Production PDF

Production  Chapter Preview n What happens to output as a firm increases the number of input(s) it uses? n To what degree is a firm able to substitute one input for another? n What happens to production as a result of technological change? ...

Production  Chapter Preview n What happens to output as a firm increases the number of input(s) it uses? n To what degree is a firm able to substitute one input for another? n What happens to production as a result of technological change?  Overview 1. Total Output – Marginal – Average Product 2. Isoquants 3. Returns to scale 4. Input Substitution 5. Technological Change 3  1. Production Functions n A firm turns inputs into outputs (black box) n A production function is a mathematical relationship between inputs and outputs ¨q = f( K, L )  Marginal Product n What happens to output as we add one more unit of one input (labor) to a fixed amount of another input (capital)? n Marginal product: additional output that can be produced by adding one more unit of some input, holding all other inputs constant MPL = DQ/DL (holding K constant) MPK = DQ/DK (holding L constant)  Total Output and Marginal Product As labor increases output The additional output gets increases but at a smaller and smaller as labor Output diminishing rate increases per week MPL L* Labor input L* Labor input per week per week  Average Product n Average product = total output / number of workers n AP tells you how productive all your workers are on average n Does not tell you how productive an extra worker is Average and Marginal Products Total Output, Average and Marginal Products TP maximized where MPL is zero. TP falls where MPL is negative; TP rises where MPL is positive.  2. Isoquants n How different amounts of capital and labor can be combined to produce output? n Isoquant is a curve that shows various combinations of inputs that will produce the same amount of output  Isoquants Capital per week KA A q=30 q=20 KB B q=10 LA LB Labor per week  Marginal Rate of Technical Substitution n Marginal rate of technical substitution (MRTS): amount by which one input can be reduced when one more unit of another input is added while holding output constant n If you used one more worker, how much less capital would you need to still produce the same level of output? n MRTS = - slope of isoquant n MRTSFROM K to L or OF L FOR K = - (change in capital) / (change in labor) = - DK/DL MRTS and Marginal Product  Isoquants Capital Along the isoquant the slope gets flatter and per week the MRTS diminishes The amount of capital that can be given up when one more unit of labor is employed gets KA smaller and smaller A KB B q = 10 LA LB Labor per week  MRTS and Marginal Product n Why must the slope of the isoquant be negative or the MRTS positive? ¨ MRTSL,K = MPL/MPK ¨ If MRTS was < 0 either MPL or the MPK would also have to be < 0 ¨ But then a firm would be paying for an input that reduced output ¨ Since no firm would do that MP’s > 0 and MRTS > 0  Isoquants Just like IC, isoquants have “always” properties: n Isoquants that represent greater output are farther from origin ¨ if you add more of both inputs, you get more output. Stated other way, if you want more output, you have to increase inputs. n Isoquants do not cross each other n Isoquants are downward sloping. ¨ there are trade-offs with inputs: you can substitute some capital with labor and produce same amount of output and vice versa. Never upward-sloping, which would imply that you could cut back on both inputs and produce same output. n Isoquants are convex  3. Returns to Scale n What would happen to production if a firm increased ALL inputs? n Returns to scale: rate at which output increases in response to proportional increase in all inputs n Two opposing effects at work as scale increases ¨ greater division of labor ¨ managerial inefficiencies and coordination problems  Returns to Scale n Constant returns to scale (CRS) ¨ If inputs increase by a factor of X, output increases by a factor equal to X n Increasing returns to scale (IRS) ¨ If inputs increase