What is a homothetic production function?
Homothetic functions are functions whose marginal technical rate of substitution (the slope of the isoquant, a curve drawn through the set of points in say labour-capital space at which the same quantity of output is produced for varying combinations of the inputs) is homogeneous of degree zero.
Is production function same as cost function?
Production and Costs. We’ve explained that a firm’s total costs depend on the quantities of inputs the firm uses to produce its output and the cost of those inputs to the firm. The firm’s production function tells us how much output the firm will produce with given amounts of inputs.
How do you show a function is homothetic?
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.
How do you tell if preferences are homothetic?
Formally, we say a preference relation is homothetic if for any two bundles x and y such that x ∼ y, then αx ∼ αy for any α > 0. questions, which is even harder. preference relation º is homothetic if and only if it can be represented by a utility function that is homogeneous of degree one.
What are the different types of production function?
The different types of production function (as shown in Figure-16).
- Cobb-Douglas Production Function: Cobb-Douglas production function refers to the production function in which one input can be substituted by other but to a limited extent.
- Leontief Production Function:
- CES Production Function:
How does production function relate to cost function?
Production function: Relates physical output of a production process to physical inputs or factors of production. marginal cost: The increase in cost that accompanies a unit increase in output; the partial derivative of the cost function with respect to output.
What is the relationship between the production function and the cost curves?
Assuming that factor prices are constant, the production function determines all cost functions. The variable cost curve is the constant price of the variable input times the inverted short-run production function or total product curve, and its behavior and properties are determined by the production function.
Are log functions homothetic?
3 A linear logarithmic utility function is both additive and homothetic; all expendi- ture proportions are constant, and elas- ticities of substitution among all pairs of commodities are constant and equal to unity.
What is homogeneous and homothetic functions?
A function is homogenous of order k if f(tx,ty)=tkf(x,y). A function is homothetic if it is a monotonic transformation of a homogenous function (note that this second function does not need to be homogenous itself).
The broad class of monotonic increasing functions of homogeneous production functions, which includes also the underlying homogeneous functions, is called homothetic. If the production function is homogeneous (of any degree), the firm’s isoclines including long-run expansion path would be straight lines from the origin.
Why is the expansion path of a homothetic production function straight line?
That is why the firm’s expansion path and its isoclines would be straight lines from the origin also for a homothetic production function, and along any such straight line with a fixed ratio of the inputs, the firm’s MRTS of L for K or the ratio of MP L to MP K would be constant.
Is Q = (1/L α k) a homogeneous or homothetic function?
For example, Q = f (L, K) = a — (1/L α K) is a homothetic function for it gives us f L /f K = αK/L = constant. But it is not a homogeneous function for it does not give us f (tL, tK) = t n Q.
What is the difference between homogeneous and homothetic?
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.