1 terminology of agricultural production economics by ...jnkvv.org/pdf/07042020170432606.pdf · 2...

1 Terminology of Agricultural Production Economics by Gourav Kumar Vani for course of AgEcon 504 & Agecon 604 Last updated on 07-04-2020 at 16:34 PM

Variables: Any quantity which can have different values in the production process.

Decision Variables: These are the quantities which are under control of the decision maker. Some variables are fixed in the short run while the rest are free to vary in the short run. These are the variables of interest in any optimization problem.

Predetermined Variables: The quantities of variables which are known to the decision maker, but he/she is unable to exercise control over their levels but can use the information regarding the same for better decision-making.

Uncertain Variables: The quantities or levels of such variables are unknown to the decision maker who is unable to exercise any control over their use.

Constant: A quantity that does not change its value in a general relation between variables.

Dependent variable: A variable the level of which is governed by the level of other variables. This is treated as the variable of interest in most of the functions.

Independent variable: A variable the level of which does not depend on the level of other variables in the system.

Function: A function is a rule that describes a relationship between two or more variables. Functions are sometimes referred to as transformations. In general, it is denoted as where Y = f (X)

is dependent variable while is an independent variable. Y X

Single Valued Function: A function satisfying a unique property that any value uniquely X determines a value. Y

Correspondence: A correspondence describes a relationship between two variables. Usually a correspondence means multi-valued function. All functions are correspondences but not all correspondences are functions.

Domain: The set of all possible values or permissible values that can take in a given context is X known as the domain of the function.

Range: The set of all values that variable can take.Y

Slope: The rate of change of a function can be interpreted graphically as the slope of the function. Mathematically, it is the first order (partial) derivative of the function when the function is dX

dY defined as and is the partial derivative when there are more than one independent Y = f (X) variables in the function as for ith independent variable when function is . ∂Y

∂X i Y = f (X , .., )1 . Xn

Slope remains constant at every point on a straight line while it changes on every point on the curve.

Equation: An equation is a function set equal to some particular number (usually zero). In general, it is denoted as . Upon solving the equation we get value of variable that satisfies this f (x) = 0 X equation.

Gourav Kumar Vani, Assistant Professor, Department of Agricultural Economics and Farm Management, JNKVV, Jabalpur First draft created at 12:43:16 AM on 09/18/18


Identity: An identity is a relationship between variables that holds for all values of the variables. Often an identity is true by definition. Instead of an equal sign, an identity involves ≡ sign.

Optimization: It is a mathematical technique of finding the extreme values of an objective function with or without constraints.

Objective Function: An objective function is the function which is the principal focus of the investigator and represents the aim of the investigation. This function is to be either maximized or minimized. A set of decision variables decides the value of the objective function.

Constraint: It places limits on level of variable(s) either through an equation or an inequality.

Lagrange Multiplier Method: this method is a mathematical technique to find the local maxima or minima of a function subject to equality constraints . Lagrange f (X , .., )1 . Xn M − g (X , .., )1 . Xn = 0 multipliers are also used very often in economics to help determine the equilibrium point of a system because they can be interested in maximizing/minimizing a certain objective function. Here, instead of using Lagrange function is used as an objective function. Lagrange function f (X , .., )1 . Xn is set up as shown below:

(λ, , .., ) [M ]L X1 . Xn = f (X , .., )1 . Xn + λ − g (X , .., )1 . Xn

where is called Lagrange multiplier which is interpreted as the change in the value of Lagrange λ function due to an infinitesimal change in constraint amount . Lagrange multiplier is sometimes M referred to as dual value, shadow price or real worth of resources in different contexts.

Production: The process which converts inputs (goods and services) into outputs.

Assumptions of Production Function Analysis: 1. Non-negative input and output quantities; 2. Non-vanishing first and second order derivatives of the single valued, continuous production functions; 3. Concave production function to input axis; 4. Convex isoquants; 5. Concave PPC; 6. Decreasing returns to scale; 7. Perfect divisibility of products and factors 8. A set of technical decisions taken by a producer determines the exact shape of the production function of the firm. 9. static parameters of production functions 10. Technical efficiency inbuilt into production function.

Output: An Output/product is any good or service that comes out of production. This is measured in physical quantities and is usually referred to as Total Physical Product (TPP).

Input: An input or resource is any good or service that goes into production.

Factors of Production: The production resources required to produce a given product. Traditionally, these are classified as land, labour, capital, and organization.

Land: “The material and forces which nature gives freely for man’s aid, in land and water, in air and light and heat.”

Labour: “Any exertion of mind or body undergone partly or wholly with a view to earn some good other than the pleasure derived directly from the work.”

Capital: “Produced means of production”



Farm Entrepreneur: The person who organizes the agricultural production as a farm business and bears the responsibility of the outcome of the business. Every farmer is a farm entrepreneur and farming is a business for him/her.

Resource: Anything that aids in production is called a resource.

Long run: It is a planning period during which all resources can vary in quantity. The supply can be fully adjusted according to demand.

Short run: The planning period during which one or more resources are fixed. In the short run, output can be varied by intensive use of fixed resources.

Fixed Resources: Resources the level of which remains unchanged with changes in level of output. The level of such resources is unaltered only in the short run but can be altered in the long run. Farmers have very little control over the use of these resources. The cost associated with a fixed resource is termed as fixed cost.

Variable Resource: Resources the level of which changes according to the level of output. Farmers have considerable control over the level of these resources. The costs associated with a variable resource is known as variable cost.

Production function: A technical and mathematical relationship among outputs and inputs which describes the manner and extent to which a particular product depends upon the quantities of inputs, at a given level of technology and time. A short run production function is denoted as following

Y = f (X |X , X , ..., X )1 2 3 n

Where is the output from a particular enterprise, is the variable resource and are Y X1 , , ..,X2 X3 . Xn fixed resources. | (Vertical bar) which is read as ‘given’, separates variable resources from fixed resources. In some places instead of separating fixed inputs by vertical bar, either a vertical line is drawn over fixed inputs or superscript is used to mark the fixed inputs .X2 X0

2

Ceteris Paribus ('key-te,ris 'pa-ri,bûs): It is a Latin phrase which means other things remaining/being the same or constant. This is a major assumption in economic analysis.

Mutatis mutandis: This is a Latin phrase which approximately translates in English as "allowing other things to change accordingly" or "the necessary changes having been made."

Average Physical Product of input (APPXi): Total physical product produced per unit of X i (Y ) the variable input keeping other inputs constant at some specified levels.X i

. Geometrically, at a point on production function, it is theAP P X i= Y

X i= X i

f (X |X ,X ,X ,..., X X ..,X )i 1 2 3 i−1, i+1, n slope of a line segment connecting that with origin. If 40 kg of nitrogen application along with other inputs results in production of 2400 kg of output then it can be interpreted that each kg of nitrogen applied on an average results in 60 kg of output, ceteris paribus. Yield: This is the average amount of output produced per unit of area under cultivation. This is usually expressed in quintal per hectare. This is a special type of average physical product.