by a factor of X, output increases by a factor greater than X n Decreasing returns to scale (DRS) ¨ If inputs increase by a factor of X, output increases by a factor less than X Returns to Scale and Cobb-Douglas Functions Returns to scale depend on sum of exponents: n ⇢ CRS n Wcx ⇢ DRS n ⇢⇢ ⇢ IRS Isoquants and Returns to Scale  Returns to Scale vs. Marginal Returns Returns to scale: all inputs are increased simultaneously Marginal Returns: Increase in quantity of a single input holding all others constant Marginal product of a single factor may diminish while returns to scale do not Returns to scale need not be the same at different levels of production  4. Input Substitution n Why degree of substitution of one input for another matters n How does firm respond to increase in price of one input? n Attempts to use less of relatively more expensive input and more of relatively less expensive input n Substituting one input for another allow firm to keep costs from rising as much  Input Substitution n Fixed proportions production function is a production function in which inputs must be used in fixed ratio to one another n There is no way to substitute one input for another  Fixed Proportions Production Capital per week If you only have K1 machines would only need L1 workers K2 If you used L2 workers your output would remain the K1 q1 same K0 q0 L0 L1 L2 Labor per week  5. Changes in Technology n A production function or isoquant reflects a firm’s current technological knowledge n What happens to production function if there is technological progress? n Technical progress is a shift in production function that allows a given output level to be produced using fewer inputs  Changes in Technology Capital per week q0 q0’ Labor per week 5. Changes in Technology vs. Input Substitution Capital With technological change the firm can per week produce the same level of output, q0, with K0 but less labor. K0 q0 q0’ L1 L0 Labor per week Changes in Technology vs. Input Substitution Capital With input substitution the firm would per week need to use K1 units of capital with L1 units of labor. K1 K0 q0 q0’ L1 L0 Labor per week  Numerical Production Example n Production of burgers at Burger Queen ¨ Hamburgers per hour = q = 10 x (KL)1/2 n What type of returns to scale? Demonstrate. ¨ q = 10K1/2L1/2 where ⍺ = ½ and β = ½ ¨ Since ⍺ + β = 1 => CRS Constant Returns to Scale at Burger Queen !"#$$%&'()& GH",-"%&'K)& L0123"4-"%&5-"&6H3"& !" !" !#" $" $" "$#" %" %" "%#" &" &" "&#" '" '" "'#" (" (" "(#" )" )" ")#" *" *" "*#" +" +" "+#" !#" !#" !##"  Average and Marginal Product at Burger Queen n Suppose that Burger Queen uses 4 units of capital ¨q = 10(KL) ½ ¨q = 10(4L) ½ ¨q = 20L ½  Total Output, Average Productivity and Marginal Product at Burger Queen What happens to q/L and MPL as more workers are used? Why? !"#$$%&'()& GH",-"%&'K)& LM12P"4-"%&5-"&LHP"&'6)& 67K& 89K& !" "#" $%&%" $%&%" '" !" "$" $(&)" #!&#" (&)" !" ")" )!&*" ##&+" *&)" !" "!" !%&%" #%&%" +&!" !" "+" !!&," "(&-" !&," !" "*" !-&%" "(&$" !&)" !" "," +$&-" ",&*" )&-" !" "(" +*&*" ",&#" )&," !" "-" *%&%" "*&,"" )&!" !" #%" *)&$" "*&)" )&$" "  Isoquant Map for Burger Queen n Suppose Burger Queen wants to make 40 burgers. ¨q = 40 = 10(KL) ½ ¨4 = (KL) ½ ¨ 16 = KL  Isoquant Map for Burger Queen !"#$%&'(&)GH(&G!,%&G-KLG 0&122)G-3LG 4,&5(&)G-6LG !"# $%&"# #$# !"# #'&"# #(# !"# #)&*# #*# !"# #!&"# #!# !"# #*&(# #)# !"# #(&+# #%# !"# #(&*# #+# !"# #(&"# #'# !"# #$&'# #,# !"# #$&%# $"# #  Isoquant Map for Burger Queen Grills 8 2 q = 40 2 8 Workers  MRTS at Burger Queen !"#$%&'(&)GH(&G!,%&G-KLG 0&122)G-3LG 4,&5(&)G-6LG !"# $%&"# #$# !"# #'&"# #(# !"# #)&*# #*# !"# #!&"# #!# !"# #*&(# #)# !"# #(&+# #%# !"# #(&*# #+# !"# #(&"# #'# !"# #$&'# #,# !"