Theory of Production PDF

Chapter two Theory of production By Shimelis.M  Cont.. Introduction 1.1. Definition ▪ The theory of production refers to the process by which firms combine various inputs to produce goods and services. ▪ It focuses on the relationship between inputs (like labor, capit...

Chapter two Theory of production By Shimelis.M  Cont.. Introduction 1.1. Definition ▪ The theory of production refers to the process by which firms combine various inputs to produce goods and services. ▪ It focuses on the relationship between inputs (like labor, capital, and raw materials) and outputs (the products produced). ▪ It is the process of converting raw material or factors of production in to output.  Cont.. The production function is a function that shows the highest output that a firm can produce for every specified combination of inputs. It is a purely technical relation which connects factor inputs to outputs. Fixed vs variable inputs ▪ In the context of production, inputs are typically categorized into two main types: fixed inputs and variable inputs. Here's a breakdown of both:  Cont.. Fixed Inputs: Fixed inputs are resources that do not change with the level of output in the short run. They remain constant regardless of how much is produced. Examples: Capital Equipment: Machinery, buildings, and tools. Land: The physical space used for production. Fixed inputs are essential for production but cannot be easily adjusted.  Cont.. Variable Inputs: Variable inputs are resources that can be changed in the short run to adjust the level of production. They vary directly with the output ▪ Examples: Labor: Number of workers or hours worked can be increased or decreased. ▪ Raw Materials: The quantity of materials used in production can vary depending on output levels. ▪ Firms can adjust variable inputs based on demand or production needs.  Cont.. In production theory, the concepts of short run and long run are crucial for understanding how firms manage their resources and respond to changes in demand. Here’s a breakdown of the differences between the two: Short Run: The short run is a production period during which at least one factor of production is fixed. This means that while some inputs can be adjusted, others cannot be changed in the short term.  Cont.. Short run is that time period which is not sufficient to change the quantities of all inputs, so that at least one input remains fixed One thing to be noted here is that short run periods of different firms have different duration. Some firms can change the quantity of all their inputs within a month while it takes more than a year to change the quantity of all inputs for another type of firms. For example, the time required to change the quantities of inputs in an automobile factory is not equal with that of flour factory. The later takes relatively shorter time.  Cont.. Long run is that time period (planning horizon) which is sufficient to change the quantities of all inputs. Thus there is no fixed input in the long - run.  2.1. Production in the short run: Production with one variable input In the short run, production with one variable input focuses on how a firm adjusts its output by changing the quantity of that single variable input while keeping other inputs fixed. Total product (TPL), marginal product (MPL) and average product (APL) ▪ In the context of production with one variable input, three important concepts are Total Product (TPL), Marginal Product (MPL), and Average Product (APL). Here’s an overview of each:  Cont.. Total product: is the total amount of output that can be produced by efficiently utilizing a specific combination of labor and capital. ▪ The TP curve, thus, represents various levels of output that can be obtained from efficient utilization of various combinations of the variable input, and the fixed input. ▪ It shows the output produced for different amounts of the variable input, labor.  