Marginal Physical Product of input (MPPXi): The change in total physical product due to X i (Y ) an infinitesimally small change in the input , keeping other inputs constant at some specified X i level. It is the slope of the total physical product curve.

or MP P X i= dY

dX i= dX i

df (X |X ,X ,X ,..., X X ..,X )i 1 2 3 i−1, i+1, n MP P X i= ∂Y

∂X i

Geometrically, at a point on a production function, it equals the slope of the tangent to the curve at that point. If upon applying an additional kg of nitrogen, total nitrogen application increases to 40 kg of nitrogen and marginal product reaches 110 kg then we interpret it as 40th kg (last one kg of nitrogen applied additionally) of nitrogen results in 110 kg of output, ceteris paribus.

Inflection Point for input : It is the point at which MPPXi reaches its maximum. At this point, X i the TPP curve changes its curvature from convex to turning concave to horizontal axis. Before this point, the TPP curve increases at an increasing rate but after this point is reached, the TPP curve increases at a decreasing rate. Let be the production function. Given that is a Y = f (x) x = x0 stationary point, then at this point either

1. is maximum if orf (x) f ′′ (x )0 < 0

2. is minimum if orf (x) f ′′ (x )0 > 0

3. then higher order derivatives must be investigated as followingf ′′ (x )0 = 0

If first derivatives are equal to zero and then by Nth derivative test this point is (n−1) =f n (x )0 / 0 an inflection point when n is an odd number.

Partial Elasticity of Production: It is the responsiveness of output to change in level of a given input keeping other inputs constant at some specified level. It is the percentage change in output for a percentage change in a given input keeping quantities of other inputs constant at some specified level. It is worked out by taking ratio of MPPXi to APPXi for input . In the first stage of X i neoclassical production, it is greater than one and at the end of the first stage, it reaches one. In the second stage, it is less than one throughout and reaches zero at the end of this stage. In the third stage, it is negative throughout. It ranges from to . It is denoted as and read as elasticity ∞− + ∞ EY

X i

of output w.r.t. input . If then it is interpreted as for one percent increase in input Y X i .85EYX i

= 0 , output increases by 0.85 percent.X i Y

Isoquant: An isoquant is locus of points representing different levels of two inputs that provides the same level of output while keeping level of other inputs at some constant level. The word “Iso” comes from a Greek ‘isos’ meaning equal and “Quant” is short for quantity. It is also called an isoproduct curve. Let be the production function then isoquant equation is denoted Y = f (X , X )1 2 by g (X , X |Y )1 2 = 0

Isoquant Map: A diagram showing the different sets of isoquant curves for a given technology is called an isoquant map. Two families of production function underlie every isoquant map. An infinite number of isoquants can be drawn on an isoquant map.



Marginal Rate of Technical Substitution (MRTS): MRTS is the slope of isoquant and is computed as ratio of marginal product for two inputs. For a convex isoquant, as we move along an isoquant from top left to bottom right then MRTS decreases. This is because of imperfect substitution between two inputs on isoquant. This means each additional unit of added input replaces less and less of other input on isoquant. MRT Sx1x2 = MP P x2

MP P x1 = − dx1

dx2

Isocline: The line joining the locus of points representing the same slope on the successively higher isoquants on an isoquant map. Isocline connects points of equal MRTS on an isoquant map. An isocline is drawn by finding the equation between two inputs by equating MRTS to a constant slope

. . All isoclines may or may not converge to a single point.K MRT Sx1x2 = MP P x2

MP P x1 = K

Ridge line: these are special isoclines with the value of slope equal to zero or infinity. which means . This will provide ridge line for input .MRT Sx1x2 = MP P x2

MP P x1 = 0 MP P x1 = 0 1x

which means . This will provide ridge line for input .MRT Sx1x2 = MP P x2

MP P x1 = ∞ MP P x2 = 0 2x

The ridge lines mark the division between stage II and III of neo-classical production function. Ridge lines exist only for those inputs w.r.t. which production has a distinct maximum output level. Thus ridge lines can be drawn only for certain types of isoquant patterns or maps. If ridge lines exist for both inputs then they intersect at Von Liebig point. Two ridge lines within their boundaries define the economically relevant, rational zone of production.

Expansion path: The line joining the locus of points representing the points of the least cost combinations on the successively higher isoquants on an isoquant map. There is only one point on the expansion path that represents the global point of profit maximization for the firm. The expansion path starts at origin and travels until the point of global output maximization is reached where MPP for each input turns zero.

For two input case expansion can be derived as following

where is price of input and is price of input .MRT Sx1x2 = MP P x2

MP P x1 = P x2

P x1 P x1 X1 P x2 X2

Hicks-Allen Partial Elasticity of substitution: The proportional change in input ratio due to a proportional change in the rate of technical substitution along a given isoquant curve. It measures how rapidly the input proportions change in response to a given change in the relative input prices. It is denoted by Greek symbol and is independent of the unit in which two inputs are measured σ and is a positive number which varies between zero and infinity. Formula to measure partial elasticity of substitution at a point on isoquant is provided as following

for σji =dln x /x( i j)

dln −dx /dx( i j)=

d x /x / x /x( i j) ( i j)d −dx /dx / −dx /dx( i j) ( i j)

=d x /x / x /x( i j) ( i j)

d(MRT Sji )/(MRT Sji ) =i / j

This formula can also be written as following according to Henderson and Quandt

where are the marginal products of input and .σji =d f /f / f /f( j i) ( j i)d x /x / x /x( i j) ( i j) and ff j i xj xi



According to R.G.D. Allen, under constant returns to scale assumption, the above formula reduces to

σji =f fj iq f ji

At point of least cost combination, equilibrium condition is and MRT Sx1x2 = MP P x2

MP P x1 = P x2

P x1 therefore, formula for partial elasticity of substitution at this point is

σji =d x /x / x /x( i j) ( i j)

d P /P / P /P( xj xi) ( xj xi)

This formula is referred to as Two-inputs, Two-prices Elasticity of Substitution or T.T.E.S. However, there exists alternative measures of partial elasticity of substitution. One among them is One-input, One-price Elasticity of Substitution (OOES). This measure is proportional to the cross price input demand elasticity evaluated at constant output. OOES between jth input quantity,

and ith input price, is thejx P xi

Φij = β dlnx( j) / (dlnP )xi

When then two inputs can not substitute each other (L shaped isoquant) while meansσji = 0 σji = ∞ two inputs are perfect substitutes (Linear downward sloping isoquant). When partial elasticity of substitution is measured between two points then it is referred to as arc measure of . It is σji measured as % change in (xi/xj)/ % change in MRT Sσji = xjxi

Returns to scale (RTS): It is the change in output due to proportionate change in all factors of production simultaneously. It is worked out as sum of partial elasticity of production for all inputs.