# #$&%# $"# # MRTS = - change in K / change in L = -(4-5.3)/(4-3) = 1.3 MRTS = - change in K / change in L = -(1.8-2)/(9-8) = 0.2  MRTS at Burger Queen n What happens to MRTS as more and more workers are used? n It falls n As more and more workers are used, firm is less able to reduce number of grills it uses  Technological Progress at Burger Queen n Suppose that due to genetic engineering hamburgers can flip themselves. n Old production function: ¨ q = 10(KL) ½ n New production function: ¨ q = 20(KL) ½ n If HH still wants to make 40 burgers ¨ 40 = 20(KL) ½ ¨ 4 = KL Technological Progress at Burger Queen Grills With the old technology: 16 = KL With the new technology: 4 = KL 4 q = 40 q = 40 1 4 Workers  Recap n Marginal product is the additional output that can be produced by adding one more unit of some input, holding all other inputs constant n As more of an input is used its marginal product falls n Isoquant shows possible input combinations that a firm can use to produce a given level of output n MRTS = rate at which one input can be substituted for another (negative slope of isoquant) n Production may exhibit constant, decreasing, or increasing returns to scale n With fixed proportions production input substitution is impossible n With technological progress a firm can produce a given level of output with fewer inputs, or greater output with given level of inputs: isoquant shifts in  Practice quiz 1. All of the following factors are likely to increase production except _________. a) inflows of migrant workers b) new regulations to improve safety c) improvements in technology d) capital accumulation  Practice quiz 1. All of the following factors are likely to increase production except _________. a) inflows of migrant workers b) new regulations to improve safety c) improvements in technology d) capital accumulation  Practice quiz 2. When a firm is technically efficient, the firm, ______. a) produces a lower amount of output in order to minimize costs b) increases the amount of labor employed c) will minimize the output produced d) produces as much output as it can given the amount of labor employed  Practice quiz 2. When a firm is technically efficient, the firm, ______. a) produces a lower amount of output in order to minimize costs b) increases the amount of labor employed c) will minimize the output produced d) produces as much output as it can given the amount of labor employed  Practice quiz 3. Marginal product of labor can be defined as the rate at which total output_____. a) increases as the quantity of labor used changes b) changes as the quantity of labor used changes c) remains the same as the quantity of labor used changes d) decreases as the quantity of labor used changes  Practice quiz 3. Marginal product of labor can be defined as the rate at which total output_____. a) increases as the quantity of labor used changes b) changes as the quantity of labor used changes c) remains the same as the quantity of labor used changes d) decreases as the quantity of labor used changes  Practice quiz 4. If a given percentage increase in the quantities of all inputs increases output by less than that specific percentage, we say that a firm has ______. a) increasing returns to scale b) constant returns to scale c) decreasing returns to scale d) none of the above  Practice quiz 4. If a given percentage increase in the quantities of all inputs increases output by less than that specific percentage, we say that a firm has ______. a) increasing returns to scale b) constant returns to scale c) decreasing returns to scale d) none of the above  Problem 1 The firm Jasmin uses the inputs of fertilizer, labor, and hothouses to produce flowers. Suppose that when the quantity of labor and hothouses is fixed, the relationship between the quantity of fertilizer and the number of flowers produced is given by the table as follows: Tons of Number of Tons of Number of fertilizer/mo flowers/mo fertilizer/mo flowers/mo 0 0 5 2500 1 500 6 2600 2 1000 7 2500 3 1700 8 2000 4 2200 a. What is the average product of fertilizer when 4 tons are used? b. What is the marginal product of the sixth ton of fertilizer? c. Does this total product function exhibit diminishing marginal returns? If so, over what quantities of fertilizer do they occur? d. Does this total product function exhibit diminishing total returns? If so, over what quantities of fertilizer do they occur? Problem 1