Production Theory and Analysis

Creation of utility Production Production refers to the transformation of inputs or resources into outputs of goods and services Creation of utility

Created by human labor and capital Characteristics of goods & services to be classified as Production are: Created by human labor and capital Satisfy human wants directly or indirectly Are comparatively scarce and have economic value Have a definite monetary price/cost

Factors of Production INPUTS Land Labor Capital Enterprise Immobile Passive Heterogeneous Active Mobile Variable productivity Structures Equipment Cap.goods Money Innovative function Risk Decision making

INPUTS CAPITAL LABOR Land & Natural Entrepreneur Workers Structures Resources Entrepreneur Workers Machinery plant & equipment

Factors of Production Inputs Short Run- At least one input is fixed Fixed Inputs Variable Inputs Short Run- At least one input is fixed Long Run - All inputs are variable The length of long run depends on industry.

Level & Scale of Production Level of production can be altered changing the proportion of variable inputs Output = Fixed inputs + Variable inputs Scale of production can be altered by changing the supply of all the inputs only in the long run Output = Total inputs(variable inputs)

Concept of Product Total Product - total volume of goods produced during a specific period of time Average Product - the per unit product of a variable factor Marginal Product - the rate at which total product increases / addition to total product resulting from a unit increase in the quantity of the variable factor

Production Function With Two Inputs K Q 6 10 24 31 36 40 39 5 12 28 42 4 3 23 33 2 7 18 30 1 8 14 L Q = f(L, K) Input & output are measured in physical units Assumption- Technology is constant the during analysis period - All units of L & K are homogenous

Production Function with Two Inputs Discrete Production Surface

Production Function with One Variable Input Total Product TP = Q = f(L) Marginal Product MPL = TP L Average Product APL = TP L Production or Output Elasticity Q/Q L/L Q/ L Q/L = EL MPL APL

Production Function with One Variable Input Total, Marginal, and Average Product of Labor, and Output Elasticity

Production Function with One Variable Input Total, Marginal, and Average Product of Labor, and Output Elasticity L Q MP AP E - 1 3 2 8 5 4 1.25 12 14 3.5 0.57 2.8 6 -2 -1

Production Function with One Variable Input

Production Function with One Variable Input Total Product 2 4 6 8 10 12 14 16 1 3 5 7 A B C D E F Law of Diminishing Returns states that when increasing amounts of Variable inputs are combined with a fixed level of another input, a point will be reached where MP will decline. TP Marginal & Average Product Labor -3 -2 -1 1 2 3 4 5 6 7 6 A’ B’ C’ D’ E’ F’ AP Labor MP

The Law of Diminishing Returns & Stages of Production Total Product 2 4 6 8 10 12 14 16 1 3 5 7 A B C D E F TP I Inflection pt. G Marginal & Average Product Labor -3 -2 -1 1 2 3 4 5 6 7 6 Stage I of Labor Stage II of Labor Stage III of Labor B’ C’ A’ D’ E’ F’ AP MP Labor

The Law of Diminishing Returns & Stages of Production Total Product 2 4 6 8 10 12 14 16 1 3 5 7 A B C D E F TP I G Marginal & Average Product Labor -3 -2 -1 1 2 3 4 5 6 7 6 Stage I of Labor Stage II of Labor Stage III of Labor B’ C’ A’ D’ E’ F’ MPL is increasing MPK is negative AP MPL & MPK Positive MPL is negative MP Labor

Optimal Use of the Variable Input Marginal Revenue Product of Labor MRPL = (MPL)(MR) Marginal Resource Cost of Labor TC L MRCL = Optimal Use of Labor MRPL = MRCL

Optimal Use of the Variable Input MP MR = P L 2.50 4 $10 3.00 3 10 3.50 2 10 4.00 1 10 4.50 10 Assumption : Firm hires additional units of labor at constant wage rate = $20

Use of Labor is Optimal When L = 3.50 Optimal Use of the Variable Input L MP MR = P MRP MRC L L L 2.50 4 $10 $40 $20 3.00 3 10 30 20 3.50 2 10 20 20 4.00 1 10 10 20 4.50 10 20 Assumption : Firm hires additional units of labor at constant wage rate Use of Labor is Optimal When L = 3.50

Optimal Use of the Variable Input $ 40 30 20 10 MRCL = w = $20 dL = MRPL 2.5 3.0 3.5 4.0 4.5 Units of Labor Used

Exercise The marginal product of labor equation for a firm is given by: MPL = 10(K/L)0.5 Currently the firm is using 49 units of capital and 100 units of labor. Capital usage is fixed, but labor can be varied. If the price of labor is $20 per unit and the firms' output sells for $4, is the firm producing efficiently in the short run? If not, explain and determine the optimal rate of labor input.