. Hence, it is also referred to as complete elasticity of production. RTS can beT SR = ∑n

i=1EY

X i

increasing, constant or decreasing. If there prevails, increasing returns to scale then increasing all inputs simultaneously in the same proportion would result in more than proportionate increase in output and distance between successive isoquants on an isoquant map would decrease. On the contrary, if there prevails decreasing returns to scale then increasing all inputs simultaneously in the same proportion would result in less than proportionate increase in output and distance between successive isoquants on an isoquant map would increase. For constant returns to scale, upon proportionate increase in input levels, there would be a proportionate increase in output level and distance between successive isoquants on an isoquant map would remain the same. For Cobb-Douglas production function, the returns to scale parameters can be worked out as sum of coefficients associated with all inputs.

Total Value Product (TVP): It is the same as Total Revenue (TR).

Marginal Value Product (MVP) of input : It is the additional revenue obtained by applying an X i additional unit of input . It is the slope of TVP w.r.t. input .X i X i PMV P X i

= ∂X i

∂T R = MP P X i Y



Value of marginal product (VMP) of input : It is the value of the incremental unit of output X i resulting from an additional unit of input , when sells for a constant market price . X i Y P 0

Y PV MP Xi = MP P Xi

0Y

Total Revenue (TR): It is the total revenue obtained by selling the output(s). If there prevails perfect competition in the output market, total revenue equals price of output times output sold in the market at that price. where is price of output. In case there are more than one R YT = P Y P Y

output in production process then .R YT = ∑s

k=1P Y k k

Marginal Revenue (MR): This is slope of total revenue curve w.r.t. output. It is the additional revenue brought about by producing an additional unit of output. RM = dY

dT R

In case of more than one output produced by production system, marginal revenue would be calculated for each output ‘j’. MRY j

= ∂Y j

∂T R

Homogeneous production function: A production function is said to be homogeneous of degree nif when each input is multiplied by some number such that , output increases by the factor t t > 0 tn

(essentially can be completely factored out of the function). For homogeneous production function,t isoclines and expansion path area linear.

Monotonic Transformation: is a way of transforming a set of numbers into another set that preserves the order of the original set, it is a function mapping real numbers into real numbers, which satisfies the property, that if , then simply it is a strictly increasing function.x > y f (x) > f (y)

Homothetic production function: A function of two or more arguments is said to be homothetic if all ratios of its first partial derivatives depend only on the ratio of the arguments, not their levels. A function which is a monotonic transformation of a homogeneous function. Homogeneous functions are a special case of homothetic functions.

Euler’s Theorem: Euler’s theorem is a mathematical relationship that applies to any homogeneous function. It states that if a function is homogeneous of degree then sumproduct of marginal n physical product of all input and its respective input level would equal times the total physical n product. or orX X Y∂Y

∂X1 1 + ∂Y∂X2 2 = n X X YMP P 1 1 + MP P 2 2 = n X X T RV MP 1 1 + V MP 2 2 = n

Thus if each input is paid according to its marginal factor productivity then sum of all payments made to factors of production would be times total revenue. Accordingly, value of must be less n n than one but greater than zero for positive profit to occur. In case shows returns to scale then n decreasing returns to scale must prevail for profit maximization to occur. When constant returns to scale would prevail then zero profit would be achieved accordingly.

(Linear) Homogeneity restriction/condition: This is the set of restrictions placed on production/profit/cost/revenue function so as to ensure homogeneity of degree zero (constant returns to scale) for the said function which is not homogeneous of degree zero by its specification. For example, Transcendental production function, is not homogeneous, then X X eY = A 1

α2β (γX +δX )1 2



parameter restriction is placed as and . If the word Linear Homogeneous is not γ + δ = 0 α + β = 1 said then second restriction would not be applicable.

Symmetry restriction/condition: Let following be the cost function (r, y) C

n(C) ln(a ) a ln(y) ln(r ) a ln(r ) ln(r ) ln(y) ln(r ) a [ln(y)]l = 0 + y + ∑n

i=1ai i + ∑

n

i=1∑n

j=1½ ij i j + ∑

n

i=1ayi i + ∑

n

i=1 yy

2

where parameter restriction is referred to as symmetry restriction/condition. a for all i = jaij = ji /

Technical Complement: An input is said to be complement to input if an increase in the use X2 X1 of former input causes the MPP of the latter input to increase, i.e., dX2

dMP P X1 > 0

Total Costs (TC): It is the sum of total fixed costs and total variable costs. TC is plotted against output to get TC curve and it starts from TFC, not from origin. . Any point on a C F C V CT = T + T cost curve shows the minimum cost at which a certain level of output may be produced.

Cost of production(CoP): The expenditure incurred on all inputs (including services) for producing a unit quantity of output is called as cost of production. This is also referred to as average cost of production. It is expressed in Rs. per quintal. If cost of cultivation and yield for a crop are already known, then cost of production equals cost of cultivation minus value of by-products divided by yield. oPC = Y ield

CoC−value of by−products

Cost of cultivation (CoC): The average expenditure incurred on all inputs (including services) for raising a crop on a unit area is referred to as cost of cultivation. This is expressed in Rs. per hectare. It is worked out as total cost incurred in raising the crop divided by area under crop.

Total Fixed Costs (TFC): The sum of all fixed cost is known as total fixed costs. TFC curve is horizontal straight line parallel to X-axis. TFC can also be worked out as area of rectangle formed by perpendicular from AFC curve to the X-axis (output axis) and to Y-axis (cost axis).

Total Variable Costs (TVC): This is the sum total of all costs incurred on variable resources. It is the sum of all variable costs incurred in the production process. TVC curve has inverse ‘S’ shape. If

there are variables inputs in production process then TVC is calculated as .n X∑n

i=1P Xi i

Cost Equation: where is the cost, is the total fixed costs and second term on XC = F + ∑n

i=1P Xi i C F

the right hand side is total variable costs.

Opportunity Cost: The returns forgone from next best alternative.This is the income that could have been received, if the input had been used in its most profitable alternative use. This is also referred to as the real cost of an input. This is not the purchase price of the input. The application of input in current use is justified only when returns in present use is greater than its opportunity cost.

Average Cost (AC): This is the cost of producing a unit quantity of output. This is the same as the cost of production. The average cost when plotted against output is ‘U’ shaped.



Average Fixed Cost (AFC): Total fixed costs per unit of output. This cost curve declines continuously with increase in output until output reaches its maximum. The curve has a hyperbola shape. F CA = Y

T F C

Average Variable Cost (AVC): Total variable costs per unit of output. It is having a ‘U’ shaped curve similar to that of AC but as output increases then distance between AVC and AC curve decreases. AVC is inversely related with APP curve.

