491;2.85;0.0099;L2;2.77199;0.55584;4.99;0.0001;DEP. VARIABLE;OBS;2.;3;2;Q = 0.04249 L + 2.77199 L;2;2;AP = 0.04249 L + 2.77199 L = AL + BL;MP = 0.12747 L2 + 5.54398 L = 2 AL2 + 3BL;3.;La = B / 2 A = (2.77199) /(2 0.04249) = 32.62 workers per day;AP (La) = 0.04249(32.62) 2 + 2.77199(32.62) = 45.21 carts per worker;AVC = w / AP = $160 / 45.21 = $3.54 per cart;4.;At maximum AP, AP = MP, so SMC = AVC, and thus SMC = $3.54.You can verify this;result by noticing that MP (at L = 32.62) is 45.21.Thus, SMC = w/MP = $160/45.21 =;$3.54, which is exactly the value found in question 3 for AVC.;5.;Lm = 21.75 workers per day, Q (21.75) = 874 carts per day;MULTIPLE CHOICE / TRUE-FALSE;1.;2.;e;d;3.;c;Empirical production and cost functions are all of these things.;To estimate a short-run production function, at least one input must be fixed, that is;usage of one of the inputs must take the same value for each observation in the;sample. A time-series on the same firm is usually the best way to accomplish this.;An S-shaped total variable cost function requires a cubic specification.;Chapter 10: Production and Cost Estimation;215;4.;e;5.;b;6.;7.;8.;d;b;b;9.;d;10.;11.;12.;13.;14.;15.;16.;b;c;b;c;b;a;T;17.;F;18.;19.;20.;T;T;F;The user cost of capital accounts for the cost of acquiring capital, and also any depreciation or capital gains/losses resulting from using and owning capital. The user cost;of capital also includes the opportunity cost of using its capital rather than renting it.;Nominal dollars are adjusted for the effects of inflation by dividing the nominal;dollars by a price index to get real (or constant) dollars.;All of these statements are true in general.;See Table 10.3 in your textbook.;If input prices vary, then they must be included in the model as explanatory;variables.;SMC = a + 2bQ + 3cQ2, where a = intercept = 43.40, 2b = 22.80 = 5.6, and 3c =;3 0.20 = 0.60.;AVCQ = 20 = 43.40 2.8020 + 0.20202 = $67.40;TCQ = 20 = (AVCQ = 20 20) + TFC = 67.4020 + 1,000 = $2,348;$62.60 = SMCQ = 12 = 43.40 5.6012 + 0.600.122;AVCmin = b/2c = (2.8)/(20.2) = 7;AVCQ = 7 = $33.60 = 43.40 2.807 + 0.2072 = $67.40;A cubic specification has an S-shape.;Correcting for inflation is not necessary because all data are for the same period in;time.;Regression through the origin is not employed in estimating a cubic short-run cost;equation but rather in estimating a cubic short-run production function.;If inflation affects all input prices equiproportionately, real input prices will not vary.;Knowing any one equation allows the other two to be derived mathematically.;Long-run cost data are used for investment decisions, while short-run cost data are;used for output and pricing decisions.;Chapter 10: Production and Cost Estimation;216;Homework Exercises;1.;A firm estimates its cubic production function of the following form;Q = AL3 + BL2;and obtains the following results;Q;RSQUARE;FRATIO;PVALUE ON F;32;0.7547;92.31;0.0001;VARIABLE;PARAMETER;ESTIMATE;STANDARD;ERROR;TRATIO;PVALUE;L3;0.0016;0.0005;3.20;0.0032;L2;0.4000;0.0950;4.21;0.0002;DEPENDENT VARIABLE;OBSERVATIONS;a.;The equations for total product, average product, and marginal product are;TP =;AP =;MP =;b.;The estimated values of A and B are statistically significant at the (exact);levels, ________ and _________, respectively.;c.;At _______ units of labor usage, marginal product of labor begins to;diminish.;When the wage rate is $300, answer the following questions. (Remember that AP =;Q/L, AVC = w/AP, and SMC = w/MP.);d.;Average product of labor reaches its maximum value at ________ units of;labor.;e.;At the output for part d, average variable cost is $______________ and;marginal cost is $____________.;f.;When the rate of labor usage is 100 units of labor, output is _______ units.;Average variable cost is $_________ and marginal cost is $__________.;Chapter 10: Production and Cost Estimation;217;2.;Suppose Heritage Corporation believes that its total variable costs follow a cubic;specification and so it estimates its average variable costs using the following;specification;AVC = a + bQ + cQ 2;The regression analysis produces the following computer output;AVC;RSQUARE;FRATIO;PVALUE ON F;45;0.6145;33.47;0.0001;VARIABLE;PARAMETER;ESTIMATE;STANDARD;ERROR;TRATIO;PVALUE;INTERCEPT;175.0;25.00;7.00;0.0001;Q;3.20;0.80;4.00;0.0003;Q2;0.08;0.01;8.00;0.0001;DEPENDENT VARIABLE;OBSERVATIONS;a.;Do the estimated coefficients have the required signs to yield a;AVC curve? Discuss the significance using the p-values.;b.;-shaped;Heritage Corporations marginal cost function is;SMC = ___________________________________.;c.;At what level of output does AVC reach a minimum? What is the value of;AVC at its minimum?;Qmin =;d.;AVC min =;Compute AVC and SMC when Heritage produces 8 units.;AVCQ=8 =;SMCQ =8 =;Chapter 10: Production and Cost Estimation;218;3.;COMPUTER EXERCISE;Use a computer regression package or Excel to work this computer exercise.;Palm Products Company has collected data on its average variable costs of;production for the past 12 months. The costs have been adjusted for inflation by;deflating with an appropriate price index. The AVC and associated output data are;presented below;obs;1;2;3;4;5;6;Q;22;31;31;25;41;41;AVC;$208;202;206;214;174;203;obs;7;8;9;10;11;12;Q;45;45;45;62;62;70;AVC;$172;158;173;170;152;175;a.;Run the appropriate regression to estimate the parameters for the empirical;cost function AVC = a + bQ + cQ 2.;b.;Using a 10 percent significance level, discuss suitability of the parameter;estimates obtained in part a. Consider both the algebraic signs and statistical;significance of the parameter estimates.;c.;Present the estimated average variable cost, total variable cost, and short-run;marginal cost functions.;d.;At what level of output does AVC reach its minimum value? What is the;minimum value of AVC at its minimum?;Qmin =;e.;AVCmin =;Compute AVC and SMC when Palm Products produces 20 units of output;AVCQ=20 =;SMCQ=20 =;Is AVC rising or falling when Palm produces 20 units? Explain.;f.;At what level of output does SMC equal AVC? How did you get this answer?;Chapter 10: Production and Cost Estimation;219


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