# Sustainable Growth and Self-Supporting Sales Increase Calculation Let's break down the problem step by step. ## Given Data - **2025 Sales (\( S_0 \))**: $4,000,000 - **Total Assets at end of 2025 (\( A_0 \))**: $2,800,000 - **Current Liabilities at end of 2025 (\( L_0 \))**: $500,000 - Notes Payable: $200,000 - Accounts Payable: $200,000 - Accrued Liabilities: $100,000 - **Asset-to-Sales Ratio for Growth**: For every $1.00 increase in sales, assets must increase by $0.70 \(\Delta A = 0.70 \times \Delta S\) - **Profit Margin (PM)**: 3% \(\text{Net Income} = 0.03 \times \text{Sales}\) - **Retention Ratio (b)**: 55% (portion of net income retained) - **Spontaneous Liabilities-to-Sales Ratio**: \( L_0 / S_0 = 500,000 / 4,000,000 = 0.125 \) So, for each $1.00 increase in sales, spontaneous liabilities increase by $0.125. ## Step 1: Calculate Addition to Retained Earnings \[ \text{Addition to Retained Earnings per \$1 increase in sales} = \text{Profit Margin} \times \text{Retention Ratio} \] \[ = 0.03 \times 0.55 = 0.0165 \] ## Step 2: Calculate Additional Assets and Spontaneous Liabilities Needed - **Additional Assets Needed per \$1 increase in sales**: $0.70 - **Additional Spontaneous Liabilities per \$1 increase in sales**: $0.125 ## Step 3: Calculate External Financing Needed (EFN) per \$1 Increase in Sales \[ \text{EFN per \$1 increase in sales} = \text{Increase in Assets} - \text{Increase in Liabilities} - \text{Addition to Retained Earnings} \] \[ = 0.70 - 0.125 - 0.0165 = 0.5585 \] To **avoid needing external funds**, set EFN = 0, and solve for maximum increase in sales (\( \Delta S \)) that can be financed internally: \[ \text{EFN} = (\Delta S) \times (\text{Increase in Assets per \$1 sales} - \text{Increase in Liabilities per \$1 sales} - \text{Addition to Retained Earnings per \$1 sales}) \] \[ 0 = (\Delta S) \times (0.70 - 0.125 - 0.0165) \] But the retained earnings is a function of **total sales**, so let's derive the correct formula. ## Step 4: General Formula for Self-Supporting Sales Increase The **self-supporting (internal) increase in sales** is found by: \[ \Delta S = \frac{ \text{Addition to Retained Earnings} }{ \text{Asset requirement per \$ sales} - \text{Liabilities per \$ sales} - \text{Addition to Retained Earnings per \$ sales} } \] But since retained earnings comes from the **increase in sales**, the formula is: \[ \text{EFN} = (\Delta S) \times (\text{Asset-to-Sales Ratio} - \text{Liability-to-Sales Ratio} - \text{Profit Margin} \times \text{Retention Ratio}) \] Set EFN = 0: \[ 0 = (\Delta S) \times (0.70 - 0.125 - 0.0165) \] \[ (\Delta S) \times 0.5585 = 0 \] But this suggests that unless there is outside funding, there can be no increase, which is not correct. Let's use the **plug** approach, expressing the retained earnings as a function of the new sales. Let \( S_1 = S_0 + \Delta S \): - **Increase in Assets**: \( 0.70 \times \Delta S \) - **Increase in Liabilities**: \( 0.125 \times \Delta S \) - **Increase in Retained Earnings**: \( 0.03 \times S_1 \times 0.55 \) - **Existing Retained Earnings are not relevant; use only new earnings.** Set EFN = 0: \[ \text{Increase in Assets} = \text{Increase in Liabilities} + \text{Increase in Retained Earnings} \] \[ 0.70 \Delta S = 0.125 \Delta S + 0.03 S_1 \times 0.55 \] Recall \( S_1 = S_0 + \Delta S \): \[ 0.70 \Delta S - 0.125 \Delta S = 0.0165 (S_0 + \Delta S) \] \[ 0.575 \Delta S = 0.0165 S_0 + 0.0165 \Delta S \] \[ 0.575 \Delta S - 0.0165 \Delta S = 0.0165 S_0 \] \[ 0.5585 \Delta S = 0.0165 S_0 \] \[ \Delta S = \frac{0.0165 S_0}{0.5585} \] Plug in \( S_0 = 4,000,000 \): \[ \Delta S = \frac{0.0165 \times 4,000,000}{0.5585} \] \[ = \frac{66,000}{0.5585} \] \[ = 118,200.14 \] ## **Final Answer** ### **The largest sales increase Paladin Furnishings can achieve in 2026 without external funds is:** \[ \boxed{118,200.14} \] **So, Paladin Furnishings can increase sales by \$118,200.14 without having to raise funds externally.** --- ### **Summary Table** | Item | Value | |----------------------------------------|---------------:| | Sales (2025) (\( S_0 \)) | $4,000,000 | | Asset Increase per $1 Sales | $0.70 | | Liability Increase per $1 Sales | $0.125 | | Profit Margin | 3% | | Retention Ratio | 55% | | Addition to Retained Earnings per $1 | $0.0165 | | **Self-supported Sales Increase** | $118,200.14 |