V CA = YT V C = Y

X∑n

i=1P Xi i

= ∑n

i=1P Xi Y

X i = ∑n

i=1P Xi

1AP P Xi

Marginal Cost (MC): It is the cost of producing an additional unit of output. It is the change in total cost due to a minute change in output level. MC curve achieves its minimum before AVC and AC curve attains its minimum. MC curve intersects AVC and AC at their minimum points. MC is inversely related with MPP.

CM = dYdT C

Marginal Input Cost (MIC) for input : It is the change in total cost (TC) resulting from a small X i increment in level of input . In case of perfect competition prevailing for input market then MIC X i for input is its price prevailing in the market.

MICXi = ∂X i

∂T C

Short-run costs: These are the costs at which the firm operates in any one period.

Long-run costs: These costs are planning costs or ex ante costs, i.e., these costs presents the optimal possibilities for expansion of the output and thus help plan future activities.

Economies of scale: This is the reduction in unit cost of production arising from an increase in scale of production or operation. This is the competitive advantage accruing to firm due to increase in scale through reduction in unit cost of production.Economies of scale are further divided into two types based on source of economies, internal and external. The internal economies are further divided into real and pecuniary economies.

Internal & External economies of scale: Internal economies are firm specific economies arising from its own action as it expands level of output independently of how other firms operate. It could arise due to better management within firm in regard to production, marketing, transport & storage, human resource and purchase/ordering. The internal economies determine the shape of the long run average cost curve (scale curve) while external economies determine the position of scale curve. The external economies arises due to changes taking place outside the firm, i.e. in industry such as advancement of technology, changes in factor prices or economic growth.

Real economies: Real economies are those associated with the reduction in the physical quantity of inputs, raw materials, various types of labour and capital. This can be further classified as production economies, selling or marketing economies, transportation and storage economies, and managerial economies.



Production economies of scale: These can arise from labour economies, technical economies, and inventory or stochastic economies. This is the economy in the use of inputs as scale of output goes up.

Labour economies: Labour economies are achieved as the scale of output increases for several reasons: (a) division of labour and specialization, (b) time-saving, (c) automation of the production process, (d) cumulative volume economies.

Technical economies: These can arise from specialization of the capital equipment (and associated labour) at large scale production facilities, from indivisibilities associated with modern industrial techniques of production, set-up costs, initial fixed costs, technical volume/input relations and reserve capacity requirements.

Indivisibility of factor of production: Indivisibility means that certain factors are available only in some minimum sizes. Certain inputs particularly machinery, management etc. are available in large and lumpy units. Such inputs cannot be divided into small sizes to suit the small scale of production.

Inventory economies: These are the economies associated with having reserve capacity in spare parts, raw materials, and final products ready for disposal. These reserve capacities built in the production facilities leads to reduction in breakdown, shortages and thus leads to reduction in cost.

Selling economies: These are related with distribution of cost associated with selling and marketing over large scale production. The main types of such economies are (a) advertising economies, (b) other large-scale economies, (c) economies from specialized arrangements with exclusive dealers/distributors, (d) model-change economies.

Managerial economies: These result from (a) specialization of management, (b) mechanization of managerial functions.

Pecuniary economies: Pecuniary economies are those realized from paying lower prices for resources used in production and distribution of the products.This is due to bulk-buying by the firm as its size increases.

Supply Curve: That portion of marginal cost curve which is above minimum average variable cost curve is the supply curve of the firm in a perfectly competitive output market. It is derived by equating marginal cost curve with price of output.

Profit: It is defined as the difference between total revenue (TR) and total cost (TC). It is denoted by Greek symbol π (pi). This is taken to be objective function to maximize in most of the economic analysis. R−T Cπ = T

Iso-cost line or price line iso-outlay line or budget line: It represents various combinations of two inputs that can be purchased with the given outlay of funds or budget. Let be the budget that is C0 available for spending on two inputs and , and budget constraint is . So X1 X2 X XC0 = P x1 1 + P x2 2 iso-cost line is which has a slope equal to negative of inverse price ratio of two − XX2 = C0

P x2 P x2

P x11

inputs. The negative slope shows iso-cost line is downward sloping and is a straight line in two inputs. With increase in budget the line would shift parallel towards right and vice-versa being true.



If prices of two input changes in different proportions then change in inverse price ratio would result in line pivoting towards left or right as the case may be.

Shut down point: The point at which price received for the output is less than average variable cost of production. Mathematically, it is a point at which .V CP Y < A

Break even point: The point where TR=TC, it is a point of no profit no loss. To find linear break even quantity ( ) for single product enterprise following formula can be usedQ

where denotes total fixed cost, denotes price per unit of output and/(P )Q = F − V F P V denotes average variable cost. This formula assumes constant returns to scale prevails in production system or cost curve has linear relationship with output.

Production Possibility Curve (PPC): The production possibilities curve represents the amount of each output that can be produced given that the available resources or inputs are taken as fixed and given. A production possibilities curve thus represents the possible alternative efficient sets of outputs from a given set of resources. A simple mathematical function for PPC is as provided below

where is the amount of fixed resources and and are the output of two X0 = g Y Y( 1, 2) X0 Y 1 Y 2 enterprises. It is also known as product transformation curve.

MRPT: Marginal rate of product transformation is the slope of PPC and is defined as the rate at which output of one enterprise has to be reduced in order to increase the output of other enterprise on a given PPC. is the rate at which substitutes for keeping the resources fixed MRP T Y 2Y 1 Y 2 Y 1 or given.

MRP T Y 2Y 1 = dY 2

dY 1 = MP P XY 2

−MP P XY 1

When is negative then two enterprises are competitive in nature. If it is positive then MRP T Y 2Y 1 two enterprises are complementary in nature. And if it is either zero or infinity then enterprise is Y 1 supplementary to the other enterprise.

Iso-revenue line: A line which represents various combinations of two enterprises which together yields the same level of total revenue. Thus on a given iso-revenue line, total revenue remains the same.

Output Expansion Path: The line joining locus of points of optimum representing optimum combination of two enterprise.

Cost function: The cost function gives the minimum cost of producing the output vector given input prices. Cost is a function of output.

Revenue function: The revenue function gives the maximum revenue that can be produced using an input vector and given output prices.

Profit function: The profit function gives the maximum profit that can be achieved given input and output prices.



Iso-profit line: A line which shows various combinations of output and input which together produce the same level of profit given output and input price.

DMU: Decision making unit, in case of agriculture, a farm-firm is considered as DMU.

Frontier: Frontier is a boundary made/drawn such that it denotes maximum (minimum) level of a variable, let’s say production (cost), resulting in all efficient DMUs to be on the frontier and less efficient DMUs below (above) frontier and no DMUs to be above (below) the frontier. One example is production possibility frontier.

