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Isoquants The possible combinations of inputs that will produce given level of output can be defined as isoquant: An isoquant; a curve that shows the efficient Combination of labor and capital that can produce a Single (iso) level of output (quantity).Thus, Q = (Lp + KP)1/P (2 - 12) The marginal rate of technical substitution for a CES isoquant is : MRTS = - (L/K)P-1 (2 - 13) At every point on a CES isoquant, the constant elasticity of substitution is: ?So, efficiency involves that isoquants do not cross.

Isoquants
The possible combinations of inputs that will produce given level of output can be defined as isoquant:
An isoquant; a curve that shows the efficient
Combination of labor and capital that can produce a
Single (iso) level of output (quantity). If the production
Function is: Q =f(L,K), then the equation of for an
Isoquant where output is held constant at Qis:
Q =f(L,K) (2-5)
lf we assume that the share of both coefficients of inputs is the same as follows: Q =f(L0.5 Ko..), the three isoquants for Q= 6, Q = 9, and Q = 12 can be depicted in figure 2.2.
these isoquants show a firm’s flexibility in producing a given level of output. They are smooth curves because the firm can use fractional units of each input.
there are many combinations of labor and capital, (L, K), that will produce 6 units, including (1, 36), (2,18), (3, 12), (4,9), (6, 6), (9, 4), (12, 3), (18, 2), (36,1). Figure 2.2 shows some of these combinations as points a to fon Q = 6 isoquant.
Properties of isoquants:
From figure 2.2 we can realize that an isoquant holds quantity constant. We also can consider four major properties of isoquants most of them resulting from efficient production by firms.
The further an isoquant is from the origin, the greater the level of output is achieved
That is, the more inputs a firm uses, the more output it gets (if it produces efficiently).
For instance, at point e in figure 2.2 a firm can produce 6 units of output using 12 workers and 3 units of capital. If the firm holds the number of workers constant and wants to increase the level of output, this entails that a firm must add 9 more units of capital as in point g.
Isoquants do not cross: Intersections are inconsistent with the requirement that the firm always produces efficiently.
For instance, If the Q=10 and Q = 14 Isoquants are crossed, the firm could produce at either output level with the same combination of labor and capital. The firm will have a sort of idiocy if it produces 10 units of output if it can produce 14 units and this indicates that the firm produces inefficiently. Thus, that labor-capital combination should not lie on the Q = 10 isoquant, which should include only efficient combinations of inputs. So, efficiency involves that isoquants do not cross.
Isoquants slope downward: If Isoquants slope upward, a firm can produce same level of output with few inputs or many inputs. Producing with many inputs will be inefficient. since isoquants show only efficient production, an upward sloping isoquant is impossible.
Isoquants must be thin: It has the same argument we just used to show that Isoquants slope downward.
Shapes of Isoquants
The shape of an isoquant shows how readily a firm can substitute one input for another. There are two extremecases where production processes can take.
The first, the inputs are perfectly substituted.
The second, the inputs cannot be substituted for each other. If inputs are perfectly substituted, each isoquant is a straight line.
Assume we can produce one kilo of a tomato paste, a, either via using one kilo of Tanta tomato xor one kilo of mansura tomato y or a bundle of Tanta & Mansura tomatoes.
This technology has a linear production function.
The isoquant for Q = one kilo of tomato paste is 1= x + y, or Y = 1-x. The slope of this straight-line isoquant is -1.
For example, we can have three isoquants with Q = 1, 2, and 3 as depicted in panel a figure 2.3.
If inputs cannot be substituted, all isoquants will have right angles. The dashed lines show that isoquants would be right angles if we included (inefficient production) as in panel b
Isoquants lie between straight lines & right angels. Along a curved isoquant, the ability to substitute an input for another varies as in panel c.
Substituting Inputs
The slope of isoquant explains the ability of a firm to replace one input with another keeping output constant. The slope is called the marginal rate of technical substitution MRTS.
MRTS = (Change in capital)/(Change in labor) = ∆K/∆L = dK/dL
ln other words, MRTS tells us how many units of capital the Firm can replace with additional unit of labor keeping output constant and vice versa. The MRTS is negative since isoquants slope is downward.
To obtain the slope of a point on an isoquant, we are in need to differentiate isoquant, Q* =f(L,K), with respect to L & K. we can also use capital as an implicit function of labor: K(L).
for a given quantity of labor, there is a level of capital such that Q’units are produced. Differentiating with respect to labor (as mentioned before, bear in mind that output does not change along the isoquant as we change labor), we have:
(dQ’)/dL = 0 = ∂f/∂L + ∂f/∂K dK/dL =MPL +MPK dK/dL (2-6)
Where MPK is the marginal product of capital. That is, as we move down & to the right along an isoquant (as in figure 2-2), we increase the amount of labor slightly, so we must decrease the amount of capital to stay on the same isoquant.
For instance, if the MP, is 2 &the firm hires one extra worker, its output rises by two units. Alike, a little extra capital increases output by MPK*. Thus, the change in output due to drop in capital in response to the increase in labor is : MPK *dK/dL.
To sum up, if we are to stay on the same isoquant – that is, hold output constant – these two effects must be offset each other: MPL = - MPK *dK/dL.
Rearranging Eq. 2-6, we find that the MRTS (the change in capital relative to the change in labor) equals the negative of the ratio of the marginal products:
MRTS = dK/dL = - MPL/MPK (2 – 7)
Example 3: What is the marginal rate of technical
Substitution for Cobb-Douglas production function in its
General form: Q=A La kb?
Answer:
Calculate MPL & MPK by differentiating the Cobb-Douglas first with respect to labor & then with respect to capital.