Cont.. Marginal Product (MPL):The MP of variable input is the addition to the TP attributable to the addition of one unit of the variable input to the production process, other inputs being constant (fixed). ▪ Before deciding whether to hire one more worker, a manager wants to determine how much this extra worker (ΔL) will increase output, ΔQ. ▪ The change in total output resulting from using this additional worker (holding other inputs constant) is the MP of the worker  Cont.. If output changes by ΔQ when the number of workers (variable input) changes by ∆L, the change in output per worker or MP of the variable input, denoted as MPL is found as ∆𝑄 𝑑𝑇𝑃 𝑀𝑃𝐿 = ∆𝐿 or 𝑀𝑃𝐿 = 𝑑𝐿. ▪ Thus, MPL measures the slope of the TP curve at a given point. ▪ In the short run, the MP of the variable input first increases reaches its maximum and then tends to decrease to the extent of being negative.  Cont.. That is, as we continue to combine more and more of the variable inputs with the fixed input, the MP of the variable input increases initially and then declines. Average Product (APL):The AP of an input is the ratio of total output to the number of variable inputs. 𝑡𝑜𝑡𝑎𝑙 𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝑎𝑣𝑒𝑟𝑎𝑔𝑒 𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝑜𝑓 𝑙𝑎𝑏𝑜𝑢𝑟 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑙𝑎𝑏𝑜𝑟. 𝑇𝑃 𝐴𝑃𝐿 = 𝐿  Cont.. The average product of labor (APL) first increases with the number of labor (i.e. TP increases faster than the increase in labor), and eventually it declines.  Graphing the short run production curves output 𝑇𝑃𝐿 Units of labor( variable input) 𝐴𝑃𝐿 𝑀𝑃𝐿 𝐴𝑃𝐿 Units of labor( variable input) 𝑀𝑃𝐿  Cont.. The relationship between MPL and APL can be stated as follows. When APL is increasing, MPL > APL. When APL is at its maximum, MPL = APL. When APL is decreasing, MPL < APL. Example: Suppose that the short-run production function of certain cut-flower firm is given by: 𝑄 = 4𝐾𝐿 − 0.6𝐾 2 − 0.1𝐿2.where Q is quantity of cut-flower produced, L is labor input and K is fixed capital input (K=5).  Cont.. A) Determine the average product of labor (APL) function. B) At what level of labor does the total output of cut-flower reach the maximum? C) What will be the maximum achievable amount of cut-flower production? Solution 𝑄 4𝐾𝐿−0.6𝐾 2 −0.1𝐿2 𝐴𝑃𝐿 = 𝐿 = 𝐿 = 2 0.6𝐾 4𝐾 − − 0.1𝐿 𝐿  Cont.. 15 20𝐿−15−0.1𝐿2 𝐴𝑃𝐿 = 20 − 𝐿 − 0.1𝐿 = 𝐿. B) When total product (Q) is maximum, MPL will be zero. 𝜕𝑄 𝜕(4𝐾𝐿 − 0.6𝐾 2 − 0.1𝐿2. 𝑀𝑃𝑙 = = 𝜕𝐿 𝜕𝐿 𝑀𝑃𝐿 = 4𝑘 − 0.2𝐿 𝑀𝑃𝐿 = 20 − 0.2𝐿 𝐿 = 100 ▪ Hence, when 100 workers are employed, total output will be maximum.  Cont.. C) Substituting the optimal values of labor (L=100) and capital (K=5) into the original production function (Q): 𝑄𝑚𝑎𝑥 = 985  The LDMR: short-run law of production The LDMR (law of diminishing marginal returns) states that as the use of an input increases in equal increments (with other inputs being fixed), a point will eventually be reached at which the resulting additions to output decreases. When the labor input is small (and capital is fixed), extra labor adds considerably to output, often because workers get the chance to specialize in one or few tasks.  Cont.