# MRTS In Economics-Marginal Rate of Technical Substitution| MPL, MRS

## Overview

MRTS in economics refers to the Marginal Rate of Technical Substitution which is termed as the slope of isoquant.

Isoquants are defined almost the same as the indifference curve with few changes. As a result, we will take a quick look at isoquants before studying MRTS in economics in detail.

## Isoquants

An isoquant is a level set of the long-run production function also known as equal product curves/iso-product curve. It is a convex line that shows all possible combinations of inputs keeping the level of output the same.

Briefly, the line is drawn through all sets of points where the same level of output is yielded while the quantity of two or more inputs is changed.

As an indifference curve locates the utility maximization matter of consumers likewise an isoquant helps in mapping the cost-minimizing factor of producers.

Furthermore, isoquants show the extent to which a firm can substitute between two inputs keeping the level of output the same. Isoquants tell alternative methods for producing different levels of output.

Furthermore, there is a difference between isoquants and isocost curves. While an isoquant depicts the possible combinations of the inputs or desired level. On the other hand, isocost curves represent the combinations of the actual capacity of the firm which they can afford.

#### Graphical Explanation

Isoquants are generally marked along the isocost curves which represent the capital-labour graphs depicting the tradeoff between both the inputs in the production function.

Moreover, the term explains holding or decreasing one input while adding one input which further decreases the marginal output reflected by the convex shape.

However, a family of isoquants can therefore be termed as an isoquant map ( showing a combination of no. of isoquants with a differentiated quantity of output).

In short, the isoquant map also indicates increasing or decreasing returns to scale resembling out of the distances between the isoquants.

Let’s say, the distance between the isoquants decreases with an increase in output, there exists increasing returns to scale where doubling of inputs results in more than twice of original output on the isoquant.

Production isoquant (strictly convex) and isocost curve (linear)

### Properties of Isoquants

• The combination of inputs or the isoquant higher or to the right of the previous isoquant always results in higher output.
• Just as an indifference curve, isoquants never intersect or cross each other. This nature can be defined by a contradiction where a single combination cannot have two outputs. For example, Point A
• Isoquants are typically convex to the origin and downward sloping.
• The marginal product decreases with an increase in output but never reaches the negative sense in the rational world will not decrease output by increasing input.

### Shapes of Isoquants

#### I) CASE A

Combined with the isocost line, isoquants are presented to solve the cost-minimizing problem for a particular level of output. The figure shown below depicts a typical case of isoquants with a convex shape.

The firm has fixed unit costs for their inputs while the isocost curves are downward sloping and linear. Therefore the point where isoquant is tangent to the isocost curve, that point depicts the cost minimization combination of inputs for production of output.

Moreover, the point connecting tangent points of all isocosts and isoquants is termed as an expansion path. Here MRTS in economics is diminishing.

#### II) CASE B

Now, in case the two inputs can be replaced for each other or are perfect substitutes, then the isoquant curve will be a straight line as represented in the graph below.

Let’s say if a boy likes both blue or red pens, so he would aim at a combination where he gets the maximum quantity of pens keeping his productivity/output the same.

This explains that in the graph input X can be exchanged for input Y at an unchanging rate. As a result, marginal rates do not change or remain constant. Here MRTS in economics is constant.

#### III) CASE C

In the third case, there arises a situation where inputs are perfect compliments which shows that they need to be used together.

For example for the sale of a pair of shoes both left and the right shoe is needed. Likewise, the graph below depicts that both inputs X and Y can be used when combined efficiently in a definite ratio.

This occurs at the kink of the isoquants to maximize the profits. Here MRTS in economics is either 0 or infinity.

## Marginal Rate of Technical Substitution(MRTS in economics)

### What is MRTS in economics ( marginal rate of technical substitution)?

MRTS in economics or Marginal Rate of Technical Substitution is a theory that explains the rate by which one factor is decreased to increase the other factor keeping the level of efficiency constant or the same.

Further MRTS in economics is the slope of the tangent line at any point on the isoquant.

MRTS in economics explains the give and take relationship between inputs to lever a constant production like labour and capital.

Moreover, this should be understood that MRTS in economics varies from MRS( marginal rate of substitution) which focuses on balancing the product rather than MRS which deals with market equilibrium.