Technical Efficiency(TE): The ability of a DMU to produce the maximum feasible output from a given bundle of inputs, or the minimum feasible amounts of inputs to produce a given level of output. The former definition is referred to as output-oriented TE, while the latter definition is referred to as input-oriented TE. It can be calculated either as the ratio of actual output of DMU to corresponding maximum feasible output for the given bundle of inputs.

Price/Allocative Efficiency(AE): The ability of a technically efficient DMU to use inputs in proportions that minimize production costs given input prices. Allocative efficiency is calculated as the ratio of the minimum costs required by the DMU to produce a given level of outputs and the actual costs of the DMU adjusted for TE. Alternatively, it can also be calculated as ratio of technically efficient output level corresponding to DMU input bundle to optimum output level.

Economic/Overall Efficiency(EE): EE of a DMU is the product of both TE and AE. Thus, a DMU is economically efficient if it is both technically and allocatively efficient. Economic efficiency is calculated as the ratio of the minimum feasible costs and the actual observed costs for a DMU. Alternatively, it can also be calculated as the ratio of actual output of DMU to optimum hi output level.

X-inefficiency: According to Prof. Harvey Leibenstein, it is the inefficiency causing society to operate at a point inside the production possibility frontier when resources are used less efficiently.

Decomposition analysis: The set of techniques of decomposing a quantity or value into its constituents is referred in totality as decomposition analysis. In agriculture, most common scheme of decomposition is by Minhas who decomposed change in production into area effect, yield effect and first order interaction of the area and yield effect.

Relationship between andAP P Xi MP P Xi

XMP P Xi = AP P Xi + ∂X i

∂AP P Xii

Relationship between MC and AC

C C YM = A + ∂Y∂AC

Decision rule for finding optimum level of input use (for input )X i

Necessary condition is or and sufficient condition is that slope of MP P Xi = P Y

P Xi MV P Xi = MICXi curve is less than zero at the point of intersection of marginal physical product and priceMP P Xi



ratio. In other words, sufficient condition can be stated as “slope of marginal value product curve for input must be less than that of marginal input cost curve at point of intersection of MVP and MIC curves”.

Decision rule for finding optimum level of output Necessary condition is and sufficient condition is slope of MR curve must be less than R CM = M that of MC curve at point of intersection of MR and MC curves. In other words, sufficient condition can be stated as “at point of equality of MR and MC, MC curve must be rising and must cut MR curve from below”.

Necessary and sufficient conditions for finding the least cost combination of two inputs

Iso-cost line must be tangent to the isoquant is a necessary condition. Mathematically, it is , i.e. marginal rate of technical substitution for two inputs must equal inverse priceMRT Sx1x2 = P x2

P x1 ratio for two inputs. Sufficient condition is states that at point of tangency of iso-cost line and isoquant, isoquant must be convex to the origin.

Necessary and sufficient conditions for optimum level of two enterprise Necessary condition is that iso-revenue line must be tangent to the PPC while second order condition states that at the point of tangency PPC must be concave to the axis.

Derivation of input demand function To derive input demand function for input , we need to solve for and resulting X i MP P Xi = P Y

P Xi X i function would be . Here, input demand is a function of the price X i = h (P , .., , .., , , )x1 . P xi . P xn P Y Y of output and inputs, and output. Upon substituting output supply function in input demand function, it would be purely a function of the price of output and inputs.

Derivation of output supply function To derive output supply function, equate MC with price of output and solve for . If input levels Y appears in output supply expression then substitute it with input demand function and solve it once again for . Alternatively, one can start with maximization of profit (difference between TR and Y TC) and solve for .Y Derivation of conditional input demand function Set up a lagrangian objective function wherein cost is minimized subject to constraints posed by production function. Take first order derivative of lagrangian objective function w.r.t. inputs and lagrangian multiplier and solve for input quantities.

Derivation of cost function To derive cost function, substitute conditional input demand functions in cost equation and solve it in terms of . .Y C = c (P , .., , .., , )x1 . P xi . P xn Y

Cost of wrong input application: It is the cost of deviation in input use from optimum level of input use defined by . This is equal to change in profit as defined below:V P ICM = M

π rof it at optimum input level rof it at current input level ΔY ΔX P E Y X)Δ = P − P = P y − P x = ( y xY − P x X

ΔX



where and is the difference between optimum and non-optimum level of input and output XΔ YΔ use, respectively. is the partial elasticity of production of input for output .Ex

Y X Y

Laws of production: The laws of production describes the technically possible ways of increasing the production. Output may increase in various ways.

Law of diminishing returns: The marginal physical product of the given input used in production process eventually begin to decline as additional units of that input are applied to the production process with other inputs remaining the same. This law is also known as Law of variable proportions on account of variation in factor proportions as level of one input is varied.

Law of equi-marginal return: The limited resources should be allocated among its alternative uses in such a way that the marginal value product of the last unit of the resource is equal in all uses.

Technology: The purposeful application of information in the design, production, and utilization of goods and services, and in the organization of human activities.

Technical progress/change: It implies improvement in techniques of production thus leading to greater production with the same amount of input or same production with reduced amount of input or both. It is an economic measure of innovation.

Technical progress can be classified into two parts:

Embodied Technical Progress: Increase in productivity due to change in the form of capital goods in use. Improved technology which is exploited by investing in new equipment. New technical changes made are embodied in the equipment.

Disembodied Technical Progress: Increase in productivity through innovations or improvements in organization of production without there being change form of capital goods used in production. Improved technology which allows increase in the output produced from given inputs without investing in new equipment.

Capital-deepening technical progress: On an isoquant map drawn between labour (L) and capital (K), a technical progress can be termed capital-deepening if, along a line, starting from origin, L

K

ratio is constant but increases. This implies that technical progress increases the marginal MRT SK,L product of capital more than that of labour.

Labour-deepening technical progress: A technical progress can be termed as labour-deepening if, along a line starting from origin (with constant ratio) decreases. This means that L

K MRT SK,L technical progress increases the marginal product of labour more than that of capital on a given isoquant map drawn between labour (L) and capital (K).

Hick neutral technical progress: The Hicks neutral technical progress happens when marginal productivity of both labour and capital increases in the same proportion thereby keeping MRT SK,Lconstant along a line through the origin on a given isoquant map drawn between labour (L) and capital (K).



Partial Factor Productivity: It is the productivity of a given input as indicated by its average physical product.

Total factor productivity (TFP): It is the ratio of total output index to total input index.

TFP growth: It denotes the growth of total production arising from non-physical inputs like technical progress and farm extension activities, etc..

Risk: A situation when all possible outcomes are known for a given management decision and probability associated with each outcome is known.

