Notes on Lagrange Multipliers

<strong>Notes</strong> on Lagrange Multipliers and the Consumer Problem
A. Understanding the Lagrangian Optimization Problem
A typical consumer problem has form
Max u(x, y)
s.t. px + qy < W.
The Lagrangian corresponding to this consumer problem is
L(x, y, λ) = u(x, y) + λ(W - px - qy).
The Lagrangian optimization problem is based on finding a combination of values
(x*, y*, λ*) that maximize the objective function L(x, y, λ) in (x, y) and minimize the
objective function L(x, y, λ) in λ subject to the requirements x, y, λ > 0.
The format of this optimization problem may seem unfamiliar; the maximization and
minimization components of the problem may appear to work at cross-purposes. The
easiest way to think about Lagrangian optimization is to consider subproblems related to
the optimal choices (x*, y*, λ*).
(1) Given (y*, λ*), the choice of x* must maximize L(x, y, λ).
(2) Given (x*, λ*), the choice of y* must maximize L(x, y, λ).
(3) Given (x*, y*), the choice of λ* must minimize L(x, y, λ).
The “solution” to the Lagrangian problem is a combination of values (x*, y*, λ*) that
solve subproblems (1) through (3) simultaneously.
The choice of λ is best illustrated in a graph of bundles relative to the budget constraint.
Figure 1 graphs the budget line y = 40 – 2x, corresponding to (say) W = 40, p = 2, q = 1.

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Figure 1:
The Lagrange Multplier and the Budget Constraint
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Cost < W
B
Cost > W
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Subproblem 3: Choosing λ given (x, y) to minimize L(x, y, λ).
Bundles in set A have cost < W, meaning px + qy < W, or equivalently W - px - qy > 0.
Since the product λ(W - px - qy) in the Lagrangian objective function is positive in set A,
λ*(x, y) = 0 minimizes L(x, y, λ) for any combination of values (x, y) in set A.
Similarly, bundles in set B have cost < W, implying that W - px - qy > 0. Since the
product λ(W - px - qy) is negative in set B, λ*(x, y) → ∞ minimizes L(x, y, λ) for any
combination of values (x, y) in set B.
Finally, bundles on the budget line have cost = W, implying that W - px - qy = 0. Since
the product λ(W - px - qy) is zero for any value of λ on the budget line, the choice of λ is
unrestricted (at least in terms of producing a minimum value of the objective function)
for any combination of values (x, y) in set B.
Combining Subproblems 1&2 with Subproblem 3.
We can simply work our way through the implications of choices (x*, y*) in set A, set B
and on the budget line to determine where the solution to the Lagrangian problem must
lie in the graph. In essence, we are using a form of the “Consistent Conjectures” (a fancy
term for “Guess and Verify”) method for identifying the properties of the solution to a
particular mathematical problem. In this case, we conjecture in turn that the solution
(x*, y*, λ*) to the Lagrangian problem has values of (x*, y*) that fall in Set A, in Set B,
or fall on the budget line. For each of these three possibilities, we attempt to verify
whether it is possible to solve subproblems 1 to 3 simultaneously with (x*, y*) in the
given region.

