Kuhn-Tucker Optimization: Maximizing Utility with Inequality Constraints

Published on March 22, 2026

Calculus

Solving Tarzan's consumption problem: maximizing U = Q1*Q2^3 subject to production (Q1^2 + Q2^2 <= 200) and environmental (Q1 + Q2 <= 20) constraints using graphical and KKT methods.

Question

please answer with detail

Answer

This is an optimization problem under inequality constraints. Tarzan wants to maximize his utility

subject to two constraints:

Production Possibility Frontier (PPF):
Environmental Constraint:
Also, we assume non-negativity constraints: and .

The problem is:

Since the utility function is strictly quasi-concave (as

is Cobb-Douglas with positive exponents, though the sum of exponents is

, implying increasing returns to scale, which is unusual for a utility function, but for optimization purposes, the Hessian matrix properties of the level sets will determine concavity/convexity, and typically for maximization problems, we assume the objective is quasi-concave, which

is in the positive orthant) and the constraints define a convex feasible set (the intersection of two disks/half-planes is convex), we can use KKT conditions to find the global maximum.

a. Solve the problem using the Graphical Method. (5 points)

The graphical method involves plotting the feasible set defined by the constraints and then drawing indifference curves (level sets of

) to find the highest attainable indifference curve that touches the feasible set.

1. Identify the Feasible Region (The Constraints):

Non-negativity: . We are in the first quadrant.
Constraint 2 (Linear): .
- The boundary is the line .
- Intercepts: and .
Constraint 1 (Circular): .
- The boundary is the circle . This is a circle centered at the origin with radius .
- The intercepts are and .

2. Determine the Binding Constraints:

We need to see which constraint is tighter (more restrictive). Let’s compare the extreme points on the axes:

On the -axis ( ):
- C1 requires .
- C2 requires .
- C1 is tighter here.
On the -axis ( ):
- C1 requires .
- C2 requires .
- C1 is tighter here.

Now, let’s check where the two constraint boundaries intersect within the first quadrant (ignoring non-negativity for a moment):
Solve:

Substitute (2) into (1):

, then

.
The intersection point is

Now we check if

satisfies the other constraint:
For C1:

. Since

, this point lies exactly on the boundary of the PPF constraint.

This means that the line

is tangent to the circle

at the point

.
Since the feasible set requires points to satisfy both

AND

, and since the line

touches the circle from the “outside” relative to the origin (i.e., points satisfying

are closer to the origin and definitely inside the circle defined by

), the constraint

is the binding constraint that defines the boundary of the feasible region in the vicinity of the optimum, as any point satisfying

automatically satisfies

(except at the point

where they are equal).

To verify this: If

, then

. This quadratic in

has a minimum when

(i.e.,

). The minimum value of

given

.
If

, the minimum of

.
Since

when

, and we require

, the only point satisfying both equality constraints is

If we consider any point where

, say

. The minimum

. So all points on the line segment

are feasible with respect to C1.

Conclusion for Feasible Region: The feasible set is defined by

, and

. The constraint

is redundant in defining the boundary of the overall feasible set because the set defined by

is entirely contained within or tangent to the set defined by

(they meet only at

3. Graphical Solution:
The problem reduces to:

We need to find the highest indifference curve tangent to the line segment connecting

and

.
For the utility function

, the slope of the indifference curve (Marginal Rate of Substitution, MRS) is:

The slope of the linear budget constraint (

) is

.
The optimum occurs where

Substitute this into the binding constraint

Solution using Graphical Method (Part a):
The optimal consumption bundle is

.
We must verify this point satisfies all original constraints:

. (Satisfied)
.

Wait!

. This means the tangency condition found above lies outside the feasible region defined by the circle

Since the unconstrained optimum on the line

is infeasible due to the PPF constraint, the optimal solution must be at a point where the indifference curve is tangent to the tighter constraint, or at a corner point.