Issue: A risk which has already occurred is considered as issue.

Opportunities: The positive risks are called Opportunities. An organization would like to take maximum advantage of these positive risks.

Risk Appetite: Amount and type of risk that an organization is prepared to seek, accept or tolerate.

Risk Tolerance: An organization’s readiness to bear the risk treatments in order to achieve its objectives.

Crisis: A situation in which an organization takes risk beyond capacity to bear or counter it.

Uncertainty: Those situations in which the parameters of the probability distribution of outcomes can not be empirically estimated.

Production Risk: This is the risk to production arising from natural causes such as pest attack, extreme weather phenomenon, breakage of/technical snag in machinery at crucial time, farm fire, etc.

Human/Personal Risk: This is the risk arising from the humans who are associated with production (owner, manager and labour). For example prolonged illness/death/non-availability of labourer/key management employee, labour strike.

Price/Market Risk: This is the risk arising from changes taking place in the market (both input as well as output). For example changes in availability of inputs, and volatility of input and output prices.

Institutional Risk: This sort of risk arises government changes in rules and regulation or policy orientation becomes affecting production and profit realization from it. For example imposing levy on some commodities, acreage restrictions, quantitative restrictions, ban on production in some region or restriction on resource utilization (for conserving groundwater).

Financial Risk: This is the risk arising from the changes taking place in financial markets. This risk is not included in market and institutional risk because firm can borrow from institutional as well as non-institutional sources, also because financial institutions are part of both institutions and markets.



Business Risk: The aggregate effect of production, personal, market and institutional risk is called Business risk. This is the risk facing the firm independent of the way in which it is financed. Hence financial risk is not part of it.

Risk-Benefit Matrix: A two dimensional plot of risk and benefits with degree of risk (low to high) on the horizontal axis and benefit/loss on vertical axis. This two dimensional representation has four quadrants

-1st quadrant shows high risk & benefit combination wherein decision maker takes calculated risk within tolerable limits. This is usually the case of farmer with cash crops and commercial farming

-2nd quadrant shows low risk & benefit combination wherein decision maker is risk averse person and wish to play it safe. In such cases the farmer grows crops for subsistence, uses less external inputs or in extreme cases rents out land for fixed rent.

-3rd quadrant shows low risk & loss combination wherein decision maker is totally risk averse and is not interested in taking up production activities, thus incurring only fixed cost in the short run.

-4th quadrant shows combination of loss and high risk which occurs when decision maker takes risk beyond capacity thus leading to crisis.

Agricultural Production Economics (APE): It is an applied field of science wherein the principles of choice are applied to the use of capital, labour, land and management resources in the farming industry.

Goals of APE: 1. to provide guidance to individual farmers in using their resources most efficiently and 2. to facilitate the most efficient use of resources from the standpoint of economy.

Subject matter of APE: Any problem of farmers’ that falls under the scope of resource allocation and marginal productivity analysis is the subject matter of agricultural production economics.

Objectives of APE:

1. to determine and outline the conditions which give the optimum use of capital, labour, land and management resources in the production of crops and livestock.

2. to determine the extent to which the existing use of resources deviates from the optimum use.

3. to analyse the forces which condition existing production pattern and resource use, and

4. to explain means and methods in getting from the existing use to optimum use of resources.

Yield Gap: This is the difference between the maximum-attainable yield and the farm-level yield for a crop variety.

IELD GAP Maximum ttainable yield arm evel yield IELD GAP I IELD GAP IIY = − a − F − l = Y + Y



IELD GAP I aximum ttainable yield ttainable yieldY = M − a − A

IELD GAP II ttainable yield arm evel yieldY = A − F − l

Maximum attainable yield: This is the yield of a crop variety on the research farm plot with no physical, biological or economic constraints and with the best known management practices at a given time and in a given ecology. Sometimes referred to as potential yield.

Attainable yield: This is the maximum yield that a sample farmer can achieve by following most of the technologies that are possible and known to the farmer and with the maximum efforts for a crop variety. This is the yield level achieved at frontline demonstration (FLD) plot of a sample farmer. Attainable yield is obtained by the farmer with his experience and knowledge.

Farm-level yield: This is the average farmer’s yield in a given target area at a given time and in a given ecology for a crop variety.

Anticipated yield: This is the yield anticipated by the farmers based on the actual efforts and technology followed by the farmers best known to them.

Observed/actual yield: This is the actual yield realized by the farmer.

Classification of Yield Gap: Yield gaps are classified according to constraints as following:

1. Agronomic gap: mainly due to biological and partly due to physical constraints such as climate/weather, soil, water, weeds.

2. Socio-economic gap: mainly due to socio-economic status, farmers’ tradition and knowledge, family size, household income/expenses/investment.

3. Institutional policy gap: mainly due to government policy, credit, input supply, land tenure, market research and development, extension.

4. Technology transfer and linkage gap: the competence and facilities of extension staffs, integration among research, farmers’ resistance to new technology; knowledge and skills, weak linkage among public, private and non-governmental extension staff.

5. Mixed gap: It is due to mix of above constraints. In this case, the socio-economic and institutional constraints should be solved before the agronomic gaps can be narrowed using improved technological packages.

Duality: The theory of duality links the production function models to the cost function models by way of a minimization or maximization framework. The cost function is derived from the production function by choosing the combination of factor quantities that minimize the cost of producing levels of output at given factor prices. Conversely, the production function is derived from the cost function by calculating the maximum level of output that can be obtained from specific combinations of inputs.



Shephard’s Lemma: Change in cost function arising from expansion path conditions with respect to the change in the price of the ith input is equal to the least cost quantity of the ith input used, when evaluated at any particular output level on the least cost (total cost) function.

Hotelling’s Lemma: Change in the indirect profit function with respect to a change in the jth input price is equal to the negative of the least cost combination quantity (optimal quantity) of the jth input. [Factor demand side]

Change in the indirect profit function arising from the output expansion path with respect to the kth product price is equal to the optimal quantity of the kth output that is produced. [Product Supply side]


Profit Function Approach by Gourav Kumar Vani, Assistant Professor, JNKVV, Jabalpur

Let Y=A X1α X2

β bet the production function representing the production process. Let PX1and

PX2be the prices of input X 1 and X2 , respectively. Then assuming no fixed cost we can write

profit equation as followingπ =PY Y−PX1

X 1−PX 2X 2 where PY is the price of output.

Now differentiating the above profit equation w.r.t. each input and equating it to zero gets us ∂π∂X 1

=PY∂Y∂ X1

−PX1=0 and ∂π

∂X2

=PY∂Y∂ X2

−PX2=0

Solving each of the above equation gets us input demand function for each input wherein optimal quantity of input demanded is a function of output price, its own price and level of output.