MRPL = MRCL = w  28 ≠ 20 Not efficient L = 196

Production With Two Variable Inputs Isoquants show combinations of two inputs that can produce the same level of output. K 6 10 24 31 36 40 39 5 12 28 36 40 42 40 Q 4 12 28 36 40 40 36 3 10 23 33 36 36 33 2 7 18 28 30 30 28 1 3 8 12 14 14 12 1 2 3 4 5 6 L

Isoquants

Economic Region of Production Firms will only use combinations of two inputs that are in the economic region of production. Ridge line W

dQ=Q/ L *dL + Q/ K *dK= 0 Marginal Rate of Technical Substitution Q = f(L,K) dQ=Q/ L *dL + Q/ K *dK= 0 dK= (-) Q/ L dL Q/ K Q/ L = MPL and Q/ K = MPK dK - MPL = MRTS dL MPK = (L) MPL = -(K) MPK

Marginal Rate of Technical Substitution K Absolute value of the slope of isoquant is called the MRTS MRTS = -(-2.5/1) = 2.5

Production With Two Variable Inputs Perfect Substitutes Perfect Complements 2K 1L Capital Capital 6 4 2 6 4 2 B C -1K 2L A 2 4 6 8 10 12 2 4 6 Labor Labor

Optimal Combination of Inputs Isocost lines represent all combinations of two inputs that a firm can purchase with the same total cost.

Isocost Lines Capital 10 8 6 4 2 Labor 2 4 6 8 10 AB C = $100, w = r = $10 10 8 6 4 2 A slope = -w/r = -1 vertical intercept = 10 1K 1L B Labor 2 4 6 8 10

Isocost Lines A’ 14 10 8 4 A A” B” B B’ B* 4 8 10 12 14 16 20 Capital AB C = $100, w = r = $10 A’B’ C = $140, w = r = $10 A’’B’’ C = $80, w = r = $10 AB* C = $100, w = $5, r = $10 A A” B” B B’ B* Labor 4 8 10 12 14 16 20

Optimal Combination of Inputs for Minimizing costs or Maximizing output Isocost Lines AB C = $100, w = r = $10 A’B’ C = $140, w = r = $10 A’’B’’ C = $80, w = r = $10 MRTS = w/r

Optimal input combinations: Slope of isoquant = Slope of isocost line (absolute)Slope of isoquant = (absolute) Slope of isocost line MRTS = w r Since MRTS = MPL/ MPK MPL = w MPK r MPL = MPK w r If MPL = 5, MPK =4, and w = r MPL > MPK w r

Profit Maximization MPL = w MPL = MPK w r MPK r MRP(input) = MRC(input) with constant input prices MRP(input) = input price To maximize Profits: MRPL = w = (MPL)(MR) MRPK = r = (MPK)(MR) MPL = w MPK r MPL = MPK w r

Returns to Scale Returns to scale refers to the degree by which output changes as a result of a given change in the quantity of all inputs used in production. Long run function Both L & K are changing

Production Function Q = f(L, K) Returns to Scale Production Function Q = f(L, K) Q = f(hL, hK) If  = h, constant returns to scale. If  > h, increasing returns to scale. If  < h, decreasing returns to scale.

Constant Returns to Scale Increasing Returns to Scale Decreasing Returns to Scale

Empirical Production Functions Cobb-Douglas Production Function Q = AKaLb For example: the inputs are doubled i.e. instead of K & L, we are using 2K & 2L, then by how much the production will increase. New Q = A(2K)a(2L)b = A(2)a+bKaLb = 2a+bAKaLb = 2a+bQ

Now Q = 2a+bQ If a + b = 1, constant returns to scale. If a + b > 1, increasing returns to scale. If a + b <1, decreasing returns to scale.