Conjectured Solution in Set A: For combinations of (x, y) in Set A, λ*(x, y) = 0.
But given λ = 0, the Lagrangian objective function is simply u(x, y), which takes its
unconstrained maximum at values (x, y) with x → ∞, y → ∞. In words, if there is no
penalty for going beyond the budget constraint, then the consumer should choose a
bundle in set B as the solution to subproblems (1) and (2) – but this contradicts the
conjecture of (x, y) in set A.
Conjectured Solution in Set B: For combinations of (x, y) in Set B, λ*(x, y) = ∞.
But given λ = ∞, the Lagrangian objective function dramatically penalizes expenditures
beyond budget W while dramatically rewarding expenditures less then budget W. In this
case, the consumer should choose x* = y* = 0 (i.e. a bundle in set A) as the solution to
subproblems (1) and (2). Once again, this contradicts the conjecture of (x, y) in set B.
Conjectured Solution on Budget Line: From our analysis above, λ*(x, y) is unrestricted
on the budget line – at least λ*(x, y) is unrestricted by subproblem 3 as any choice of λ
will solve subproblem 3 when (x, y) is on the budget line. In this case, it is possible to
find a specific value of λ that will induce the consumer to choose values of x*, y* in
subproblems (1) and (2) that are on the budget line.
In sum, we tried out three different conjectures about where solutions (x*, y*) to the
Lagrangian problem could lie in the graph in Figure 1 relative to the budget line. We
conclude that combinations of (x*, y*) below the budget line (Set A) and above the
budget line (Set B) cannot be part of simultaneous solution to subproblems (1) to (3).
However, a combination of (x*, y*) on the budget line can be part of a simultaneous
solution to subproblems (1) to (3) for an appropriate choice of λ.
The tricky element of this solution (x*, y*, λ*) to the Lagrangian problem is that the
value of λ is chosen to support a particular solution of x*, y* (i.e. a combination on the
budget line) rather than to solve subproblem 3. However, this is still consistent with the
mathematical condition of solving subproblems 1 to 3 simultaneously. Section B
provides further interpretation of the value λ*.

B. Interpreting λ* as a Shadow Price or Penalty Term
The technical advantage of the Lagrangian formulation is that it converts a constrained
maximization problem into an unconstrained maximization problem. That is, the
Lagrangian problem formally allows choices of (x, y) that are not in the budget set.
For the Lagrangian objective function to match the objective function u(x, y), the term
λ(px + qy - W) must end up equal to zero. That is, either λ = 0 or px + qy = W; this
combination of two possibilities is sometimes called a complementary slackness
condition. Further since we know that the solution to the consumer problem is on the
budget line, we want to find a Lagrangian solution that satisfies px + qy = W
Intuitively, the new term in the Lagrangian objective function must provide incentives for
the consumer to choose (x, y) on the budget line. Let’s rewrite the Lagrangian with a
negative sign on λ, meaning that we also have to reverse the signs on W, px, and qy.
L(x, y, λ) = u(x, y) - λ(px + qy - W).
This format for L(x, y, λ) encourages the interpretation of λ as a “penalty” per unit of
expenditure beyond W. But since px + qy – W can be negative, λ also serves as a
“reward” per unit of expenditure less than W.
Naturally, if λ is too large, the consumer will choose (x, y) below the budget line, while if
λ is too small, the consumer will choose (x, y) above the budget line. The only way to
balance rewards and penalties and induce the consumer to stay on the budget line is to
choose λ* = [∂u / ∂x] / p = [∂u / ∂y] / q for a particular point on the budget line. For the
bundle (x*, y*) that is the best choice in terms of utility on the budget line, this value of
λ* provides exactly the right incentives for the consumer.
One last comment is that this entire construction is tied to small changes in bundle from
(x*, y*) on the budget line. Reducing or increasing expenditures for either x or y by a
small amount ε would reduce change the value of the utility function u(x, y) by λ* units,
but this change would be offset by a reward or penalty from the Lagrangian λ term.
What about a much larger change in either x or y from (x*, y*)? Partial derivative
approximations no longer apply for changes in quantities that are more than infinitesimal.
But because of the decreasing returns condition (essentially by the second-order
condition for maximization in (x, y)), we know that the effect on the utility function from
marginal changes in x and y decreases as x and y increase. That is, a large increase in
expenditure on x or y from (x*, y*) would yield an increase in u(x, y) of less than λ* per
additional $ spent, but with constant cost λ* units per $ spent from the Lagrangian term.
In words, the first-order condition indicates that utility gains and losses from small
changes in expenditure from (x*, y*) are exactly offset by appropriate choice of λ* in the
Lagrangian term. In addition, the second-order condition (decreasing MRS) indicates
that utility gains and losses from large changes in expenditure are more than offset by the
appropriate choice of λ* in the Lagrangian term.