Single-input production

By way of review, we examine a basic application of calculus in economics.  Suppose a competitive firm's production function is Y = 100X - X² where Y is its output and X is its variable input.  The firm purchases input X at constant unit price w, has fixed cost K, and sells its output Y at constant unit price p.  So the firm's profit function can be written out mathematically:

R = pY - wX - K = p(100X - X²) - wX - K. 

The profit-maximizing quantity of X to use can be determined by taking the derivative of this profit function with respect to X, setting that equal to zero, and solving for X.

The derivative of this profit function is R' = p(100 - 2X) - w.  If we set this equal to zero we get p(100-2X) = w. 

Note that the expression 100 - 2X is the derivative of the production function, i.e. the marginal physical product of X.  Multiplying this by p yields the marginal value product of X.  So the first-order condition p(100-2X) = w restates a principle you learned in your first microeconomic class: a profit-maximizing a firm should use the quantity of input where the MVP of X equals w, the marginal factor cost of X.  Also note that the fixed cost, the constant K, dropped out of the derivative and has no bearing on the marginal productivity or use of X.

Solving this first-order condition for X yields X = 50 - w/p, which is the firm's factor demand for X.  The firm's responses to changes in w and/or p are logical: if w increases, the firm uses less X and produces less Y, while if p increases, the firm uses more X and produces more Y. 


Cobb-Douglas production

The production function for a two-input firm is often modeled in Cobb-Douglas form, e.g., Z = aX^bY^c where a, b and c are constants.  If the exponents b and c sum to one, this implies that production exhibits constant returns to scale: if you double both inputs X and Y, you produce double the amount of Z.  If a + b < 1 then production exhibits decreasing returns to scale. 

Suppose a competitive firm has Cobb-Douglas production Z = 10X^0.4Y^0.4 where inputs X and Y cost $v and $w per unit respectively and Z sells for $p per unit.  The firm's objective is to maximize its profit function R  = pZ - vX - wY = p[10X^0.4Y^0.4] - vX - wY. 

To maximize profit with respect to both X and Y, we take the partial derivatives with respect to each input, set both derivatives equal to zero, and solve this pair of equations for the optimal values of X and Y:

R_X = 4pX^-0.6Y^0.4 - v = 0   and   R_Y = 4pX^0.4Y^-0.6 - w = 0, or rearranged:

4pX^-0.6Y^0.4 = v   and   4pX^0.4Y^-0.6 = w  

Multiplying both sides of the first equation by X, and both sides of the second equation by Y yields:

4pX^0.4Y^0.4 = vX = 4pX^0.4Y^0.4  = wY    which simplifies to   Y/X = v/w.

Again, this restates a basic principle of microeconomics.  If a two-input firm is maximizing its profits, its marginal rate of technical substitution (MRTS) between X and Y, Y/X, must equal the factor price ratio v/w.  This is logical too: if v, the price of input X, increases relative to w, the price of input Y, then the profit-maximizing firm would use relatively less X and relatively more Y.   A particular feature of this Cobb-Douglas production function is that the firm always spends an equal amount on X and Y regardless of their prices, since vX = wY. 

To obtain the factor demands for X and Y, substitute Y = vX/w into one of the first order equations and solve for X:

4pX^-0.6(vX/w)^0.4 = v    so    X^0.2 = 4pv^-0.6w^-0.4   and   X = 1024p⁵v^-3w^-2.     Similarly, Y = 1024p⁵v^-2w^-3

Note that these factor demands respond to relative price changes, but if p, v and w all change in the same proportion there is no change in X, Y or output, since the sum of the exponents is zero.

These factor demand functions have constant elasticities: 
E_X = (dX/dv)(v/X) = -3(1024p⁵v^-4w^-2)v/(1024p⁵v^-3w^-2) = -3
E_Y = (dY/dw)(w/Y) = -3(1024p⁵v^-2w^-3)w/(1024p⁵v^-2w^-3) = -3.


Constrained optimization

The calculus applications discussed up to this point have been straightforward unconstrained optimization problems.  But constrained optimization problems in economics are generally more interesting.  For example, consider the results from a simple unconstrained utility experiment in which a 10-year-old female subject reported her marginal utilities for successive spoonfuls of Edy's Double Chocolate Fudge ice cream (her favorite) slightly past her point of satiation.  The subject's cumulative utility is approximated by the quadratic function U = 0.26(60X-X²), and her total utility is maximized at X* = 30 spoonfuls. 