### CALCULATION OF MRTS in economics

MRTS in economics (L, K) i.e. substitution of labour for capital measures the rate at which quantity of capital can be reduced for every one unit increase in labour, holding the quantity of output constant. Briefly, it indicates the firm the amount of capital needed to replace a single unit of labour keeping output the same.

Formula for MRTS in economics(l, k) = -dk/dl = MPl/MPk = – ∆K/∆L(for fixed q)                                                                                                    = slope of IQ curve                                                                                                                                          Where, K = capital                                                                                                                                                        L = labor                                                                                                                                                            MP = marginal product

• (Assumption: Typically assume that the labour is increased by one unit)

#### Graphical representation

For a better understanding of MRTS in economics, we will take a graphical illustration. Here capital is presented on Y-axis and labour on X-axis termed as dK/dL. Also as the shape of isoquant or input values changes as studied above, the slope or MRTS changes.

• In Case A, since the isoquant is convex in shape therefore MRTS in economics is diminishing.

For example: MRTS at B point = 12-8 / 2-1 = 4/1 = 4
C Point = 8-5 / 3-2 = 3 /1 = 3
D Point = 5-3/ 4-3 = 2/1 = 2                                         where the figure is decreasing

• In Case B, since the inputs are perfect substitutes as a result the MRTS in economics is constant.

For example: MRTS at        B point = 12-9/2-1 = 3                                                                                                                                    C point = 9-6/3-2 = 3                                                                                                                                      D point = 6-3 / 4-3 = 3                                                                                      where MRTS remains the same.

• In Case C, where the inputs are complements, MRTS in economics is either 0 or ∞ .

For example: MRTS                         B point = 12-9 / 1-1 = 3/0 = ∞                                                          D point = 3-3 / 3-2 = 0/1 = 0

Therefore, the graph explains that to increase output both the inputs need to be increased in the mentioned ratio.

### Applications of MRTS in economics

1. MRTS in economics helps to display the slope of the isoquant which defines the rate at which one input can be substituted for the other keeping the level of output unchanged.
2. In the concept of producer equilibrium, the producers aim at minimum cost to gain the maximum amount of profits. By stating a combination of inputs that incur the least money to produce maximum output, the producer attains an equilibrium.
3. As a result, the decision to determine a combination is related to the substitution principle and MRTS in economics.
4. MRTS in economics helps to choose between the factors. If there are two factors of production in the plant and one which gives a higher quantity of output than the other. As a result, the producer would invest comparatively more in the factor which gives better results.

## Elasticity Of Substitution

A measure of how easy it is for a firm to substitute one factor for another. It is equal to the percentage change in (k/l) ratios to the percentage change in MRTS(l,k) holding output constant. It is denoted by ‘ 𝞂 ‘.

FORMULA for the elasticity of substitution:

𝞂 = % Δ (K/L) / % Δ[MRTS(l,k)]   = d(K/L) / (K/L) ÷ d(MRTS) / [MRTS(l,k)]   ≥ 0

• If MRTS in economics does not change at all for change in K/L, we might say that substitution is easy because the ratio of MP’s does not change. Alternatively, if MRTS in economics does change rapidly for a small change in (K/L), we say that substitution is difficult.
• The elasticity of substitution is also a measure of the curvature of isoquant, while lesser curvature means higher 𝞂.
1. MRTS(l,k) at point A ≈ 4
2. MRTS(l,k) at point B ≈ 1
3. K/L at point A = 20/5 = 4
4. K/L at point B = 10/10 = 1

𝞂 = (1-4) / 4 ÷ (1-4)/4 = 1

### Special Production functions

#### 1. Linear production function ( perfect substitutes ) – constant MRTS

• Production function : Q = aL + bK
• MP(k) = b       ,        MP(l) = a         ,        MRTS(l,k) = a/b (constant)
• Constant returns to scale prevail
• Isoquants are parallel straight lines with slopes = -a/b (downward sloping)
• Since MRTS does not change with change in K/L, therefore, 𝞂 = ∞

For example:- a given amount of fuel say natural gas can always be substituted for each litre of petrol.