We established that the feasible region is defined by

, and

AND

. Since the intersection of the boundaries

and

, the feasible set is the area within the circle

that also satisfies

. As shown before, any point satisfying

already implies

(unless they are both equal at

).
Let’s re-examine the geometry:
If

, then

.
If

.
The constraint

For a point

to be feasible, it must satisfy both.
If

, then

. This doesn’t help determine the relationship easily.
However, we know

satisfies both constraints exactly.
Any point satisfying

is closer to the origin than

in a certain sense, but not necessarily inside the circle.

Let’s compare the boundaries at a point other than

on the line

, say

. This point violates

.
Let’s compare boundaries at a point other than

on the circle, say

. This point satisfies

The feasible set is the intersection of the interior of the circle centered at

with radius

and the region below the line segment connecting

and

.
Since

, the maximum intercepts of the circle are at

and

. Since

, the circular constraint is the binding constraint that limits the feasible region. The linear constraint

is redundant because any point satisfying

(in the first quadrant) automatically satisfies

(since the maximum sum occurs when

, giving

, so

, and any other point on the circle has a sum less than 20).

The effective constraint is .

We maximize

subject to

.
We use the tangency condition derived earlier:

The slope of the tangent to the circle

is found by implicit differentiation:

.
For tangency, the absolute slope of the indifference curve must equal the absolute slope of the boundary:

Substitute this into the binding constraint

Then

Graphical Solution Result (Corrected): The optimal consumption is

.
We check the environmental constraint:

. Since

, this solution is feasible.

b. Formulate the Kuhn-Tucker first-order conditions (5 points)

We use the Lagrange function incorporating the inequality constraints

and

, and multipliers

and

. We maximize

subject to

and

The Lagrangian is:

The Kuhn-Tucker (KKT) First-Order Conditions (FOCs) are:

Stationarity (Derivatives w.r.t. and equal zero):

Primal Feasibility (Constraints must be satisfied):

Dual Feasibility (Lagrange multipliers must be non-negative):

Complementary Slackness (The multiplier times the constraint slack must be zero):

c. Using the Kuhn-Tucker Method, find the optimal consumption of and . (10 points)

Based on the graphical analysis in part (a), we strongly suspect that the optimal solution lies on the boundary of the PPF constraint (

) and that the environmental constraint (

) is not binding (

Case: Only Constraint 1 is binding () and Constraint 2 is not binding ().

and

(binding

):
KKT1:

(since

)
KKT2:

(since

)

From KKT2, dividing by

Substitute

into the expression derived from KKT1:

Since

, we can divide by

(This is the same condition derived from MRS matching the circle slope in part a).

Substitute this into the binding constraint

We must now check the non-binding constraint (KKT4 and KKT8):

Since

. Thus,

must be zero, which satisfies KKT8.

We also need to ensure

. Using

. This satisfies KKT5.

Since we found a solution where

, and all primal and dual feasibility conditions are met, this is the optimal solution.

Optimal Consumption:

The maximum utility is:

The optimal solution found is

.
This point lies exactly on the boundary of the production possibility frontier:

So,

The solution also satisfies the environmental constraint:

So,

Graphical Justification:

Feasible Set: The feasible region is constrained by , and the intersection of the circle and the half-plane . As established in part (a), because , the circular boundary is tighter than the linear boundary everywhere in the first quadrant except at the point where and . Therefore, the effective outer boundary of the feasible set is the arc of the circle that lies in the first quadrant.
Tangency: The optimal solution occurs where an indifference curve is tangent to this binding constraint arc ().
Condition: We found the tangency condition by setting the slope of the indifference curve (MRS) equal to the slope of the tangent to the circle ():
Verification: The point satisfies this tangency condition:Since , the indifference curve passing through is tangent to the PPF boundary at that point. Since the objective function is well-behaved (quasi-concave) and the constraint set is convex, this tangency point represents the global maximum within the effective boundary.
Non-binding constraint: Because , the optimal point lies strictly inside the region defined by the environmental constraint, confirming that this constraint is not active at the optimum.

Therefore, graphically, the highest indifference curve touches the feasible region exactly at the point of tangency with the circle boundary, confirming

as the optimum.