Producer has to decide on….. How much to produce What capacity to be installed What combination of inputs to be employed to maximise profits and minimise costs At what price to sell
Production function A production function is a functional specification that provides the most efficient combination of input with which a chosen target level of output can be produced. It is specific to each industry and technology. Decisions that producers need to make: – To meet increased demand, should the firm go in for capacity expansion or stretch the existing production facilities. – How should they handle existing idle capacity.
Production Function Decision variable involved in production decisions are – Inputs and Output Input is anything that the firm employs in the production process Output is what the firm produces making use of inputs. Production functions change when the technical process of production change leading to an entirely different set of input combinations related in an entirely different manner.
Production Function with two variable inputs Q = f (K , L) where K is capital and L is labour. Given a target level of output, this function gives us the highest level of output that can be produced from a given combination of inputs. Production function can take many forms like F(L,K) = 3L + 2K2, or, 5K0.5 L0.5, or any other form
Production function… Consider the following combination of inputs for the production of a given level of output, say 160 cars. Given f(L,K) = 5K0.5L0.5 Combination L K Output A 50 20.5 160 B 40 25.6 160 C 30 34.13 160 D 20 51.2 160
Production function… Consider the following combination of inputs for the production of different levels of output Given f(L,K) = 5K0.5L0.5 Combination L K Output A 50 20.5 160 B 40 40 200 C 30 120 300 D 20 128 400
Isoquant An isoquant is a curve on which every point satisfies the production function and thus, all combination of L and K on an isoquant are technically efficient combination with which the given level of output can be produced. Each isoquant corresponds to a different level of output.
Production Function with Isoquant MapY O U T P U T8 37 60 83 96 107 117 1277 42 64 78 90 101 110 1196 37 52 64 73 82 90 975 31 47 58 67 75 82 894 24 39 52 60 67 73 793 17 29 41 52 58 64 692 8 18 29 39 47 52 651 4 8 14 20 27 24 21X 1 2 3 4 5 6 7
Isoquant map The preceding table represents a production function with two inputs, X and Y It can be observed that combinations (2,6), (3,4), (4,3), (6,2)yield same level of output, that is 52. By connecting the combinations we get the isoquant corresponding to output level 52 Similar combinations for different levels of output can be produced can be extracted from the table.
Isoquants Graphical representation of production function A curve drawn through the technically feasible combinations of inputs to produce a target level of output K Output 260 Output 200 Output 160 Marginal rate of technical substitution L is the substitution of one input for another. MRTS = -change in K/change in labor
Properties of Isoquants They are downward sloping – That is as you employ more and more of the input on the X axis, you necessarily employ less of the input on the Y axis in order to maintain the same level of output. Employing more of both inputs would lead to a higher isoquant. They are convex to the origin – This happens as the power to substitute diminishes, called as the marginal physical product as we employ more and more of a factor. However, in case of perfect substitutes the isoquant is a downward sloping straight line with a constant slope, while for perfect complements the isoquant is a L shaped curve.
Properties of Isoquants They do not intersect each other Application of Isoquants Isoquants enable the decision maker to conceptualize the trade-offs involved in substitution between inputs. Managers consider the costs and benefits of substituting one input for another and select that one where the net benefits are maximised. Isoquant model helps the decision maker to figure out the increase /decrease in output with a change in input.
Production function with one variable input Short run is defined as a period during which only one of the inputs can be varied. Long run is defined as a period during which no factor is fixed and all input factors can be varied. Average Product : Q /L Marginal Product : MP = dQ/dL ie it is the increase in output for a unit increase in the variable input.
Production function Law of diminishing returns or the law of variable proportions - According to this relationship, in a production system with fixed and variable inputs (say factory size and labor), beyond some point, each additional unit of variable input yields less and less output Increasing return is the stage where with each additional unit of variable input employed, the marginal product increases. O U 1 – Stage of increasing returns; Ex>1 T 1 2 – Stage of decreasing returns,0<Ex<1 P 3 3 – Stage of negative returns; Ex<0 2 U T
Production function Diminishing return is the stage where with each additional unit of variable input employed, the output increases but at a decreasing rate. The stage where with increase in variable input, the decreasing marginal product becomes negative, resulting in a decline of total output. It is the stage of negative returns. At this point the variable factor becomes counter productive. O U 1 – Stage of increasing returns; Ex>1 T 1 2 – Stage of decreasing returns,0<Ex<1 P 3 3 – Stage of negative returns; Ex<0 2 U T
Production function with one variable input Total Product: Q = 30L+20L2-L3 Average Product : Q /L Marginal Product : MP = dQ/dL = 30+40L-3L2
Production Elasticity Production elasticity is the proportionate change in output due to a proportionate change in input. ∆Q / ∆X * X / Q = MPx * 1 / APx Production elasticity greater than one indicates that output increases by a proportion greater than the increase in input. In cases where the elasticity is zero, there is no change in output due to a change in input. For a value of production elasticity less than zero indicates that output decreases with a given increase in input.
