To solve the optimization problem described, we first need to translate the natural language description into a symbolic representation. This involves defining variables for each item (postage stamps, paper clips, and headsets), formulating the objective function using these variables, and then listing all constraints as semi-algebraic expressions.

Let's define:
- \(x_1\) as the number of postage stamps,
- \(x_2\) as the number of paper clips,
- \(x_3\) as the number of headsets.

The objective function to minimize is given by \(7.71x_1 + 6.19x_2 + 5.07x_3\).

Constraints:
1. \(4.49x_1 + 1.38x_3 \geq 50\) (spend at least $50 on postage stamps and headsets),
2. \(3.37x_2 + 1.38x_3 \geq 38\) (spend at least $38 on paper clips and headsets),
3. \(4.49x_1 + 3.37x_2 + 1.38x_3 \geq 67\) (spend at least $67 on all items),
4. \(6x_1 - 6x_2 \geq 0\) (six times the number of postage stamps minus six times the number of paper clips is at least zero),
5. \(4x_1 - 5x_3 \geq 0\) (four times the number of postage stamps minus five times the number of headsets is at least zero),
6. \(x_1, x_2, x_3 \in \mathbb{Z}^+\) (all variables must be positive integers).

Given this setup, we can now formulate the problem in a symbolic representation as requested:

```json
{
  'sym_variables': [('x1', 'postage stamps'), ('x2', 'paper clips'), ('x3', 'headsets')],
  'objective_function': '7.71*x1 + 6.19*x2 + 5.07*x3',
  'constraints': [
    '4.49*x1 + 1.38*x3 >= 50',
    '3.37*x2 + 1.38*x3 >= 38',
    '4.49*x1 + 3.37*x2 + 1.38*x3 >= 67',
    '6*x1 - 6*x2 >= 0',
    '4*x1 - 5*x3 >= 0'
  ]
}
```

Now, let's implement this problem using Gurobi in Python:

```python
from gurobipy import *

# Create a new model
m = Model("Optimization_Problem")

# Define the variables
x1 = m.addVar(vtype=GRB.INTEGER, name="postage_stamps")
x2 = m.addVar(vtype=GRB.INTEGER, name="paper_clips")
x3 = m.addVar(vtype=GRB.INTEGER, name="headsets")

# Set the objective function
m.setObjective(7.71*x1 + 6.19*x2 + 5.07*x3, GRB.MINIMIZE)

# Add constraints
m.addConstr(4.49*x1 + 1.38*x3 >= 50, name="constraint_1")
m.addConstr(3.37*x2 + 1.38*x3 >= 38, name="constraint_2")
m.addConstr(4.49*x1 + 3.37*x2 + 1.38*x3 >= 67, name="constraint_3")
m.addConstr(6*x1 - 6*x2 >= 0, name="constraint_4")
m.addConstr(4*x1 - 5*x3 >= 0, name="constraint_5")

# Optimize the model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Postage Stamps: {x1.x}")
    print(f"Paper Clips: {x2.x}")
    print(f"Headsets: {x3.x}")
    print(f"Objective Function Value: {m.objVal}")
else:
    print("No optimal solution found")
```