… Eventually, however, the LDMR operates: when the number of workers increases further, some workers will inevitably become ineffective and the MPL falls  Efficient Region of Production in the short- run We are now not in a position to determine the specific number of the variable input (labor) that the firm should employ because this depends on several other factors than the productivity of labor such as the price of labor, the structure of input and output markets, the demand for output, etc. However, it is possible to determine ranges over which the variable input(labor) be employed. To do best with this, let’s refer back to fig 3.1 and divide it into three ranges called stages of production.  Cont.. Stage I - ranges from the origin to the point of equality of the APL and MPL. Stage II - starts from the point of equality of MPL and APL and ends at a point where MP is equal to zero. Stage III - covers the range of labor over which the MPL is negative  Cont.. Obviously, a firm should not operate in stage III because in this stage additional units of variable input are contributing negatively to the total product (MP of the variable input is negative) because of overcrowded working environment i.e., the fixed input is over utilized Stage I is also not an efficient region of production though the MP of variable input is positive. The reason is that the variable input (the number of workers) is too small to efficiently run the fixed input; so that the fixed input is underutilized(not efficiently utilized).  Cont.. Thus, the efficient region of production is stage II. At this stage additional inputs are contributing positively to the total product and MP of successive units of variable input is declining (indicating that the fixed input is being optimally used). Hence, the efficient region of production is over that range of employment of variable input where the MP of the variable input is declining but positive.  2.2 Long run Production: Production with two variable inputs Long run is a period of time (planning horizon) which is sufficient for the firm to change the quantity of all inputs. For the sake of simplicity, assume that the firm uses two inputs (labor and capital) and both are variable. The firm can now produce its output in a variety of ways by combining different amounts of labor and capital. With both factors variable, a firm can usually produce a given level of output by using a great deal of labor and very little capital or a great deal of capital and very little labor or moderate amount of both  Cont.. In this section, we will see how a firm can choose among combinations of labor and capital that generate the same output. To do so, we make the use of isoquant. So it is necessary to first see what is meant by isoquants and their properties.  Isoquants An isoquant is a curve that shows all possible efficient combinations of inputs that can yield equal level of output. If both labor and capital are variable inputs, the production function will have the following form. 𝑄 = 𝑓(𝐿, 𝐾) ▪ Given this production function, the equation of an isoquant, where output is held constant at q is q=𝑓(𝐿,𝐾)  Cont.. Isoquant maps: when a number of isoquants are combined in a single graph, we call the graph an isoquant map. An isoquant map is another way of describing a production function. Each isoquant represents a different level of output and the level of out puts increases as we move up and to the right. The following figure shows isoquants and isoquant map.  Cont.. Figure 2.1. Isoquant map Capital 𝑄3 𝑄1 𝑄𝑜 Labor  Properties of isoquants Isoquants have most of the same properties as indifference curves. The biggest difference between them is that output is constant along an isoquant where as indifference curves hold utility constant. Most of the properties of isoquants, results from the word ‘efficient’ in its definition. 1. Isoquants slope down ward. Because isoquants denote efficient combination of inputs that yield the same output, isoquants always have negative slope. Thus, efficiently requires that isoquants must be negatively sloped.  