Is Product of Two Convex Functions Necessarily Convex? A Case of the MRP Curve

The curvature of the marginal revenue product curve plays an important role in most theoretic microeconomic models since it determines the size of profit contribution to an employer and optimality conditions of solutions. There are many well established introductory and intermediate microeconomic textbooks portray marginal revenue product curves as linear or concave to the origin. In nearly all cases, the MRP cannot be linear, nor can it be concave. In this analysis, most of the well-known production functions generate convex MRP curves.


Introduction
Input hiring decisions play an important role in microeconomic theory.These decisions are based on the monetary contribution of a variable input to the firm compared with its cost.In a competitive output market, it is known as the value of the marginal product (VMP), whereas in a monopoly market, the additional monetary contribution is often referred to as the marginal revenue product (MRP).These two concepts, VMP and MRP, are equal in the competitive output market as its output price equals marginal revenue.This paper shows first that the MRP curve cannot be linear in input factor although it is often drawn as such in many microeconomics textbooks.As is shown by Yang, Means, and Moody [1], for a given domain, the concavity or convexity property may not be preserved beyond functional addition.The MRP curve cannot be concave to the origin, due to the fact that even the product of two concave functions may not necessarily be concave.
In this paper, we analyze the shape of the MRP curve mathematically, then present a set of simulations for a variety of well-known production functions frequently used in microeconomics textbooks.The non-concavity of the MRP curve (especially in production stage II) is witnessed in these simulations.The curvature of the MRP curve plays an important role in most standard microeconomic models (see e.g.Takayama [2], Baumol and Klevorick [3]) in determining the characteristics of optimal solutions.

Shapes of MRP Curves
Given that the marginal revenue curve of a linear demand is twice as steep, the curvatures of the MRP and VMP are qualitatively similar.Thus, our paper focuses on the shape of MRP curves.The quintessence of the MRP curve is the product of two functions: Marginal product of labor L MP and marginal revenue MR in the output market.To investigate the shape of the MRP curve, we start with MR P QP   or where  and indicates decreasing marginal product of labor and . The curvature of the MRP curve can be determined by the second derivative with respect to labor or where is the second derivative of L MP .Note that the sign is not easily amenable to the analysis: It requires , , and . Tables 1 and 2 illustrate the shapes of the MRP curve in the cases of linear demand and constant elasticity demand.Given a linear demand function and a standard concave producion function Q, with some fixed capital, K, the shape of t function which is convex with respect to L, and the L MP function.Since many well-known production funct such as the Cobb-Douglas [4], CES (see Arrow, Chenery, Minhas and Solow [5]), VES (see Mukerji [6] and Revankar [7]) and Translogarithm (see Christensen, Jorgenson and Lau [8]), have rather convex ions, L MP , it is not likely that the product of two convex func is linear or concave curves.
For a constant elasticity demand   In order to explore the possible shapes of MRP curves, the following production functions are examined: Cobb-Douglas

Quadratic asticity of Substitution (CES)
Variable Elasticity of Substitution (VES)    In general, the MRP decreases as the input increases.Strictly speaking, a linear MRP curve from a quadratic production function cannot reflect characteristics of the well-behaved production functions.Using the parameters of well-known production functions 2 (Cobb-Douglas, Constant and Variable Elasticity of Substitution (CES, VES) and Translogarithm) and assuming the simplest case of linear demand function 3 , Figures 1-7 illustrate specific VMP (MRP) curves.These curves cannot be concave to the origin for most empirically relevant parameters.

Case II: MRP Curves in Imperfectly
Competitive Output Markets rkets is The MRP of labor in imperfectly competitive ma the product of  1 Differentiate Equations ( 3)-( 9) with respect to L, we could obtain Equations ( 10)-( 16). 2 Simulation of Figure 8 is based on well-known estimation by Douglas [9].The parameters on quadratic and cubic production functions are quite common, i.e., e ≤ 0 and c ≤ 0 as in many textbooks.The CES production functions are from Arrow, Chenery, Minhas and Solow [5]; VES production functions are from Lovell [10] and the translogarithmic production functions are from Humphrey and Moroney [11]. 3The demand functions used with the corresponding production functions to calculate the MRP are the following: is the case of a constant L MP nd f ear (linear production function) with a linear dema unction.It is to be pointed out that product of a lin MR function (convex with respect to L) and a convex L MP (in most well-known cases) is not likely to be concave.A concave MRP reflects logical inconsistency in nearly all cases 5 .
Examination of undergraduate textbooks that cover microeconomics indicates a range of approaches in discussions of the MRP curves.Of the 61 texts surveyed, most texts present the logically erroneous and empirical infeasible concave MRP curve.Only 15 of 61 texts (24.59%) illustrate the MRP curves correctly, as convex to the origin 6 .In particular, of the prominent texts including th red by Nobel laureates in Eco omics, Price Theory by Milton Friedman [12] and Econ ics by illiam ose autho n om Paul A. Samuelson and W D. Nordhaus [13] draw MRP curves correctly.

Conclusions
We have simulated VMP and MRP curves with empirically relevant parameters of demand and production,   which indicate that most of MRP curves are convex to east two reasons.The first is simple accuracy and consistency.A concave VMP (MRP) curve is possible only with the cubic function in perfect competetion.As for MRP curves, a linear or concave one is a mathematical oddity or impossibility.Even the MRP curve of the cubic production function is convex in the efficient production range where the marginal product is relatively small.The second has to do with the rhetoric of economics.As McCloskey [14] discusses at length, there are many rhetorical devices used in economics just to simplify a concept.Potentially damaging rhetoric exists where we try too hard to "twist" the curves to illustrate optimum labor hiring.the origin.This finding is quite in contrary to the straight or concave MRP curves often found in textbooks.
The fact of errors in some 75.41% of textbooks is important for at l   As the article makes clear, under reasonable parametric limits, the MRP curve should be convex to the origin.Finally, from the viewpoint of policy implications, a convex MRP may well translate into hiring fewer workers if wage rate is relatively high: a disturbing phenomenon facing developed economies nowadays.
can be tions shown readily that 0 P  , P 0 0   , and P  .Without the knowledge o rod functio of P and P alone are not sufficient to determine the shape of M P curve.The MRP curves, for almost all possible cases, can be ne n the p uction n, signs R varia of even two linear functions (one is the function of the ither linear nor concave to the origin in terms of the ble input.They cannot be linear because the product other) cannot be linear.

. 1 . 8 )
Case I: MRP Curves in a Perfectly Competitive Output Market In perfect competition, the VMP or MRP is simply the (constant price or MR and L MP .ence, the MRP has the exact curv As a consequ a f th ture o e L MP .Only when L MP is linear in L, can one draw a corresponding linear MRP curve.The L MP for a given K corresponding to the previously specified production functions have the following forms 1 linear MRP (VMP) curve is lo sistent except in the case of a quadratic production function (nction in L. In such a case, a liner MP times a constant price gives rise to a linear VMP (MRP) curve.Notice that except in the case of the roduction function with a negative coefficient on the cubic terms in which AP L and L MP are concave to the origin, all other MRP (VMP) curves are convex.

L MP and 4 MR
. As is true with the es, most MRP curves cannot be li VMP curv near.The fundar ant.mental difference from the previous case is that MR curve is no longe a const It is a linear function of output, which is concave to L. The MRP curves are shown in Figures 8-14.Again, the only exception found , ,