How do you find the homothetic function?
How do you find the homothetic function?
- You can know if a function is homothetic simply by looking at the marginal rate of substitution of the function.
- Graphically, the marginal rate of substitution of a homothetic function are on the same ray of line from the origin of a graph.
- A homothetic function is a monotonic transformation of a homogeneous function.
What is a homogeneous production function?
Definition: The Linear Homogeneous Production Function implies that with the proportionate change in all the factors of production, the output also increases in the same proportion. Such as, if the input factors are doubled the output also gets doubled. This is also known as constant returns to a scale.
How do you know if a utility function is homogeneous?
A function f: C → R is homothetic if for every x, y ∈ C and t > 0, f(x) ≥ f(y) if and only if f(tx) ≥ f(ty). One consequence of the definition of homotheticity is that f is equivalent to g defined by g(x) = f(tx). Any homogeneous utility function is also homothetic.
What is homothetic function in economics?
In consumer theory, a consumer’s preferences are called homothetic if they can be represented by a utility function which is homogeneous of degree 1.
What is the meaning of homothetic?
: similar and similarly oriented —used of geometric figures.
What is the difference between homogeneous and homothetic?
Homogeneous Production Function – If x and x` produce y units of output then 2x and 2x` produce 2y units of output. Homothetic Production Function- If x and x` produce same level of output y then 2x and 2x` produce same level of output, but not necessarily 2y.
What is a linearly homogeneous function?
What makes an equation homogeneous?
A first‐order differential equation is said to be homogeneous if M( x,y) and N( x,y) are both homogeneous functions of the same degree. The method for solving homogeneous equations follows from this fact: The substitution y = xu (and therefore dy = xdu + udx) transforms a homogeneous equation into a separable one.
What is homogeneous function with example?
For example, a function is homogeneous of degree 1 if, when all its arguments are multiplied by any number t > 0, the value of the function is multiplied by the same number t. Here is a precise definition.
What is CES in economics?
Constant elasticity of substitution (CES), in economics, is a property of some production functions and utility functions. Specifically, it arises in a particular type of aggregator function which combines two or more types of consumption goods, or two or more types of production inputs into an aggregate quantity.
What is the definition of homothetic production function?
This implies that if the production function is to be homothetic, then the ratio of the input quantities would be a constant at the points of tangency, i.e., the points of tangency lie on a ray from the origin. In other words, homotheticity requires that the firm’s expansion path coincides with such a ray.
How do you know if a production function is homogeneous?
A homogeneous production function is also homothetic—rather, it is a special case of homothetic production functions. In Fig. 8.26, the production function is homogeneous if, in addition, we have f (tL, tK) = t n Q where t is any positive real number, and n is the degree of homogeneity.
In other words, homotheticity requires that the firm’s expansion path coincides with such a ray. A homogeneous production function is also homothetic—rather, it is a special case of homothetic production functions.
How do you know if a transformation is homothetic?
R+is called homotheticif it is amonotonetransformationof a homogeneous function. Put more formally, if there is a monotonictransformation such thaty7!f(y)2R+and a homogeneous functiong: Rn7!