Cont.. As employment of one factor increases, the employment of the other factor must decrease to produce the same quantity efficiently. 2. The further an isoquant lies away from the origin, the greater the level of output it denotes. ▪ Higher isoquants (isoquants further from the origin) denote higher combination of inputs. ▪ The more inputs used, more outputs should be obtained if the firm is producing efficiently. Thus efficiency requires that higher isoquants must denote higher level of output.  Cont.. Isoquants do not cross each other. This is because such intersections are inconsistent with the definition of isoquants. Consider the following figure 3.2. capital Q=20 Q=50 𝐾∗ Labor 𝐿∗  Cont.. This figure shows that the firm can produce at either output level (20 or 50) with the same combination of labor and capital (L* and K*). The firm must be producing inefficiently if it produces q = 20, because it could produce q = 50 by the same combination of labor and capital (L* and K*). Thus, efficiency requires that isoquants do not cross each other.  Shape of isoquants Isoquants can have different shapes (curvature) depending on the degree to which factor inputs can substitute each other. 1-Linear isoquants ▪ Isoquants would be linear when labor and capital are perfect substitutes for each other. In this case the slope of an isoquant is constant. ▪ As a result, the same output can be produced with only capital or only labor or an infinite combination of both. Graphically,  Cont.. Figure capital 10 Q=100 8 Labor 12 14 2. Input output isoquant ▪ It is also called Leontief isoquant. This assumes strict complementarities or zero substitutability of factors of production.  Cont.. ▪ In this case, it is impossible to make any substitution among inputs. ▪ Each level of output requires a specific combination of labor and capital: Additional output cannot be obtained unless more capital and labor are added in specific proportions. ▪ As a result, the isoquants are L-shaped. See following figure  Cont.. Leontief isoquant capital 𝐾 is constant 𝐿 Q2 Q1 Q0 labor ▪ When isoquants are L-shaped, there is only one efficient combination of labor and capital of producing a given level of output.  Cont. To produce q1 level of output there is only one efficient combination of labor and capital (L1 and K1). Output cannot be increased by keeping one factor (say labor) constant and increasing the other (capital). To increase output (say from q1 to q2) both factor inputs should be increased by equal proportion  Cont.. 3. Kinked isoquants ▪ This assumes limited substitution between inputs. Inputs can substitute each other only at some points. Thus, the isoquant is kinked and there are only a few alternative combinations of inputs to produce a given level of output. These isoquants are also called linear programming isoquants or activity analysis isoquants. See the figure below  Cont.. Kinked isoquant capital A B labor In this case labor and capital can substitute each other only at some point at the kink (A, B,) Thus, there are only two alternative processes of producing q=100 output.  Cont.. 4. Smooth, convex isoquants ▪ This shape of isoquant assumes continuous substitution of capital and labor over a certain range, beyond which factors cannot substitute each other. Basically, kinked isoquants are more realistic: There is often limited (not infinite) method of producing a given level of output. However, traditional economic theory mostly adopted the continuous isoquants because they are mathematically simple to handle by the simple rule of calculus, and they are approximation of the more realistic isoquants (the kinked isoquants).  