Definition
A function f is a homothetic if for all (x, y) in domain D,
f(x) = f(y), t > 0 implies f(tx) = f(ty)
A homogeneous function of any degree k is homothetic, but not all homothetic functions are homogeneous.
Homothetic production function
A production function is homothetic if when output level goes up, the multiple inputs are required at the fixed level. So we can decide whether a production function is homothetic, by checking if the ratio of conditional factor demand is fixed when output level changes, or when q changes, if /equation.png)
is fixed, the production function is homothetic.
Homothetic utility function
A utility function is homothetic, if when wealth increases, the ratio of requirements of two goods are fixed.
Note: here we can regard the utility function as a special production function, which uses commodities as "inputs" to "produce" utility.
Features of homothetic utility function:
- homogeneous at degree of 1.
- When wealth increases, ratio between consumption of goods do not change, or increase of wealth changes the quantity of consumption. but not the pattern of consumption
- In other words, taste do not change with time
The homothetic utility function is easy to use, but may not satisfy real world cases. That's why we would like to represent non-homothetic utility