Constrained Optimization with Lagrangian Multipliers: Maximizing Cobb-Douglas Utility

Published on March 22, 2026

Calculus

This problem involves maximizing an audit quality function (Cobb-Douglas form) subject to a budget constraint using the Lagrangian method and verifying maximization using the Bordered Hessian Matrix.

Question

please answer

Answer

This is a constrained optimization problem. We will follow the steps requested in parts a, b, c, and d.

The given production function (audit quality) is:

where

is the working hours of Partners and

is the working hours of Staff.

Hint: Use a unit of a hundred thousand Rupiahs (Rp 100,000) to simplify calculations.

Original values:
Total Budget

Partner Salary Rate

per hour
Staff Salary Rate

per hour

In units of Rp 100,000:
Budget

Partner Salary Rate

Staff Salary Rate

a. Formulate the problem of maximizing audit quality at the accounting firm along with its constraint functions. (5 points)

We want to maximize

subject to the budget constraint. We assume

and

Objective Function (to Maximize):

Constraint Function (Budget Constraint):
The total cost must equal the budget:

The optimization problem formulation is:

b. Write the Lagrangian equation and calculate and that produce the maximum quality of audit results. (8 points)

1. Lagrangian Equation
The Lagrangian function

is defined as:

2. First-Order Conditions (FOCs)
We set the partial derivatives of

with respect to

, and

to zero.

3. Solving for and
From (1) and (2), we can solve for

From (1):

From (2):

Equating the two expressions for

Rearrange to find the relationship between

and

Substitute (4) into the budget constraint (3):

Now find

using (4):

The optimal hours are

(Partner hours) and

(Staff hours), in units of 100,000 Rupiahs worth of work time.

c. Determine the quality of the audit expressed by . (5 points)

Substitute

and

into the quality function

We can rewrite 50 and 400 to simplify the calculation if possible, noticing that

.
Also,

.
Since

, this is not easily simplified by the sum of exponents being 1 (Cobb-Douglas optimization usually leads to

where

). Here

Let’s calculate the numerical value:

Alternatively, using properties of exponents:

Since

. This is also hard to simplify without a calculator.

Since

is between 20 and 25.

Using the calculated approximation:

(Note: Given the context of typical economics problems, if an exact simplification was intended, the exponents might sum to 1 or the numbers would be powers of common bases. Since , we rely on numerical evaluation unless the problem implies a specific rounding or format.)
We state the result:

d. Create a Bordered Hessian Matrix and check whether the second-order sufficient condition for maximization is met. (7 points)

For constrained maximization using the Lagrangian method, we check the sign of the determinants of the leading principal minors of the Bordered Hessian Matrix (

). If we have

variables (

) and

constraint, we check minors

. Since

and

, we only check

For maximization, we require the sign of the determinants to alternate starting with the negative sign for the smallest relevant order. We need

The Bordered Hessian matrix

is constructed as:

1. Calculate first partial derivatives of the constraint :

2. Calculate second partial derivatives of the Lagrangian :
Note that

because the constraint terms disappear upon the second differentiation with respect to

3. Evaluate derivatives at the optimal point

Self-Correction/Simplification Check: For Cobb-Douglas functions where

, it is well known that

are negative (concave) and

is positive. This is necessary for a maximum.

4. Construct the Bordered Hessian

5. Calculate the Determinant
Using cofactor expansion along the first row:

Since

Substitute the approximate numerical values:

6. Check Second-Order Sufficient Condition (SOSC)
For maximization, the SOSC requires that the sign of the determinant of the

-th leading principal minor of the Bordered Hessian alternates starting with

, where

is the number of constraints.
We check

.
We require

to have the sign

Since

, the condition for a local maximum is met.

Final Answer Summary

a. Formulation:

b. Lagrangian and Optimal Values:
Lagrangian Equation:

Optimal Hours:

c. Optimal Quality:

d. Bordered Hessian and SOSC:
The Bordered Hessian Matrix

evaluated at the optimal point is:

The required sign for maximization is

for the

-th minor, where

and

. Thus, we require

.
The calculated determinant is:

Since

, the second-order sufficient condition for maximization is met.

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Constrained Optimization with Lagrangian Multipliers: Maximizing Cobb-Douglas Utility

Question

Answer

a. Formulate the problem of maximizing audit quality at the accounting firm along with its constraint functions. (5 points)

b. Write the Lagrangian equation and calculate and that produce the maximum quality of audit results. (8 points)

c. Determine the quality of the audit expressed by . (5 points)

d. Create a Bordered Hessian Matrix and check whether the second-order sufficient condition for maximization is met. (7 points)

Final Answer Summary

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