Cont.. From now on we use the smooth and convex isoquants to analyze the long run production. capital Q Labor Smooth isoquant  Cont.. The slope of an isoquant: marginal rate of technical substitution (MRTS) ▪ The slope of an isoquant (-ΔK/ΔL) indicates how the quantity of one input can be traded off against the quantity of the other, while output is held constant. ▪ The slope of the isoquant (dK/dL) defines the degree of substitutability of the factors of production. ▪ This slope decreases (in absolute terms) as we move downwards along the isoquant, showing the increasing difficulty in substituting L for K.  Cont.. The slope of the isoquant is called the rate of technical substitution, or the marginal rate of technical substitution (MRTS) of factors: −𝑑𝐾 𝑀𝑅𝑇𝑆𝐿𝐾 = 𝑑𝐿. Which is slope of an isoquant. ▪ MRTSL,K is defined as the amount of K that the firm must sacrifice in order to obtain one more unit of L so that it produces the same level of output. It is the slope of an isoquant. ▪ It can be proved that the MRTS is equal to the ratio of the marginal products of the factors.  Cont.. 𝜕𝑋 −𝑑𝐾 𝑀𝑃𝐿 That is 𝑀𝑅𝑇𝑆𝐿𝐾 = 𝑑𝐿 = 𝜕𝐿 𝜕𝑋 = 𝑀𝑃𝐾 𝜕𝐾 Proof Production function can be written as 𝑄 = 𝑓 𝐿, 𝐾 = 𝐶. ▪ It is equal to C because along an isoquant the TP is constant. ▪ The slope of a curve is the slope of a tangent line at that point. The slope of a tangent line is defined by the total differential.  Cont.. The total differential (dQ) is zero along an isoquant since the TP is constant. 𝜕𝑄 𝜕𝑄 𝑑𝑄 = 𝜕𝐿. 𝜕𝐿 + 𝜕𝐾. 𝜕𝐾 =0 𝑀𝑃𝐿. 𝑑𝐿 + 𝑀𝑃𝐾. 𝑑𝐾 = 0 𝑀𝑃𝐿. 𝑑𝐿 = −𝑀𝑃𝐾. 𝑑𝐾 𝑀𝑃𝐿 −𝑑𝐾 𝑀𝑃𝐾 = 𝑑𝐿  Cont.. Elasticity of substitution ▪ MRTS as a measure of the degree of substitutability of factors has a serious defect. It depends on the units of measurement of factors. ▪ A better measure of the ease of factor substitution is provided by the elasticity of substitution, δ. The elasticity of substitution is defined as 𝐾 𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝐿 𝛿 = 𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑀𝑅𝑇𝑆 𝐿𝐾  Cont.. 𝐾 𝐾 𝑑( )/ ൗ𝐿 𝐿 = 𝑑𝑀𝑅𝑇𝑆𝑘𝐿/𝑀𝑅𝑇𝑆𝐿𝐾 ▪ The elasticity of substitution is a unit free independent of unit of measurement of capital and labor since the numerator and denominator are measured with the same unit. Factor intensity ▪ Factor intensity of any production process is measured by the slope of a line through the origin representing the particular process. Thus, the factor intensity is the capital-output ratio.  Cont.. Consider the following figure, Capital 𝑲𝟏 𝑲𝟐 Clearly, > 𝑳𝟏 𝑳𝟐 K1 K2 Q Labor L1 L2 ▪ Here, production process P1 is more capital intensive than process P2.  Cont.. Example Let us illustrate the above concepts with a specific form of production function, namely the Cobb-Douglas production function. This form is the most popular in applied research, because it is easier to handle mathematically. It is of the form: 𝑄 = 𝐴𝐿𝛼 𝐾𝛽 1. The marginal product of the factors  Cont.. 𝜕𝑄 𝛼−1 𝛽 𝐴𝐿𝛼 𝐾 𝛽 𝑄 𝑀𝑃𝐿 = 𝜕𝐿 = 𝛼𝐴𝐿 𝐾 =𝛼 𝐿 = 𝛼( 𝐿 ) = 𝛼(𝐴𝑃𝐿 ) Marginal product of capital 𝛼 𝛽 𝜕𝑄 𝐴𝐿 𝐾 𝑀𝑃𝐾 = = 𝛽𝐴𝐿𝛼 𝐾𝛽−1 = 𝛽 𝜕𝐾 𝐾 𝑄 = 𝛽( ) 𝐿 = 𝛽(𝐴𝑃𝐾 )  Cont.. 2. Marginal rate of substitutions 𝑄 𝑀𝑃𝐿 𝛼( 𝐿 ) 𝛼 𝑘 𝑀𝑅𝑇𝑆𝐿𝐾 = = = 𝑀𝑝𝐾 𝛽(𝑄 ) 𝛽 𝐿 𝐾 3. Elasticity of substitutions (𝐾ൗ 𝑑 𝐿)൙ 𝐾ൗ 𝛿= 𝐿 =1 𝑑(𝑀𝑅𝑇𝑆𝐿𝑘 ) ൗ𝑀𝑅𝑇𝑆 𝐿𝐾  Cont.. 4. Factor intensity. In a Cobb-Douglas function 𝛼 factor intensity is measured by the ratio 𝛽. The higher the ratio the more labor intensive the technique is and vice versa. The efficiency of production. 5. The efficiency in the organization of factors of production is measured by the coefficient A. ▪ The more efficient firms will have a higher A than the less efficient one.  