Q: How does a firm choose the combination of input to maximize output?

Transcription

1 Page 1 Ch. 6 Inputs and Production Functions Q: How does a firm choose the combination of input to maximize output? Production function =maximum quantity of output that a firm can produce given the quanities of inputs that it might employ. Q = f(k,l) If the firm is producing at technically inefficient points like A and B, then the firm is not maximizing output and producing efficiently. of L Labour requirements function inverts the production function to show the min amount required to produce a given level of output. Caves and Barton found that the typical manufacturer produced only 63% of their potential output given their stocks of L and K. Firms that were more likely to be efficient faced more competition: 1. More competition from abroad 2. More competition domestically

2 Page 2 3. Had better transportation infrastructure Total Product Functions =Single input production functions.

3 Page 3 Notes: TOTAL PRODUCT The TP function above shows output as a function of one input holding constant the quantity of the other input. (This represents a Short-run scenario. Usually, we say that L is variable and K is fixed in the SR. ) In long run, all inputs are variable. Points to note about the graphs: 1. Marginal Product of Labour (MPL) = slope of TP curve = ΔQ/ΔL = dq/dl (derivative of TP curve) 2. Average product of Labour (APL) = Q/L (output per unit of L) Note: AP is the slope (Qo/Lo) of a ray from the origin to the TP curve 3. TP curve maximized where MPL=0. 4. The production function displays diminishing returns. decreases as Law of diminishing returns = the marginal product of an input eventually the input increases if other inputs are held constant. 5. Demonstrates 3 Properties of all MP and AP curves: -AP increases when MP > AP - AP decreases when MP <AP -AP reaches a max when MP=AP 6. Two notions of Productivity: a. AP reported in news = output per unit labour hours b. MP amount of additional output produced when firm hires an extra unit of labour. c. An increase in technical efficiency will shift the TP curve up d. Labour productivity (output per hour of labour) decreased in the U.S. from 1995 due to oil supply shocks. Labour productivity (APL) increased from due to internet and communications technology. Example 1: Problem 6.3 p. 240

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7 Page 7 NOTE: At maximum AP, AP=MP. Is this true above? Yes. MPL=APL 12L 3(L) 2 = 6L L 2 L = 3 where maximum AP occurs.

8 Page 8 Production Functions with more than one Input Q=f(K,L) Total product Hill

9 Page 9 Partial derivatives: the marginal product curves are calculated as partial derivatives holding constant the other input: The TP hill is like the total utility hill in the consumer optimization problem. 1. TP function is like the Utility function of consumer theory. The height of the TP represents a given level of output just like the height of the total utility curve represented a fixed level of utility.

10 P a g e The isoquants are like indifference curves from consumer theory. Collapsing this hill into 2dimensions gives an isoquant map like the indifference curve map from consumer theory. 3. Isoquants represent the combination of inputs K and L that produce a fixed level of output. 4. Higher isoquants represent higher quantitative levels of output, not output rankings. 5. Important difference: Isoquants are cardinal, utility curves are ordinal. 6. Typical isoquants are convex because we assume firm does not operate in the uneconomic regions. In the yellow regions, the MP of one of the inputs is negative, and we can say that the firm will not produce in these regions (say at A) where the MPL is negative if it is cost-minimizing because it could produce the same level of output on the isoquant at lower cost. 7. Isoquants are downward sloping.

11 P a g e MRTS LK is the negative of the slope of the isoquant. So a negatively sloping isoquant has a positive MRTS and vice versa. to keep = ΔK/ΔL, the amount of K needed to compensate for a decrease in one L hour production Q constant. = MPL/MPK Proof: If Q is constant along a particular isoquant, then Solving, we see that the slope of the isoquant -ΔK/ΔL = MPL/MPK (= MRTSLK) Example 1: What does a MRTSLK = 6 mean? MRTS is a measure of the substitutability of inputs given the firm s particular production technology (production function and isoquant) MRTS = 6/1 machine =MPL/MPK -The marginal product of L is six times higher than that for capital. -An additional unit of labour will yield six times the output of an additional of K at the current input combination. -Another man-hour of L is six times more productive than an additional hour of K. output =ΔK/ΔL -If we lose one unit of L, we must replace it with 6 units of K to keep constant (on the same isoquant). unit

12 P a g e Shape of the isoquant tells you the technology of the firm, or the way in which K can be substituted for L. Shapes of Isoquants: 1. Perfect Substitutes 2. Perfect Complements (Leontieff) 3. Cobb-Douglas 4. CES (incorporates 1-3 above) Example 2: Find the MRTS for Q= K.5 L.75 MPL = dq/dl = 0.75K 0.5 L MPK= dq/dk = 0.5K-0.5 L 0.75 MRTSLK = MPL/MPK = (0.75K 0.5 L -0.25)/( 0.5K-0.5 L 0.75) = (3/2)(K/L) Elasticity of Substitution σ Elasticity of substitution σ: is another measure of the substitutability of inputs.