For example:- A firm needs to store 200 GB of data so can choose between two options i.e a high capacity drive starting 20 GB and another hard drive with 10 GB capacity. Here one 20 GB drive is a perfect substitute for two 20 GB drive

#### 2. Fixed proportion production function ( perfect compliments )

• Also known as Leontief production function and is given by Q = min{aL,b K}
• In this type of production function inputs are combined in a fixed proportion.

For example, One molecule of water requires two atoms of hydrogen and one unit of an oxygen atom. Hence water = ( H/2, O)

• Isoquants are L-shaped with kink point given by aL = bK, and MRTS=0 or ∞
• The elasticity of substitution 𝞂 = 0, because at kink of isoquant, for a very small change in K/L causes an infinite change in MRTS in economics say from horizontal to vertical or vice-versa.

#### 3. Cobb-Douglas production function

• Q = f(L,K) AKaLb  where A ,a , b > 0
• Isoquants are convex and downward sloping
• Returns to scale:

f(tL,tK) = A(Kt)a(Lt)b CRS for a+b=1
= ta+b(AKaLb) DRS for a+b<1                                                                                                                    = ta+bf(K,L) IRS for a+b>1

• MRTS(l,k) = MP(l) / MP(k) = AKabLb-1 / AaKa-1Lb = bK/aL
• Elasticity of substitution : log MRTS(l,k) = log (b/a) + log (K/L)

Dlog MRTS / Dlog (K/L) = 1

𝞂 = dlog(K/L)/dlogMRTS(l,k)  = 1

## Marginal Product of Labor(MPL)

Keeping all the other inputs constant, marginal product of labor(MPL) refers to increase in total production when the labor is increased by one unit. It explains the additional output that can be produced by an additional unit of factor(labor) keeping all other factors same.

MPL is higher than the original cost of the worker since the additional output is generated by the new worker. However, the marginal product of labour decreases with an increase in total production in the short-run due to the problem of optimum utilization.

### Calculation of MPL formula

• As a result, a discrete case:            MPl of nth unit = TPn -TPn-1 = Y/L

For continuous case:                        MPl = y(.) / l = fl

### Graphical representation with an example

Let’s consider an example in which the total production changes with change in the labour while assuming the other factors to remain constant.

The above two can be better understood with the graph depicting total production and the marginal product of labour.

Now, there is a concept that states that even if labour is increased and another input or capital is held constant then also the marginal product of labour decreases because of the unbalance between optimum production ratios. Furthermore, the production is less efficient where workers are more and machines are less.

Furthermore, after talking about the production the firm focuses on how the revenue can be maximised which can be studied in detail in Baumol’s sales maximisation model.

## Marginal Rate of Substitution

Although the concepts studied above are related to the production function however marginal rate of substitution is a term introduced under the consumption function.

Just like isoquants, the indifference curve represents a different combination of goods that offers the consumer the same level of satisfaction/cardinal utility.

In the same way as MRTS in economics, we have MRS which represents the amount of Y whose loss can be compensated by one unit of X.

In other words, MRS(x,y) represents the amount of Y which the consumer has to give up for the gain of one additional unit of X so that level of satisfaction remains the same.

Formula for MRS

I slope of IC I = -dY/dX = MRS(x,y) = MUx/ MUy = – Y / X

Now, the character of diminishing marginal rate of substitution can be explained briefly. As the consumer increases the consumption of X, the marginal utility of the X decreases ( due to diminishing marginal utility).

However, due to the sacrifice of Y its marginal utility increases. According to the given equation above, MRS decreases since the numerator or MU(x) decreases, and the denominator increases.

Hence the diminishing marginal rate of substitution arises due to the convex shape of the indifference curve or the reduction in the willingness to sacrifice Y to increase one unit of X.

### Marginal rate of substitution in different preferences

1. Firstly, in the case of perfect substitutes, the indifference curve is linear whereas MRS = constant. The consumer is indifferent between both the commodities and gives him the same level of satisfaction.
2. Secondly, in the case of perfect complements, the indifference curve is L-shaped where MRS = 0 or infinity. Here the consumer wants both goods equally with each other together.
3. Thirdly, the indifference curve has a typical convex shape by COBB-Douglas where the diminishing marginal rate of substitution prevails. Moreover, the consumer chooses the combination which gives the maximum level of satisfaction.
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