Cont.. The law of returns to scale ▪ The law of returns to scale refers to the long run analysis of production. ▪ In the long run, where all inputs are variable output can be increased by changing all factors by the same proportion. The rate at which output increases as inputs are increased by the same proportion is called returns to scale. ▪ We have three cases of returns to scale: increasing, constant and decreasing returns to scale.  Cont.. Suppose we start from an initial level of inputs and output 𝑋0 = 𝑓 𝐿, 𝐾 and we increase all the factors by the same proportion 𝑘. We will clearly obtain a new level of output X*, higher than the original level 𝑋0 , X* = 𝑓(𝑘𝐿, 𝑘𝐾). If X* increases by the same proportion k as the inputs, we say that there are constant returns to scale. If X* increases less than proportionally with the increase in the factors, we have decreasing returns to scale. If X* increases more than proportionally with the increase in the factors, we have increasing returns to scale.  Cont.. Returns to scale and homogeneity of the production function Suppose we increase both factors of the function 𝑋0 = 𝑓 𝐿, 𝐾 by the same proportion k, and we observe the resulting new level of output X* = 𝑓(𝑘𝐿, 𝑘𝐾) ▪ If k can be factored out (that is, may be taken out of the brackets as a common factor), then the new level of output X* can be expressed as a function of k (to any power v) and the initial level of output  Cont.. 𝑋 ∗ = 𝑘 𝑣 𝑓(𝐿, 𝐾) Or 𝑋 ∗ = 𝑘 𝑣 𝑋0 and the production function is called homogeneous. If k cannot be factored out, the production function is non-homogeneous. Thus: ▪ A homogeneous function is a function such that if each of the inputs is multiplied by k, then k can be completely factored out of the function. ▪ The power v of k is called the degree of homogeneity of the function and is a measure of the returns to scale:  Cont.. If v = 1 we have constant returns to scale. This production function is sometimes called linear homogeneous. If v < 1 we have decreasing returns to scale. If v > 1 we have increasing returns to scale. Given a Cobb-Douglas production function , 𝑄 = 𝐴𝐿𝛼 𝐾𝛽 returns to scale is measured by the sum of the power of the factors. That is, If 𝛼 + 𝛽 = 1, constant returns to scale if 𝛼 + 𝛽 < 1, decreasing returns to scale If 𝛼 + 𝛽 > 1,increasing returns to scale.  Cont.. proof ▪ Let capital and labor increased by 𝑘, then the new output is. 𝑋 ∗ = 𝐴(𝑘𝐿)𝛼 (𝑘𝐾)𝛽 𝑘 𝛼+𝛽 (𝐴𝐿𝛼 𝐾𝛽 ) 𝑘 𝛼+𝛽 (𝑋 ∗ ) Thus, 𝑣 = 𝛼 + 𝛽  Equilibrium of the firm: choice of optimal combination of factors An isoquant denotes efficient combination of labor and capital required to produce a given level of output. But, this does not mean that the monetary cost of producing a given level of output is constant along an isoquant. That is, though different combinations of labor and capital on a given isoquant yield the same level of output, the cost of these different combinations of labor and capital could differ because the prices of the inputs can differ.  Cont.. Thus, isoquant shows only technically efficient combinations of inputs, not economically efficient combinations. Technical efficiency takes in to account the physical quantity of inputs where as economic efficiency goes beyond technical efficiency and seeks to find the least cost (in monetary terms) combination of inputs among the various technically efficient combinations. Hence, technical efficiency is a necessary condition not efficient condition for economic efficiency  Cont.. To determine the economically efficient input combinations we need to have the prices of inputs and consider following simplifying assumptions. 1. Profit maximization is the goal of the firm, 𝜋 = 𝑇𝑅 − 𝑇𝐶. 2. The price of product given by 𝑃ത𝑥 and prices of factors given by 𝑤ഥ and 𝑟.