Exercise Do the following production functions have constant, increasing or decreasing returns of scale? ( K, L, M are inputs) a. Q = 0.5X + 2Y + 40Z b. Q = 3L + 10K + 500 c. Q = K/L d. Q = 4A + 6B + 8AB e. Q = 10L 0.5 K 0.6

A. constant returns to scale. B. diminishing returns to scale. C. Decreasing returns to scale D. increasing returns to scale. E. increasing returns to scale

Economies of scale Diseconomies of scale Technology – cost effective at high level of production Specialization of labour Diseconomies of scale Transportation cost Difficult to manage

Exercise Medical Testing Labs, Inc., provides routine testing services for blood banks in the Los Angeles area. Tests are supervised by skilled technicians using equipment produced by two leading competitors in the medical equipment industry. Records for the current year show an average of 27 tests per hour being performed on the Testlogic-1 and 48 tests per hour on a new machine, the Accutest-3. The Testlogic-1 is leased for $18,000 per month, and the Accutest-3 is leased at $32,000 per month. On average, each machine is operated 25 eight-hour days per month. a. Does Medical Testing Lab usage reflect an optimal mix of testing equipment? b. If tests are conducted at a price of $6 each while labor and all other costs are fixed, should the company lease more machines?

a) (27*25*8)/ 18000 = (48*25*8) / 32000 = 0.3 In both instances, the last dollar spent on each machine increased output by the same 0.3 units, indicating an optimal mix of testing machines. b) For each machine hour, the relevant question is Testlogic-1 27 ×(25×8)× $6 > $18,000 or $32,400 > $18,000. Accutest-3 48 ×(25×8)× $6 > $32,000 or $57,600 > $32,000. In both cases, each machine returns more than its marginal cost (price) of employment, and expansion would be profitable.

Exercise The marginal product of labor for international trading is given by the equation MPL = 10K0.5/L0.5 Currently the firm is using 100 units of capital and 121 units of labor. The capital stock is constant but the labor can be varied. If the price of labor is 10/- and price of output is Rs. 2/- per unit, is the firm operating efficiently in the short run? If not, determine the optimal rate of labor input.

Answer: not optimally, L = 400

The production function is : Q = 20K0.5L0.5 With marginal product functions MPK = 10L0.5/K0.5 MPL = 10K0.5/L0.5 If the price of capital is Rs. 5/- and price of labor is Rs. 4/- per unit, determine the expansion path for the firm. The firm currently is producing 200 units of output per period using input rates of L = 4 and K =25. is this an efficient input combination? Why or why not? If not, determine the efficient input combination for producing an output rate of 200. If the price of labor increases from Rs 4 to Rs 8 per unit, determine the efficient input combination for an output rate of 200. What is the capital –labor ratio now?

Answer: K= 0.8L, L= 11.18 and K= 8.94, L = 7.905 and K = 12.65

Suppose the price of one unit of labor is $10 and price of one unit of capital is $2.50. Use this information to determine the isocost equations corresponding to a total cost of $200 and $500. Plot these two isocost lines on a graph If the price of labor falls from $10 per unit to $8 per unit, determine the new $500 isocost line and plot it on the same diagram used in part (b) Answer: K = 80- 4L and K = 200 – 4L, K = 200 – 3.2L

4. Given the production function Q = 30K0.7L0.5 and input prices r = 20 and w = 30. Determine an equation for the expansion path What is the efficient input combination for an output rate of Q = 200? For 500? Answer: K = 2.1L, for 200: L = 3.15 and K = 6.62, for 500: L = 6.765 and K = 14.207

Answer: CPAs – 1.67 and bookkeepers – 2.29 The revenue dept. of a state govt. employs certified public accountants (CPAs) to audit corporate tax returns and book keepers to audit individual returns. CPAs are paid $31200 per yr, while the annual salary of a bookkeeper is $18200. Given the current staff of CPAs and bookkeepers, a study made by the dept’s economist shows that adding one year of a CPA’s time to audit corporate returns results in an additional tax collection of $52000. In contrast, an additional bookkeeper adds $41600 per year in additional tax revenue. If the dept’s objective is to maximize tax revenue collected, is the present mix of CPAs and bookkeepers optimal? Explain If the present mix of CPAs and bookkeepers is not optimal, explain what re-allocation should be made. That is, should the department hire more CPAs and fewer bookkeepers or vice versa. Answer: CPAs – 1.67 and bookkeepers – 2.29