The trick to incorporating a constraint into an optimization problem is to express it as a zero-valued term.  In this case suppose there are only N spoonfuls of ice cream, so N - X >= 0.  Using a technique developed by French mathematician Joseph-Louis Lagrange (1736-1813), we multiply the constraint expression N - X by a Lagrangean multiplier denoted  λ and add this expression into the optimization problem, e.g.: max L =  0.26(60X-X²) + λ(N - X).  Now we are simply maximizing the same objective function plus a zero-value expression representing the constraint. 

One of three possible conditions must apply: either (1) the constraint is non-binding (e.g., N = 40 so N - X* = 40 - 30 > 0), and the multiplier λ must equal zero; or (2) the constraint is only just binding (e.g.., N = 30 so N - X* = 30 - 30 = 0), and the multiplier λ also equals zero; or (3) the constraint is strictly binding (e.g.., N = 20 so N - X* = 20 - 20 = 0), and the multiplier λ is non-zero.   (These are known as Kuhn-Tucker conditions.)


To maximize L =  0.26(60X-X²) + λ(N - X). we take the first partial derivatives with respect to X and λ, set them equal to zero and solve for the utility-maximizing valeus of X and λ: L_X = 15.6 - 2X - λ = 0 and  L_λ = N - X >= 0.   In this case, the variable λ represents the marginal utility of X.  If N >= 30, then the consumer reaches satiation at X = 30 where marginal utility of X is zero.  But if N < 30, the consumer cannot reach satiation, and the marginal utility of X remains positive. 


Unconstrained profit maximization versus constrained cost minimization. 

A less trivial application of constrained optimization contrasts the profit-maximizing firm with a cost-minimizing firm that is required to meet a specific production quota with the same production function. 

As before, the firm has Cobb-Douglas production Z = 10X^0.4Y^0.4 where inputs X and Y cost $v and $w per unit respectively.  The firm's objective is to minimize its cost function C  = vX + wY subject to the production requirement 10X^0.4Y^0.4 >= Z₀.  We rewrite the production requirement as the inequality  Z₀ - 10X^0.4Y^0.4 <= 0  and incorporate it into the cost-minimization problem by attaching the multiplier to it:  L = vX + wY + λ(Z₀ - 10X^0.4Y^0.4).

as before, the final term always equals zero.  Either  (a) Z₀ - 10X^0.4Y^0.4 = 0  and λ > 0; or (b) Z₀ - 10X^0.4Y^0.4 < 0  and λ = 0; or (c) both Z₀ - 10X^0.4Y^0.4 = 0  and λ = 0.

Taking the first partials with respect to X, Y and λ, and setting them equal to zero, we obtain:
L_X = v - 4λX^-0.6Y^0.4 = 0;   L_Y = w - 4λX^0.4Y^-0.6 = 0;   and L_λ = Z₀ - 10X^0.4Y^0.4 = 0 .

The first two first-order conditions can be rearranged and combined to obtain the same condition Y/X = v/w.  Both the cost-minimizer and the profit-maximizer equate MRTS with the factor price ratio. 

But the factor demands are different.  Substituting Y = vX/w into the third first-order condition and solving for X:
Z₀ = 10X^0.8v^0.4w^-0.4   so   X' = Z₀^1.2510^-1.25v^-0.5w^0.5.    Likewise  Y' = Z₀^1.2510^-1.25v^0.5w^-0.5.

Note that the pre-determined level of output, rather than output price, drives these factor demands. 

The cost-minimizing firm's factor demands also have constant elasticities: 
E_X' = (dX'/dv)(v/X) = -0.5(Z₀^1.2510^-1.25v^-1.5w^0.5)v/(Z₀^1.2510^-1.25v^-0.5w^0.5) = -0.5
E_Y' = (dY'/dv)(v/Y) = -0.5(Z₀^1.2510^-1.25v^0.5w^-1.5)w/(Z₀^1.2510^-1.25v^0.5w^-0.5) = -0.5.

Note that the cost-minizing firm is far less responsive to input price fluctuations than the profit-maximizing firm:  E_X' = -0.5 < E_X = -3 and E_Y' = -0.5 < E_Y = -3.