Three Stages of Production Stage 1: AP is increasing, MP is increasing and Production Elasticity is > 1. This stage corresponds to output levels that indicate underutilization of capacity. Stage 2: AP and MP are decreasing, until MP=0 and Production Elasticity is 0 < Prod.Elas < 1. Producer’s optimal choice of employment of variable input lies in this stage.
Three Stages of Production Stage 3: MP and AP continue to decrease and MP <0; leading to decrease in total product with increasing units of input and Production Elasticity < 0. No rational producer would want to be in this stage. This stage corresponds to output levels that indicate overutilization of capacity.
Relationship between Marginal physical product and marginal rate of technical substitution MRTS is equal to slope of the isoquant ie =-∆K/ ∆L MRTS can also be expressed algebraically. While moving from A to B or B to C on an isoquant, the following condition is to be satisfied: MP x ∆L + MP x ∆K = 0 l k Therefore - ∆K/ ∆L = MP /MP l k
Returns to Scale The rate at which output increases to a proportionate change in all inputs is known as the degree of returns to scale. Increasing output given constraints of technology means moving from one isoquant to another. Increasing Returns to Scale - When output increases by a proportion greater than the proportionate increase in all inputs. Specialization and division of labor assists in being in this stage. Decreasing returns to scale – Output increases by a proportion less than that of the increase in inputs. Constant returns to scale – Output increases by the same proportion as the increase in inputs.
Estimation of production functionsThere are a variety of production functions Linear production functions: Q = a + bX, these are subject to constant returns only Quadratic prod. Functions: Q = a + bX – cX2 , these capture the diminishing returns phase ( in –cX2 ) but not the increasing returns to scale. Cubic form pfs.: Q = a + bX + cX2 -dX3 , these capture both the increasing and the diminishing returns to scale Power function : Q =aXb
Cobb – Douglas Production Function Q = A Lα Kβ Q = total production (the monetary value of all goods produced in a year); L = labor input ;K = capital input ; A = total factor productivity; α and β are the output elasticities of labor and capital, respectivelyWhere, α + β indicates Returns to ScaleIf > 1, it exhibits Increasing Returns to ScaleIf < 1, it exhibits Decreasing Returns to ScaleIf = 1, it exhibits Constant Returns to Scale
Optimal Input Levels Level of output determines the demand for inputs and employment of inputs is related to revenue generated from the output. Stage of decreasing – diminishing returns is where the producer should be operating with one variable input. Total revenue product is total product multiplied by the market price of output
Optimal Input Levels Marginal revenue product is the change in total revenue product due to a unit change in the variable input = MP * Price Total variable cost is total cost of the variable input and is arrived at by multiplying quantity of variable input by its price. Marginal variable cost is the change in total variable cost due to a unit change in the variable input.
Optimal Input Levels With one variable input, producer produces in stage of decreasing returns produces upto the point where Marginal Revenue Product (MRP) = Marginal Variable Cost. MRP = MP * MR=MVC With many variable inputs: MPL / PL = MPK / PK = ……….
Optimal Combination of Inputs Given that each combination of inputs have a different cost attached, the producer needs to arrive at the least cost combination. The concept of the ‘isocost’ line is used for arriving at the optimal combination. An isocost line gives the combination of inputs that can be purchased in the market at the going market prices with a tentative budget. All combinations on this line result in the same total cost.
Optimal Combination of Inputs The point of tangency between an isocost line and an isoquant gives the least cost combination of inputs with which the output level corresponding to the isoquant can be produced. Isocost line is linear as input prices are constant. For a given target level of output, the combination of inputs on the isocost line, which is tangential to the isoquant is the optimal combination. The optimum point must be technically efficient (lie on the relevant isoquant) and has to satisfy the cost constraint).
Optimal Combination of Inputs Isoproduct curve : This curve gives us the combination of outputs that two levels of constant input quantities can produce, assuming production is most efficient and the inputs are completely exhausted. The curve is concave assuming diminishing returns occur when specialization increases. This curve is also called the production frontier as it indicates the highest levels of the two outputs that can be produced with the given inputs. The economy can go beyond the frontier only when the frontier shifts outward. This happens when: More is produced as the resource base has expanded or technological change has occurred. If there is trade between economies, then it is possible for an economy to consume more than it has produced.