X 1=PYPX1

α Y and X 2=PYPX2

β Y

Now substituting input demand function for each input in place of input quantities in production function would get us

Y=A [PYPX1

α Y ]α

[PYPX2

β Y ]β

Solving the above equation for Y would get us

Y=(Aα α β β )(1 /1−α−β )(PYPX1

)(α /1−α−β )

(PYPX2

)(β /1−α−β )

This is the output supply function wherein the optimum quantity of output supplied is a function of prices of inputs and price of output. Now, cost equation can be written as C=PX1

X 1+PX2X2

Now substituting input functions in place of respective input quantities gets us

C=PX1

PYPX 1

α Y +PX2

PYPX2

β Y

which upon simplifying yields us C=PYα Y+PY β Y=(α +β )PY Y

Now substituting the simplified cost equation into profit function in place of cost of inputs and simplifying the terms yields usπ =PYY−(α +β )PYY=(1−α−β )PYY

Now substituting output supply function in place of output quantity yields us following after simplifying terms in the equation π =(1−α−β )(Aαα β β )(1 /1−α −β )PY

(1/1−α −β )PX 1

(−α /1−α−β )PX2

(−β /1−α −β )

This is profit function wherein profit of the firm is function of price of output and prices of inputs. Upon differentiating this profit function with respect to price of output, price of each input would yield output supply function and negative of respective input demand function, i.e.

∂ π∂PY

=Y=(Aαα β β )(1 /1−α −β )(PYPX 1

)(α /1−α−β )

(PYPX2

)(β /1−α −β )

∂π∂PX1

=−X 1=−α (Aα α β β )(1 /1−α−β )(PYPX1

)(1 /1−α−β )

(PX 1

PX 2

)(β /1−α −β )

Profit Function Approach by Gourav Kumar Vani, Assistant Professor, JNKVV, Jabalpur

∂π∂PX2

=−X2=−β (Aα α β β )(1 /1−α−β )(PYPX 2

)(1 /1−α−β )

(PX2

PX1

)(α /1−α −β )

Production Functions in Agricultural Economics, Created by Gourav Kumar Vani, Assistant Professor, JNKVV, Jabalpur.

Production Functions in Agricultural Production Economics 1. First degree polynomial or Linear Production function

XY = ao + ∑

ai i

2. Polynomial production function of degree ‘bi’

X x xY = a0 + ∑

ai i

bi + ∑

∑

aij i

bijbi

2.1 When then the polynomial is of degree 2, i.e. quadratic production functionbi = 1 2.2 When then the polynomial is of degree 1/2, i.e. square root production functionbi = 2

1 3. Generalized power function

eY = a0 ∏

X if i(X1,X2,...,Xn) g(X1,X2,...,Xn)

where, and can be any function.if (.) g (.) 3.1 When turns out to be constant for ith variable and equals zero (0) then the generalized power function becomesif (.) g (.)

Cobb-Douglas production function. Let then Cobb-Douglas production function can be written as which upon logif (.) = ai Y = ao ∏

X iai

transformation becomes .nY lnXl = lna0 + ∑

ai i

3.2 When and then generalized power function becomes Transcendental production function as whichif (.) = ai Xg (.) = bi i eY = ao ∏

X iai b Xi i

upon log transformation becomes .nY lnX Xl = lna0 + ∑

ai i + ∑

bi i

3.3 When and then generalized power function becomes Translog production function asif (.) = ai g (.) = 21 ∑

aij (lnX )i lnX( j)

which upon log transformation becomes (i=1,...,n and j=1,...,n).eY = ao ∏

X iai (lnX ) lnX2

1 ∑

aij i ( j) nY lnX l = lna0 + ∑

ai i + 2

1 ∑

∑

aij (lnX )i lnX( j)

4. Generalized ´Constant Elasticity of Substitution´ (CES) production function or SMAC (Sollow, Minhas, Arrow and Chennery) Production Function

δX Y = A[ 1−ρ + (1−δ)X2

−ρ] ρ−k

5. Variable Elasticity of Substitution (VES) production function [Liu and Hildebrand (1965)]

1

(1−δ) X X Y = A[ X2−ρ + δ 2

−ρm1−ρ(1−m)] ρ

−1

6. Spillman production function

or − RY = M ∑

Ai i

X i Y = a0 ∏

1−a( i

X i + bi)

7. Resistance fucntion

(b )Y −1 = a0 + ∑

ai i + X i

−1

8. Constant Marginal Share (CMS) production function X −mXY = AX1

α2(1−α)

2 9. Stone Geary Production Function

Y = A∏

(X −a )i i

γi

10. Generalized Production Function [Zellner and Revankar Production Function] e pY θY = ch h

The above production functions are subject to following restrictions: 1. The production period is sufficiently long to allow for completion of necessary technical progress. 2. The production period is sufficiently short enough so that decision maker is unable to exercise any discretion on changing level of fixed factors employed in production process. 3. The production period is sufficiently short so that technological changes does not affect the shape of production function during production process.

Assignment 1. Let the estimated production functions be as following

1.27 0.3X .3X −0.32X −0.44X .13X XY = 2 + 1 1 + 4 221

22 + 0 1 2

Find out 1. APP, 2. MPP, 3. Existence of inflection point 4. partial elasticity of production w.r.t. each input when and .7.38X1 = 1 .02X2 = 6 2. For the production function provided above in question number 1, find out expression for isoquant, isocline, ridge line and partial elasticity of substitution between and .X1 X2 3. For the production function provided above in question number 1, find out profit maximizing level of input use of both inputs if price of input and are respectively, Rs. 6 per unit and Rs. 14.4 per unit. Also the price of output is Rs. 12 per unit.X1 X2 Y 4. Derive profit function from the quadratic production function provided in question no. 1 and price information provided in question no. 3. 5. What is the profit maximizing level of output corresponding to the information provided in question no. 3.

2

6. Derive the corresponding cost function for the production function provided in question no. 1 when and are the prices of inputw1 w2 X1and , respectively.X2 7. Find out the cost minimizing level of output for cost function derived in question no. 6. 8. Derive the input demand function from the cost function derived in question no. 6. 9. Derive output supply function from the cost function derived in question no. 6 when price of output price is .P y 10. Find out whether the two inputs in the production function provided in question no. 1 are economic substitutes or complements ? 11. Find out the least cost combination of input and when price of inputs and production function are as provided in question no. 1X1 X2 and the budget amount is Rs. 40,000. 12. Whether the production function in question no. 1 follows law of diminishing marginal returns?

