To solve the given optimization problem using Gurobi, we first need to understand and translate the problem into a mathematical formulation. The objective is to minimize the cost function subject to several constraints.

Given:
- Objective function: Minimize \(7.71 \times \text{postage stamps} + 6.19 \times \text{paper clips} + 5.07 \times \text{headsets}\)
- Constraints:
  1. Dollar cost for each item: postage stamps ($4.49), paper clips ($3.37), headsets ($1.38)
  2. Spend at least $50 on postage stamps and headsets.
  3. Spend at least $38 on paper clips and headsets.
  4. Spend at least $67 on all three items together.
  5. \(6 \times \text{postage stamps} - 6 \times \text{paper clips} \geq 0\)
  6. \(4 \times \text{postage stamps} - 5 \times \text{headsets} \geq 0\)
  7. All items must be whole numbers.

Let's denote:
- \(x_0\) as the number of postage stamps,
- \(x_1\) as the number of paper clips,
- \(x_2\) as the number of headsets.

The objective function can be written as: Minimize \(7.71x_0 + 6.19x_1 + 5.07x_2\)

Constraints:
1. \(4.49x_0 + 1.38x_2 \geq 50\)
2. \(3.37x_1 + 1.38x_2 \geq 38\)
3. \(4.49x_0 + 3.37x_1 + 1.38x_2 \geq 67\)
4. \(6x_0 - 6x_1 \geq 0\)
5. \(4x_0 - 5x_2 \geq 0\)

All variables are integers.

Now, let's write the Gurobi code for this problem:

```python
from gurobipy import *

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

# Create variables
postage_stamps = m.addVar(vtype=GRB.INTEGER, name="postage_stamps")
paper_clips = m.addVar(vtype=GRB.INTEGER, name="paper_clips")
headsets = m.addVar(vtype=GRB.INTEGER, name="headsets")

# Set the objective function
m.setObjective(7.71*postage_stamps + 6.19*paper_clips + 5.07*headsets, GRB.MINIMIZE)

# Add constraints
m.addConstr(4.49*postage_stamps + 1.38*headsets >= 50, "postage_and_headsets")
m.addConstr(3.37*paper_clips + 1.38*headsets >= 38, "paperclips_and_headsets")
m.addConstr(4.49*postage_stamps + 3.37*paper_clips + 1.38*headsets >= 67, "all_items_together")
m.addConstr(6*postage_stamps - 6*paper_clips >= 0, "postage_vs_paperclips")
m.addConstr(4*postage_stamps - 5*headsets >= 0, "postage_vs_headsets")

# Optimize the model
m.optimize()

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