ҧ Now before we go to the discussion of optimal input combination (or economically efficient combination), we need to know the isocost line, because optimal input is defined by the tangency of the isoquant and isocost line.  Cont.. Isocost line Isocost lines have most of the same properties as that of budget lines, an isocost line is the locus points denoting all combination of factors that a firm can purchase with a given monetary outlay, given prices of factors. Suppose the firm has C amount of cost out lay (budget) and prices of labor and capital are w and r respectively. The equation of the firm’s isocost line is given as:  Cont.. 𝐶 = 𝑤𝐿 + 𝑟𝐾 𝐶 𝑤 𝑟𝐾 = 𝐶 − 𝑤𝐿 , 𝐾 = 𝑟 −𝑟𝐿 ▪ Given the cost outlay C , the maximum amounts of capital and labor that the firm can 𝐶 𝐶 purchase are equal to 𝑟 and 𝑤. 𝐶 𝑟 Isocost line 𝐶 𝑤  Cont.. Now we are in a position to determine the firm’s optimal in put combination. However, the problem of determining optimal input combination (economic efficiency) takes two forms. Sometimes, situations may happen when a firm has a constant cost outlay and seek to maximize its output, given this constant cost out lay and prices of inputs. Still, there are also situations when the goal of the firm is to produce a predetermined (given) level of output with the least possible cost. Under we will discuss the two situations separately. Case 1. Maximization of output subject to cost constraint Suppose a firm having a fixed cost out lay (money budget) which is shown by its iso-cost line. Here, the firm is in equilibrium when it produces the maximum possible output, given the cost outlay and prices of input. The equilibrium point(economically efficient combination) is graphically defined by the tangency of the firm’s iso-cost line (showing the budget constraint) with the highest possible isoquant.  Cont.. 𝑊 At this point, the slope of the iso cost line( 𝑟 ) 𝑀𝑃𝐿 is equals to the slope of the isoquant (𝑀𝑃𝐾). Capital The condition of equilibrium 𝑊 𝑀𝑃𝐿 Under this case = 𝑟 𝑀𝑃𝐾 K e Q3 Q2 Q1 Labor L  Cont.. 𝑀𝑃𝐿 𝑀𝑃𝐾 Which is 𝑤 = 𝑟. This is the first order (necessary) condition. The second order (sufficient) condition is that isoquant must be convex to the origin.  Mathematical derivation A rational producer seeks the maximization of its output, given total cost outlay and the prices of factors. That means: maximize X = f (K, L) Subject to C = wL + rK. This is a constrained optimization which can be solved by using the lagrangean method. The steps are: A) Re write the constraint as 𝑤𝐿 + 𝑟𝐾 − 𝐶 = 0 B) Multiply the constraint with λ. 𝜆(𝑤𝐿+𝑟𝐾−𝐶)=0  Cont.. C) Form the composite function, ∅ = 𝑋 − 𝜆(𝑤𝐿 + 𝑟𝐾 − 𝐶) D) Partially derivate the function and then equate to zero. ▪ The partial derivative of the above function with respect to 𝐿, 𝐾 𝑎𝑛𝑑 𝜆 are: 𝜕∅ 𝜕𝑋 = − 𝜆𝑤 = 0 −−−−−− − 1 𝜕𝐿 𝜕𝐿 𝜕∅ 𝜕𝑋 = − 𝜆𝑟 = 0 −−−−−− − 2 𝜕𝐾 𝜕𝐾 𝜕∅ = 𝑤𝐿 + 𝑟𝐾 − 𝐶 −−−−−− −(3) 𝜕𝜆 Solving the above two equation for 𝜆  Cont.. we obtain, 𝜕𝑋 𝑀𝑃𝐿 𝜕𝐿 = 𝜆𝑤 or 𝑤 = 𝜆 𝜕𝑋 𝑀𝑃𝐾 𝜕𝐾 = 𝜆𝑟 or 𝑟 = 𝜆 The two expression must be equals. thus, 𝑀𝑃𝐿 𝑤 = 𝑀𝑃𝐾 𝑟 ▪ This firm is in equilibrium when it equates the ratio of the marginal productivities of factors to the ratio of their prices.  Cont. It can be shown 1 that the second-order conditions for equilibrium of the firm require that the marginal product curves of the two factors have a negative slope. The slope of the marginal product curve of labor is the second derivative of the production function: 𝜕2𝑋 The slope of 𝑀𝑃𝐿 𝑐𝑢𝑟𝑣𝑒 = 𝜕2𝐿 𝜕2𝑋 The slope of 𝑀𝑃𝐾 𝑐𝑢𝑟𝑣𝑒 = 𝜕2𝐾  Cont.. The second order condition. Thus. 𝜕2𝑋 = 𝜕2𝐿 < 0 and 𝜕2𝑋 = 𝜕2𝐾 𝜕𝐿𝜕𝐾 2 These conditions are sufficient for establishing the convexity of the isoquants.  