13 P a g e 13 NOTE: K/L ratio = slope of ray from origin to isoquant. 1. σ measures the curvature of the isoquant. when the output. Defn- σ is measuring how the substitutability of inputs (or MRTS) changes firm operates at different K/L ratios given a constant level of Substitutable for L at a constant rate equal to the slope of the isoquant (σ=infinity).

14 P a g e σ 0 : The value of σ ranges from 0 (no substitutability) to infinity. σ=low value<1: σ=high value:

15 P a g e Example: Motor vehicle manufacturing σ =.10 Food industry σ = Labour is more substitutable in the food industry. 4 Types of Isoquants: 1) Perfect Substitutes Q = al + bk - MRTS constant = b/a - σ = perfect substitutability of inputs Ex: Q = 10L + 20H H = high capacity computer (20GB) L = low capacity computer (10GB) MRTS = -slope of isoquant = ½

16 P a g e 16 2) Fixed proportions production (Leontief) Q = min(al, bk) MRTS doesn t exist because isoquant is not continuous. σ = 0 no substitutability between inputs. Ex: Chemical Industry. To make water H20 needs two H and one O. Q = min(h/2, O)

17 P a g e 17 3) Cobb-Douglas Production α Q= A L K β MRTS = (constant)(k/l) σ=1 4) CES Production Function most general form. It incorporates (1) (3):

18 P a g e 18 [ Q= al σ 1 σ + bk σ σ 1 σ 1 σ ] where 0 σ perfect substitutes ( σ =, fixed proportions (σ = 0) Cobb-Douglas (σ = 1) Returns to Scale =how the percentage of output changes when all inputs are increased by the same amount. 1) Increasing Returns F(λK,λL) > λf(k,l) Increasing inputs by the same percentage increases output by a bigger percentage. Due to economies of scale (decreasing long run average costs LRAC). Usually one firm in this market: utilities, oil pipelines, etc.

19 P a g e 19 2) Constant Returns F(λK,λL) > λf(k,l) Increasing inputs by the same amount will increase output by the same percentage. 3) Decreasing returns F(λK,λL) < λf(k,l) Due to diseconomies of scale (increasing LRAC). Output increases less than the same percentage increase in all inputs. To determine whether or not a production function displays IRS,DRS, or CRS, we increase all inputs by some percentage λ and try to manipulate to function to see how λ affects the original production function. Example 2: What are the returns to scale for Q = ak + bl? A(λK) +b(λl) = λ (ak + bl) = λq. Constant returns to scale if λ >0. If we were to scale up all inputs by a factor (that is, replace K by K, and L by L), the resulting output would equal Q. Therefore a linear production function has constant returns to scale. Returns to Scale differs from Marginal Returns (MP) Returns to scale measures what happens to output when the firm increases ALL inputs the same amount. Marginal returns measure what happens when you only increase one input.

20 P a g e 20 increases by Above: Holding K fixed at 10 units, there are diminishing returns to L as L increments of 10 units. When fixed at K=10: L=10, Q=100 L=20, Q=140 L=30, Q=170 However, there are constant returns to scale if all inputs L and K increase by 10 units. K=10 and L=10 Q=100 K and L double K=20 and L=20 Q=200 Output doubles Technological Progress = a change that allows a firm to achieve more output from a fixed quantity of inputs or, equivalently, the same amount of output with fewer inputs. Neutral Technological Progress MRTS is remains the same along a ray from the origin. Labour-saving technological progress MRTS decreases along a ray from the origin as the isoquant shifts inward (fewer inputs to make same amount of output). A decrease in MRTS, means that MPK relative to MPL. Ex: Robotics, advances in computers, increase the productivity of capital relative to L. Capital-saving technological progress MPL more than MPK. Ex: education, job-training.

