Question

Describe, using diagrams or algebra where appropriate, the following concepts/terms: homogeneous production function; and, elasticity of substitution.

Answer 1

A operate is said to be homogeneous of measure n if the multipli­cation of all the unbiased variables by the same constant, say λ, results in the multiplication of the elegant variable by λn. Therefore, the operate

Y = X2 + Z2

is homogeneous of measure 2 when you consider that

(λX)2 + (λZ)2 = λ2 (X2 + Y2) = λ2Y

A operate which is homogeneous of degree 1 is alleged to be linearly homogeneous, or to display linear homogeneity. A creation operate which is homogeneous of degree 1 shows regular returns to scale for the reason that a doubling all inputs will lead to an special doubling of output. So, this form of creation function reveals regular returns to scale over the entire range of output. Ordinarily, if the construction perform Q = f (okay, L) is linearly homogeneous, then

F (λk, λL) = λf (ok ,L) = λQ

for any mixture of labour and capital and for all values of λ. If λ equals three, then a tripling of the inputs will result in a tripling of output.

There are more than a few examples of linearly homogeneous services.

Two such examples are the next:

Q = aK + bL

and Q = A kα L1-α zero < α < 1

The second instance is known as the Cobb-Douglas production perform. To see that it is, certainly, homogeneous of measure one, feel that the organization originally produces Q0 with inputs K0 and L0 after which doubles its employ­ment of capital and labour.
Houses:

There are quite a lot of intriguing properties of linearly homoge­neous creation functions. First, we can categorical the operate, Q = f (okay,L) in either of two substitute types.

(1) Q = Kg (L/k) or,

(2) Q = Lh (okay/L)

This property is traditionally used to show that marginal merchandise of labour and capital are functions of handiest the capital-labour ratio.

In distinctive, the marginal merchandise are as follows:

MPk = g (L/ok) (L/ok) g (L/okay)

and MPL = (L/ok)

the place (L, k) denotes the by-product of g (L/k). The significance of that is that the marginal products of the inputs do not trade with proportionate raises in both inputs. On account that the marginal price of technical substitution equals the ratio of the marginal merchandise, which means the MRTS does not change alongside a ray by way of the starting place, which has a steady capital- labour ratio. Seeing that the MRTS is the slope of the isoquant, a linearly homo­geneous construction perform generates isoquants that are parallel alongside a ray through the starting place.

Growth path:

If a corporation employs a linearly homogeneous creation operate, its expan­sion course can be a straight line. To confirm this point, allow us to start from an preliminary factor of cost minimisation in Fig.12, with an output of 10 models and an employment (utilization) of 10 units of labour and 5 items of capital. Now, think, the corporation needs to develop its output to 15 units. On the grounds that input costs do not exchange, the slope of the new iso­quant have to be equal to the slope of the long-established one.

Elasticity of substitution is the elasticity of the ratio of two inputs to a creation (or utility) perform with respect to the ratio of their marginal merchandise (or utilities). In a competitive market, it measures the percentage exchange in the ratio of two inputs used in line with a percentage change of their costs. It measures the curvature of an isoquant and thus, the substitutability between inputs (or goods), i.E. How handy it's to replacement one input (or good) for the other.