Numerical example Example: suppose the production function of the 1 1 firm given as 𝑋 = 𝐿 𝐾. Prices of labor and capital 2 2 are given as birr 5 and birr 10 respectively and the firm has a constant cost out lay of $ 600.Find the combination of labor and capital that maximizes the firm’s output and the maximum output. Solution 𝑀𝑃𝐿 𝑤 The condition for equilibrium is 𝑀𝑃𝐾 = 𝑟  Cont. 𝜕𝑋 1 −1 1 𝑀𝑃𝐿 = 𝜕𝐿 = 2 𝐿 2 𝐾2 𝜕𝑋 1 1 −1 𝑀𝑃𝐾 = 𝜕𝐾 = 2 𝐿2𝐾 2 −1 1 1 𝑀𝑃𝐿 𝐿 2 𝐾2 𝐾 5 2 𝑀𝑃𝐾 = 1 1 −1 = 𝐿 = 10 𝐿2𝐾 2 2 𝐿 = 2𝐾 We have constrained function 5𝐿 + 10𝐾 = 600 Substitute,𝐿 = 2𝐾 in to the constraint function. We get  Cont.. 5 2𝐾 + 10𝐾 = 600 20𝐾 = 600 𝐾 = 30 We have 𝐿 = 2𝐾. Then, 𝐿 = 60 Thus, the firm should use 60 units of labor and 30 units of capital to maximize its production (output). (Check the second order condition). The maximum output can be found by substituting 60 and 30 for L and K in the production process.  Case 2: minimization of cost for a given level of output The conditions for equilibrium of the firm are formally the same as in Case 1. That is, there must be tangency of the (given) isoquant and the lowest possible isocost curve, and the isoquant must be convex. However, the problem is conceptually different in the case of cost minimization. The entrepreneur wants to produce a given output (for example, a bridge, a building, or X tons of a commodity) with the minimum cost outlay.  Cont.. In this case we have a single isoquant which denotes the desired level of output, but we have a set of isocost curves ( as shown in the following figure) Curves closer to the origin show a lower total- cost outlay. The isocost lines are parallel because they are drawn on the assumption of constant prices of factors: since w and r do not change, all the isocost curves have the same 𝑤 slope 𝑟. are equal.  Cont.. Thus the firm minimizes its cost by employing the combination of K and L determined by the point of tangency of X isoquant with the lowest possible isocost line. Points below e are desirable because they show lower cost but are unattainable for output X. points above e show higher costs. Hence point e is the least cost point  Cont.. Figure— Capital e K X Labor L ▪ Formally, min 𝐶 = 𝑓 𝑋 = 𝑤𝐿 + 𝑟𝐾 Subject to 𝑋ത = 𝐿, 𝐾  Cont.. Rewrite the constraint function as 𝑋ത − 𝑓 𝐿, 𝐾 = 0 Pre multiply the constraint by the Lagrangian multiplier 𝜆. 𝜆 (𝑋ത − 𝑓 𝐿, 𝐾 = 0 ▪ Form the 'composite' function as ∅ = 𝐶 −𝜆 (𝑋 ̅−𝑓(𝐿,𝐾) ∅ = 𝑤𝐿 + 𝑟𝐾 − 𝜆 (𝑋 ҧ − 𝑓 𝐿, 𝐾 ▪ Take the partial derivatives of c/J with respect to L, K and A. and equate to zero:.  Cont. 𝜕∅ 𝜕𝑓(𝐿,𝐾) 𝜕𝐿 = w −𝜆 𝜕𝐿 = 0 𝑀𝑃𝐿 𝑤 = 𝜆𝑀𝑃𝐿 , 𝑤 =𝜆------------(1) 𝜕∅ 𝜕𝑓(𝐿,𝐾) 𝜕𝐾 = r −𝜆 𝜕𝐿 = 0 𝑀𝑃𝐾 𝑟 =𝜆------------(2) 𝜕∅ = −[ (𝑋 ҧ − 𝑓 𝐿, 𝐾 ]---------(3). 𝜕𝜆 From the first two expressions we obtain 𝑀𝑃𝐿 𝑤 = 𝑀𝑃𝐾 𝑟  Cont.. Example Suppose a certain contractor wants to maximize profit from building one bridge. The contractor uses both labor and capital, and efficient combinations of Labor and capital that are sufficient to make a bridge is by the function 1 1 0.25𝐿 𝐾.If the prices of labor (w) and capital 2 2 (r) are $ 5 and $ 10 respectively. Find the least cost combination of L and K, and the minimum cost.  Cont.. solution The contractor wants to build one bridge. Thus, the constraint equation can be written as 1 1 1 − 0.25𝐿 𝐾. 2 2 Now our composite function be like, 1 1 ∅ = 𝑤𝐿 + 𝑟𝐾 +𝜆(1 − 0.25𝐿 𝐾 ) 2 2 Take partial derivatives as 𝜕∅ 1 −1 1 𝜕𝐿 =𝑤 − 𝜆.8 𝐿 2 𝐾 2 =0  Cont.. 𝜕∅ 1 1 −1 = 𝑟 − 𝜆. 𝐿2 𝐾 2 =0 𝜕𝐾 8 1 1 𝜕∅ = 0.25𝐿 𝐾 = 1 2 2 𝜕𝜆 From the above first two questions we get, 𝐾 5 = 𝐿 10 𝐿 = 2𝐾 Substitute, 𝐿 = 2𝐾 in the constraint equation. 1 1 0.25(2𝐾) 𝐾 = 1, 0.25. 2. 𝐾 = 1 2 2 1 4 4 8 𝐾= = and 𝐿 = 2 = 0.25. 2. √2 2 √2  Cont.. The least cost will be, 𝑤𝐿 = 𝑟𝑘 = 𝐶 8 4 80 5 + 10 = 2 2 √2 The end