So far, we simply used the Lagrange function without asking where it comes from. This chapter will o¤er a derivation of the Lagrange function and also an economic interpretation of the Lagrange multiplier. In maximization problems employing a utility function, the Lagrange multiplier can be understood as a price measured in utility units. It is often called a shadow price.
2.3.1 Where the Lagrangian comes from I
The maximization problem
Let us consider a maximization problem with some objective function and a constraint, maxx1;x2
F (x1; x2) subject to g(x1;x2) = b: (2.3.1)
The constraint can be looked at as an implicit function, i.e. describing x2 as a function of x1; i.e. x2 = h (x1) : Using the representation x2 = h (x1) of the constraint, the maximization problem can be written as
maxx1
F (x1;h (x1)) : (2.3.2)
The derivatives of implicit functions
As we will use implicit functions and their derivatives here and in later chapters, we brie‡y illustrate the underlying idea and show how to compute their derivatives. Consider a function f (x1; x2; :::; xn) = 0: The implicit function theorem says stated simply -that this function f (x1; x2; :::; xn) = 0 implicitly de…nes (under suitable assumptions concerning the properties of f (:) - see exercise 7) a functional relationship of the type x2 = h (x1; x3; x4; :::; xn) : We often work with these implicit functions in Economics and we are also often interested in the derivative of x2 with respect to, say, x1:
In order to obtain an expression for this derivative, consider the total di¤erential of f (x1; x2; :::; xn) = 0;
df (:) = @f (:)
@x1 dx1+ @f (:)
@x2 dx2+ ::: + @f (:)
@xn dxn= 0:
When we keep x3 to xn constant, we can solve this to get dx2
dx1 = @f (:) =@x1
@f (:) =@x2: (2.3.3)
We have thereby obtained an expression for the derivative dx2=dx1 without knowing the functional form of the implicit function h (x1; x3; x4; :::; xn) :
For illustration purposes, consider the following …gure.
Figure 2.3.1 The implicit function visible at z = 0
The horizontal axes plot x1 and, to the back, x2:The vertical axis plots z: The increas-ing surface depicts the graph of the function z = g (x1; x2) b: When this surface crosses the horizontal plane at z = 0; a curve is created which contains all the points where z = 0: Looking at this curve illustrates that the function z = 0 , g (x1; x2) = bimplicitly de…nes a function x2 = h (x1) : See exercise7 for an explicit analytical derivation of such an implicit function. The derivative dx2=dx1 is then simply the slope of this curve. The analytical expression for this is - using (2.3.3) - dx2=dx1 = (@g (:) =@x1) = (@g (:) =@x2) :
First-order conditions of the maximization problem
The maximization problem we obtained in (2.3.2) is an example for the substitution method: The budget constraint was solved for one control variable and inserted into the objective function. The resulting maximization problem is one without constraint. The problem in (2.3.2) now has a standard …rst-order condition,
dF
dx1 = @F
@x1 + @F
@x2 dh
dx1 = 0: (2.3.4)
Taking into consideration that from the implicit function theorem applied to the con-straint,
dh
dx1 = dx2
dx1 = @g(x1; x2)=@x1
@g(x1; x2)=@x2; (2.3.5) the optimality condition (2.3.4) can be written as @x@F
1
@F=@x2
@g=@x2
@g(x1;x2)
@x1 = 0: Now de…ne the Lagrange multiplier @F=@x@g=@x2
2 and obtain
@F
@x1
@g(x1; x2)
@x1 = 0: (2.3.6)
As can be easily seen, this is the …rst-order condition of the Lagrangian
L = F (x1;x2) + [b g (x1;x2)] (2.3.7) with respect to x1:
Now imagine we want to undertake the same steps for x2;i.e. we start from the original problem (2.3.1) but substitute out x1: We would then obtain an unconstrained problem as in (2.3.2) only that we maximize with respect to x2: Continuing as we just did for x1
would yield the second …rst-order condition
@F
@x2
@g(x1; x2)
@x2 = 0:
We have thereby shown where the Lagrangian comes from: Whether one de…nes a La-grangian as in (2.3.7) and computes the order condition or one computes the …rst-order condition from the unconstrained problem as in (2.3.4) and then uses the implicit function theorem and de…nes a Lagrange multiplier, one always ends up at (2.3.6). The Lagrangian-route is obviously faster.
2.3.2 Shadow prices
The idea
We can now also give an interpretation of the meaning of the multipliers . Starting from the de…nition of in (2.3.6), we can rewrite it according to
@F=@x2
@g=@x2 = @F
@g = @F
@b:
One can understand that the …rst equality can “cancel” the term @x2 by looking at the de…nition of a (partial) derivative: @f (x@x1;:::;xn)
i = lim4xi!0 f (x1;:::;xlim i+4xi;:::;xn) f (x1;:::;xn)
4xi!0 4xi : The
second equality uses the equality of g and b from the constraint in (2.3.1). From these transformations, we see that equals the change in F as a function of b. It is now easy to come up with examples for F or b: How much does F increase (e.g. your utility) when your constraint b (your bank account) is relaxed? How much does the social welfare function change when the economy has more capital? How much do pro…ts of …rms change when the …rm has more workers? This is called shadow price and expresses the value of b in units of F:
A derivation
A more rigorous derivation is as follows (cf. Intriligator, 1971, ch. 3.3). Compute the derivative of the maximized Lagrangian with respect to b,
@L(x1(b) ; x2(b))
The last equality results from …rst-order conditions and the fact that the budget constraint holds.
The Lagrange multiplier is frequently referred to as shadow price. As we have seen, its unit depends on the unit of the objective function F: One can think of price in the sense of a price in a currency, for example in Euro, only if the objective function is some nominal expression like pro…ts or GDP. Otherwise it is a price expressed for example in utility terms. This can explicitly be seen in the following example. Consider a central planner that maximizes social welfare u (x1; x2) subject to technological and resource constraints,
max u (x1; x2) subject to
x1 = f (K1;L1) ; x2 = g (K2;L2) ; K1+ K2 = K; L1 + L2 = L:
Technologies in sectors 1 and 2 are given by f (:) and g (:) and factors of production are capital K and labour L: Using as multipliers p1; p2; wK and wL; the Lagrangian reads
Here we see that the …rst multiplier p1 is not a price expressed in some currency but the derivative of the utility function with respect to good 1, i.e. marginal utility. By contrast, if we looked at the multiplier wK only in the third …rst-order condition, wK = p1@f =@K1; we would then conclude that it is a price. Then inserting the …rst …rst-order condition,
@u=@x1 = p1;and using the constraint x1 = f (K1;L1)shows however that it really stands for the increase in utility when the capital stock used in production of good 1 rises,
wK = p1 @f Hence wK and all other multipliers are prices in utility units.
It is now also easy to see that all shadow prices are prices expressed in some currency if the objective function is not utility but, for example GDP. Such a maximization problem could read max p1x1 + p2x2 subject to the constraints as above. Finally, returning to the discussion after (2.1.5), the …rst-order conditions show that the sign of the Lagrange multiplier should be positive from an economic perspective. If p1 in (2.3.9) is to capture the value attached to x1in utility units and x1 is a normal good (utility increases in x1;i.e.
@u=@x1 > 0), the shadow price should be positive. If we had represented the constraint in the Lagrangian (2.3.8) as x1 f (K1;L1)rather than right-hand side minus left-hand side, the …rst-order condition would read @u=@x1+ p1 = 0 and the Lagrange multiplier would have been negative. If we want to associate the Lagrange multiplier to the shadow price, the constraints in the Lagrange function should be represented such that the Lagrange multiplier is positive.