3

Properties Linear Quadratic Square root Cobb-Douglas

Transcendental Translog Spillman CES

Elasticity of substitution ∞ (Infinity) Variable Variable Always one

Variable over finite level of input use

Variable Variable Constat

Elasticity of production

Varies with level of input use



Constant and =β of each input

Depends on level of input use




Inflection point

Does not exists

Does not exists

Does not exists

Does not exists

Yes exists when βi>0, 0<αi<1 or βi<0, αi>1

Does not exists

Does not exists

LDMR Never Follows Follows Always follows Follows Follows Follows

Isoquant Linear Elliptical ------- Rectangular hyperbola

Concentric rings Rectangular

hyperbola Depends on value of σ

Ridge line No ridge lines exits

Rectangular and intersects at Von-Leibig point

Non-linear No ridge lines exits

Exists when βi>0, 0<αi<1 or βi<0, αi>1

Identical to x-& y-axis

Identical with input axes

Isocline Undefined Linear and may pass through origin

Non-linear Only IInd stage for Σβi<1

Non linear, passes through origin

Non-linear, passes through origin

Linear, pass through origin

Stages of production Only Ist stage Either stage I

or IInd & IIIrd IInd & IIIrd Linear, passes through origin

All three when βi<0, αi>1

Usually second stage

Only IInd stage

Usually only IInd stage

Homogeneous No No No Yes No No No Yes

4

Estimation OLS OLS OLS OLS OLS OLS MLE MLE Particulars Factor-Product Relationship Factor-Factor Relationship Product-Product Relationship Management Problem How much to produce How to produce What to produce

Objective Optimization of resource use Cost minimization Profit maximization

Goal Determination of optimum input to use and optimum level of output to produce

Determination of least cost combination of resources

Determination of optimum combination of enterprises

Concepts used Production function and iso-profit line Isoquant and iso-cost line Production possibility curve and iso-revenue line

Necessary condition for optimum

Iso-profit line must be tangent to the production function

Iso-cost line mut be tangent to isoquant

Iso-revenue line must be tangent to production posssibility curve

Sufficiency condition for optimum

At the point of tangency, production function must be concave to the x-axis.

At the point of tangency, isoquant must be convex to origin

At the point of tangency, production possibility curve must be concave to origin

MPP X = P XP Y

MRTSX X1 2= P x2

P x1 MRPSY Y1 2= P Y 1

P Y 2

Difference and Similarity between Returns to Scale and Economies of Scale

Difference/Similarity Returns to Scale Economies of Scale Existence Returns to scale exists when all inputs used by firm

in production process are varied in same proportion then output may vary in any proportion.

Economies of scale exists when the scale of production and/or operation undertaken by firm and/or industry is increased then the resulting cost of production/operation decreases.

Short/Long run A long run concept A long run concept

5

Types Increasing, Constant and Decreasing Internal and External

Extent Production All operations undertaken by firm including production Input Demand Functions, Output Supply Function & Profit Function Let be a production function with , , and being price of output , inputs , and , respectively. Then profit can be X XY = A 1

α2β P P 1 P 2 Y X1 X2

worked out as . Now substituting production function we get .Y −P X −P Xπ = P 1 1 2 2 AX X −P X −P Xπ = P 1α

2β

1 1 2 2

Now taking first order partial derivative of profit w.r.t. each input and setting it equal to zero we get

and thereafter, and finally we get α −P β −P ∂Y∂X1

= P YX1 1 = 0 ∂Y

∂X2= P Y

X2 2 = 0 α Pβ P YX1

= P 1YX2

= P 2 Y X Y X1 = ( PP 1 ) α 2 = ( P

P 2 ) β

Now substituting expression obtained for and into production function and after simplifying terms, we obtain output supply function X1 X2

as

.Here, output supply is a function of price of output and inputs.Y = Aα β( α β)( −α−β)1

1

( PP 1 )

( −α−β)1α

( PP 2 )

−α−β( 1β )

Now substituting output supply function in for in expression of and , we obtain input demand function as Y X1 X2

and . Here, input demand is a function of price ofX1 = Aα β( 1−β β)( −α−β)1

1

( PP 1 )

1− −α−β( 1β )( P

P 2 )−α−β( 1

β ) X2 = Aα β( α 1−α)

( −α−β)11

( PP 1 )

( −α−β)1α

( PP 2 )

(1− −α−β)1α

output and inputs. Input demand function has positive relationship with output price meaning that as output price would increase the profit

maximizing level of input would increase further. Increase in input price would reduce optimum level of input use.

Now to derive profit function, substitute the initial expression obtained for and into profit equation and simplify it in terms of . This X1 X2 Y

leads to . Now substitute output supply function for Y in the profit equation to Y −P X X Y − Y Yπ = P 1 1 + P 2 2 = P (α )+ β P = (1−α )+ β P

obtain profit function as . When all prices and profits are normalized by price of (P ) (P ) Pπ = (1−α )+ β Aα β( α β)( −α−β)1

1

1( −α−β)1

−α

2−α−β( 1

−β ) ( −α−β)11

output then the resultant profit function is referred to as normalized profit function.

6

Conditional Input Demand Functions and Cost Function

Let be the cost equation then the constrained output maximization problem can be solved through Lagrangian multiplier X XC = P 1 1 + P 2 2

method.

The Lagrangian objective function for this problem is . Now taking first order partial derivative of L w.r.t. X X L = A 1α

2β + λ C−P X X[ 1 1 + P 2 2]

each input and also , and setting it equal to zero, we get followingλ

which upon simplifying terms leads to .−P ; −P ; C−P X −P X ∂L∂X1

= α YX1 1 = 0 ∂L

∂X2= β Y

X2 2 = 0 1 1 2 2 = 0 ; X ; C X X X1 = α YP 1

2 = β YP 2

= P 1 1 + P 2 2

Now substituting expressions for and into cost equation which yields . This is the initial expression for cost function. X1 X2 C = (α )+ β Y

Now cost expression can be solved for output as .This expression is now substituted into expression for and which yields Y = C(α+β) X1 X2

conditional input demand function for each input as following

. Now, substitute conditional input demand function into production function as shown following: X X1 = ( αα+β) C

P 1 2 = ( βα+β) C

P 2

. Upon simplifying terms and solving for cost we get cost function as followingC P P Y = A( CP 1

αα+β)

α( CP 2

βα+β)

β= A α+β( α

α+β)α( β

α+β)β

1−α

2−β

This is the final expression of cost function wherein cost is a function of output, and price of inputs.C = ( AY )( +β)α

1 ( αP 1 )( +β)α

α

( βP 2 ) +β( α

β )(α )+ β

Now once we substitute cost function into conditional input demand function we have following final expression for input demand

functions

. Here, conditional input demand function is function of output, and X X1 = ( AY )( +β)α

1 ( αP 1 ) +β( α

−β )( βP 2 ) +β( α

β )2 = ( A

Y )( +β)α1 ( α

P 1 )( +β)αα

( βP 2 )( +β)α

−α

price of inputs.

1 terminology of agricultural production economics by ...jnkvv.org/pdf/07042020170432606.pdf · 2...

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