MPL = ∂Q/∂L = a A La-1 Kb = aQ/L

MPK = ∂Q/∂K = b A La-1 kb-1 = bQ/K

Substitute the expression for MP, and MPx into equation 2.7 to determine the MRTS as follows:



         MRTS = dK/dL = - MPL/MPK = - (a Q⁄L)/(b Q⁄K) = - a/b  K/L         (2 – 8)

Thus, the MRTS for a Cobb-Douglas production function is a Constant, -a /b, times the capital labor ratio K/L.

Diminishing Marginal Rate of Technical Substitution
To interpret how MRTS changes along an Isoquant we will estimate that Q=4 = L0.5 K0.5 isoquant for a fabrics plant as in figure 2.4
Setting a = b = 0.5 in equation 2.8, we find that the slope along this isoquant is MRTS = -K/L
At point d in figure 2.4 where K = 8 and L = 2, the MRTS = -4
The dashed line that is tangent to the Isoquant at that point has the same slope.
The MRTS will be -1 at point e when K= 4 and L= 4, and the MRTS will be – 0.25 at f when K= 2 and L= 8.
As we move down and to the right across this curved isoquant, the slope becomes flatter – the slope gets closer to zero- as the ratio K/L moves closer to zero.
Recall again, in special case where isoquants are straight lines, isoquants don’t show diminishing MRTS as neither input becomes more valuable in the production process (Figure 2.3 a).
In other special case of fixed proportions, where isoquants are right angles (more precisely single points as in figure 3.3 b), no substitution is possible.
The Elasticity of Substitution
We have observed that the MRTS varies as we move along a curved isoquant.
so, it is useful to have a measure of this curvature, which reflects the ability of the firm to substitute capital for labor.
The well – known measure of the ease of substitution is the elasticity of substitution, σ( Greek letter sigma), which is percentage change in the capital – labor ratio divided by percentage change in the MRTS:
σ = ((d(K/L))⁄(K⁄L))/(dMRTS⁄MRTS) = (d(k/l))/(d MRTS) × MRTS/(K/L) (2 – 9)
This measure tells us how the input factor ratio changes as the slope of the isoquant changes. If the elasticity is large (a small change in the slope results in a big increase in the factor ratio), the isoquant is relatively flat.
As elasticity falls, isoquant becomes more curved. We also will see that as we move along the isoquant, both K/L and the absolute value of MRTS change in the same direction (as in figure 2.4), so elasticity is positive. Since the factor ratio, K/L, and absolute value of MRTS, MRTS, are positive numbers, so the logarithm of each is significant.
lt is useful to write elasticity of substitution as logarithm derivative:
σ = (dIn(K/L))/dIn|MRTS| (2 – 10)
Constant Elasticity of Substitution Production Function:
Generally, the elasticity of substitution differs along an isoquant, but an exception is the constant elasticity of substitution (CES) production function;
Q = (aLP + bKP )d/p (2 – 11)
Where p is a positive constant and for simplicity we presume that a = b = d = 1. Thus,
Q = (Lp + KP)1/P (2 – 12)
The marginal rate of technical substitution for a CES isoquant is :
MRTS = - (L/K)P-1 (2 - 13)
At every point on a CES isoquant, the constant elasticity of substitution is:
σ = 1/(1-ρ ) (2 - 14)
The linear, fixed-proportion & Cobb-Douglas production function are special cases of constant elasticity production function.
Linear Production Function: By setting ρ = 1 on equation 2-12, we have linear production function Q = L + K. We can realize that at every point along a linear isoquant elasticity of substitution is infinite (a = 1/(1- ρ)) = 1/0. This indicates that the two inputs are perfect substitutes for each other.

Cobb-Douglas Production Function:
as ρ reaches zero, CES isoquants are Cobb-Douglas isoquants and hence CES production function approaches Cobb-Douglas production function. Consistent with equation 2-13 for CES production function, MRTS = -(K/L)p-1 . When ρ reaches zero, MRTS = -(L/K).
Likewise, setting a = b in eq. 2.8, we find that the Cobb-Douglas MRTS is the same. The elasticity of substitution is σ=1/1-ρ=1/1=1 at every point along Cobb-Douglas isoquant.

Fixed-Proportion Production Function:
as ρ reaches negative infinity, this production function reaches Fixed-Proportion Production Function, which has right-angle isoquants or, more precisely single-point isoquants
The elasticity of substitution σ = 1/ (1/ - ∞), which approaches zero. Thus, substitution between the inputs is impossible.