Demand theory

Demand theory models consumers as little happiness factories where consumer goods are inputs for the production of utility.   Maximizing utility subject to a budget constraint is another class of optimization problems solvable via the Lagrangean method.

For example, suppose utility has the same Cobb-Douglas form U = 10X^0.4Y^0.4, where X and Y are quantities of two consumer goods, Px and Py are the unit prices of X and Y respectively, and the consumer's budget constraint is M >= PxX + PyY where M is her total money budget.

Restating the budget constraint as M - PxX - PyY  >= 0, we can incorporate it into the Lagrangean maximization problem
max L =  U + 10X^0.4Y^0.4 + λ[M - PxX - PyY]
where λ is the Lagrangean multiplier.  The three first-order conditions are:

L_X = 4X^-0.6Y^0.4 - λPx = 0;    L_Y = 4X^0.4Y^-0.6 - λPy = 0;    and  L_λ = M - PxX - PyY = 0.

The expressions 4X^-0.6Y^0.4 and  4X^0.4Y^-0.6 are the marginal utilities of X and Y respectively.

Rearranging and combining the first two equations, we get MUx/MUy = Y/X = Px/Py:  The marginal rate of substitution between X and Y equals the price ratio or marginal rate of transformation.  Rearranged, PxX = PyY: the budget shares of X and Y are equal. 

The demands for X and Y can be solved by substituting Y = PxX/Py into the budget expression:  M = PxX + Py(PxX/Py) = 2PxX  so  X = M/(2Px)
and likewise Y = M/(2Py). 

The own-price demand elasticity for X is Ex = [dX/dPx][Px/X] = [-M/(2Px²)][Px(/M/(2Px))] = -1;  likewise Ey = -1.  Cross-price elasticities are zero.  Income elasticities are +1. 

Now contrast these consumer demands derived from the utility-maximization problem with demands derived from the dual problem where we minimize the expenditure necessary to meet a pre-determined utility constraint U₀ <= 10X^0.4Y^0.4.  The dual problem is set up as:

min L = PxX + PyY + µ[U₀ - 10X^0.4Y^0.4]

and the first-order conditions are L_X = Px - µ4X^-0.6Y^0.4 = 0;    L_Y = Py - µ4X^0.4Y^-0.6 = 0   and   L_µ = U₀ - 10X^0.4Y^0.4 = 0.

Rearranging and combining the first two equations, we get MUx/MUy = Y/X = Px/Py, as before.  

Substituting Y = PxX/Py into the utility constraint yields  U₀ = 10X^0.4[PxX/Py]^0.4 = 10X^0.8Px^0.4Py^-0.4 which is solved for X' = 10^-1.25U₀^1.25Px^-0.5Py^0.5.
Likewise, Y' = 10^-1.25U₀^1.25Px^0.5Py^-0.5.

As with the comparison between the profit-maximizing and cost-minimizing firms, the more constrained demands exhibit lower own-price responsiveness. 
Here the own-price elasticities of demand are equivalent to those of the cost-minimizing firm: Ex' = (dX'/dPx)(P/X') = -0.5  and Ey' = -0.5.
The cross-price elasticities are both +0.5. 

The demand functions derived from the utility-maximization problem are conventional demand schedules X = X(Px,Py,M) and Y = Y(Px,Py,M), although in this simple Cobb-Douglas formulation the cross-price terms drop out.  The demand functions derived from the expenditure minimization problem are income-compensated demand schedules X = X'(Px,Py,U₀) and Y = Y'(Px,Py,U₀): the budget is implicitly adjusted to hold utility constant as prices change, so these demands show only the substitution effects of price changes.  


Econometric modeling of Cobb-Douglas functions

Demand functions such as these can be converted to linear form by taking tle logarithms of both sides.  For example, a profit-maximizer's Cobb-Douglas factor demand X = Kp^av^bw^c. can be transformed to logX = log(K) + alog(p) + blog(v) + clog(w).  Given data on X, p, v and w, you could convert these data to log form and then estimate the parameters log(K), a, b and c  by conventional OLS regression.   Just be aware that the additive residual from the estimated log formulation is actually multiplicative in the original function: this could cause bias problems.