21 P a g e 21 Example 3: Q = K 1/2L1/2 initially then P.F. changes to Q = LK1/2. What type of tech progress is this? a) More Q can be produced with the same amount of K and L. tech progress b) MRTS increases capital-saving. Extra Practice Use the following to answer question 1: Quantity Total Product 35,000 30,000 25,000 20,000 15,000 10,000 5, Labor A) B) C) D) 1. Marginal product reaches a maximum when labor equals Use the following to answer question 2: L Q

22 P a g e 22 A) B) C) D) 2. Marginal productivity is maximized with the worker. second third fourth sixth Q = 100( al + bk ). 3. A) B) C) D) Let a firm's production function be The production function then Q = 500(aL + bk ). becomes Which of the following statements is true? Neutral technological progress has occurred. Labor-saving technological progress has occurred Capital-saving technological progress has occurred. Economies of scale have increased. Use the following to answer question 4: L=0 L=1 L=2 L=3 L=4 A) B) C) D) K= K= K= K= K= Holding capital constant at 3 units, the marginal productivity of the second laborer is Suppose every molecule of salt requires exactly one sodium atom, NA, and one chlorine atom, CL. The production function that describes this is A) Q = NA + CL

23 P a g e 23 B) C) D) A) B) C) D) Q = NA x CL Q = min(na, CL) Q = max(na, CL) 6. A production manager notices that when she triples all of her inputs simultaneously, her output doubles. The production manager determines that for this range of output, the production function exhibits increasing returns to scale. constant returns to scale. decreasing returns to scale. undefined returns to scale Suppose that a firm s production function is given by Q = KL + K, with MPK = L + 1 and MPL = K. At point A, the firm uses K = 3 units of capital and L = 5 units of labor. At point B, along the same isoquant, the firm would only use 1 unit of capital. a) Calculate how much labor is required at point B. b) Calculate the elasticity of substitution between A and B. Does this production function exhibit a higher or lower elasticity of substitution than a Cobb Douglas function over this range of inputs? a) At point A, the firm produces 18 units of output. Therefore, since B is on the same isoquant, it must be that L = 17 at B. b) The capital-to-labor ratio at A is 3/5 and MRTSL,K = ½. At B, the capital-to-labor ratio is 1/17, and MRTSL,K = 1/18. Therefore the elasticity of substitution is ( KL ) MRTS MRTS ( KL ) σ= (1 / 17 3 / 5) /(3 / 5) 69 =. (1 / 18 1 / 2) /(1 / 2) 68 = A Cobb-Douglas production function has an elasticity of substitution of 1. Therefore this production function has a slightly higher elasticity of substitution, indicating a slightly greater ease of substitutability of inputs.

24 P a g e Let B be the number of bicycles produced from F bicycle frames and T tires. Every bicycle needs exactly two tires and one frame. a) Draw the isoquants for bicycle production. b) Write a mathematical expression for the production function for bicycles. a) This isoquants for this situation will be L-shaped as in the following diagram Tires 6 Q=3 4 Q=2 2 Q= Frames These L-shaped isoquants imply that once you have the correct combination of inputs, say 2 frames and 4 tires, additional units of one resource without more units of the other resource will not result in any additional output. b) Mathematically this production function can be written 1 Q = min( F, T ) 2 where F and T represent the number of frames and tires Consider a CES production function given by Q = (K0.5 + L0.5)2. a) What is the elasticity of substitution for this production function? b) Does this production function exhibit increasing, decreasing, or constant returns to scale? c) Suppose that the production function took the form Q = (100 + K0.5 + L0.5)2. Does this production function exhibit increasing, decreasing, or constant returns to scale? a) For a CES production function of the form

25 P a g e 25 Q = al σ 1 σ the elasticity of substitution is form σ + bk σ 1 σ σ σ 1. In this example we have a CES production function of the 2 Q = K L0.5. σ /(σ 1) = 2 (σ 1) / σ = 0.5 To determine the elasticity of substitution, either set solve for σ or and. σ 1 = 0.5 σ σ 1 = 0.5σ 0.5σ = 1 σ = 2. In either case, the elasticity of substitution is 2. b) Qλ = (λ K )0.5 + (λ L)0.5 2 Qλ = (λ 0.5 )( K L0.5 ) Qλ = λ K L0.5 Qλ = λ Q. 2 2 Since output goes up by the same factor as the inputs, this production function exhibits constant returns to scale. c)

26 P a g e 26 Qλ = (λ K )0.5 + (λ L)0.5 Qλ = λ 0.5 ( K L0.5 ) Qλ = λ K L0.5 < λq. λ When the inputs are increased by a factor of than λ λ, where λ >1 output goes up by a factor less implying decreasing returns to scale. K L Intuitively, in this production function, while you can increase the and inputs, you cannot increase the constant portion. So output cannot go up by as much as the inputs.