Answer to Question #242987 in Microeconomics for Mehak

Question

Homothetic production function includes homogeneous production function as a special case . Justify this statement.

Answer 1

Homothetic refers to a comprehensive category of monotonic rising homogeneous production functions that encompasses the underlying homogeneous functions. The firm's isoclines, including the long-run expansion path, would be straight lines from the origin if the production function is homogenous (to any degree).

The isoquants of the homothetic production function are the same as those of the underlying homogeneous function, however with different quantity indices in most cases.

That is why, for a homothetic production function, the firm's growth route and isoclines would be straight lines from the origin, and along any such straight line with a fixed ratio of inputs, the firm's MRTS of L for K or the ratio of MPL to MPK would be constant.

This is because the MRTS is a function of the ratio of the input quantities for both the underlying homogeneous function and its monotonic modifications. In other words, the ratio of MPL to MPK would be determined by relative rather than absolute input values.

Therefore, the homothetic production function would give us

Slope of IQ₁ at A₁ = Slope of IQ₂ at A₂ and

Slope of IQ₁ at B₁ = Slope of IQ₂ at B₂.

where A₁ and A₂ are points on two separate rays from the origin, and B1 and B2 are points on two different rays from the origin. That is, along any straight line from the origin, the slope of the IQs would be constant. Now, if the slopes of IQs are equal along any ray, MPL/MPK must not change with a proportionate change in L and K at any point in the input space.

From the other side, the iso-cost lines (ICLs) for various cost levels are parallel since the input price ratio is constant. As a result, at locations of tangency between the ICLs and IQs, the slope of the IQs, MRTS, or MPL/MPK would be constant, equal to the slope of the ICLs.

This means that if the production function is homothetic, the ratio of the input numbers will be constant at the points of tangency, which are locations on a ray from the origin. To put it another way, homotheticity necessitates that the firm's expansion path parallels such a beam.

A homogeneous production function is homothetic as well-indeed, it is a subset of homothetic production functions. The production function in Fig. is homogeneous if it also has f(tL, tK) = tnQ, where t is any positive real integer and n is the homogeneity degree.

Any homogeneous function is a homothetic function, but any homothetic function is not a homogeneous function, as shown above. Q = f (L, K) = a - (1/LK) is a homothetic function since it yields fL/fK = K/L = constant. It is not, however, a homogeneous function because it does not yield f (tL, tK) = tnQ.