A firm that purchases electricity from the local utility is considering installing a steam generator. A large generator costs $280,000 whereas a small generator costs $140,000. The cost of operating the generator would be $200,000 per year for the large and $220,000 for the small. Either generator will last for five years. The cost of capital is 9%.

For each generator option, assume immediate installation, with purchase and operating costs in the current year and operating costs continuing for the next four years. Assume payments under both options at the start of each year (i.e., immediate, one year from now,..., four years from now).

**What is the net present value of the more attractive generator?**

Please round your answer to the nearest dollar. Report the NPV of cost as a negative number.

The Net Present Value (NPV) of costs allows us to determine which generator option is more cost-effective for the firm. The NPV calculation will consider both the initial purchase cost and the annual operating costs, discounted at the cost of capital (9%).

### Step 1: Formula for NPV of the Costs
The NPV for each generator is calculated using the following formula:

\[
\text{NPV of Costs} = C_0 + \sum_{t=1}^{4} \frac{C_t}{(1+r)^t}
\]

Where:
- $ C_0 $ is the initial purchase cost,
- $ C_t $ is the operating cost in year $ t $,
- $ r $ is the discount rate (9% or 0.09),
- $ t $ is the time in years.

### Step 2: Calculating NPV for the Large Generator
- **Initial cost $ C_0 $**: $280,000
- **Operating cost $ C_t $**: $200,000 per year for the next 4 years

The NPV of costs for the large generator will be:

\[
\text{NPV}_{\text{large}} = 280,000 + \frac{200,000}{1.09^1} + \frac{200,000}{1.09^2} + \frac{200,000}{1.09^3} + \frac{200,000}{1.09^4}
\]

### Step 3: Calculating NPV for the Small Generator
- **Initial cost $ C_0 $**: $140,000
- **Operating cost $ C_t $**: $220,000 per year for the next 4 years

The NPV of costs for the small generator will be:

\[
\text{NPV}_{\text{small}} = 140,000 + \frac{220,000}{1.09^1} + \frac{220,000}{1.09^2} + \frac{220,000}{1.09^3} + \frac{220,000}{1.09^4}
\]

I'll now compute the NPVs for both generators to determine which is more attractive.The NPVs of costs for each generator are as follows:

- NPV of the large generator: $-927,944$ dollars
- NPV of the small generator: $-852,738$ dollars

Since the NPV of the small generator is less negative, the small generator is the more attractive option. Therefore, the net present value of the more attractive generator (the small generator) is $-852,738$ dollars.

Would you like more details or have any questions?

Here are some related questions to consider:
1. How does the discount rate affect the NPV calculations?
2. What happens to the NPV if the operating costs change after a few years?
3. How could a longer generator lifespan influence the decision?
4. What is the break-even discount rate at which both generators have the same NPV?
5. How does inflation impact the NPV of future operating costs?

**Tip**: When evaluating investments, always compare NPVs to assess long-term cost-effectiveness.

A firm that purchases electricity from the local utility is considering installing a steam generator. A large generator costs $280,000 whereas a small generator costs $140,000. The cost of operating the generator would be $200,000 per year for the large and $220,000 for the small. Either generator will last for five years. The cost of capital is 9%. What is the net present value of the more attractive generator? Please round your answer to the nearest dollar. Report the NPV of cost as a negative number.

This detailed solution calculates the Net Present Value (NPV) for a firm deciding between two steam generators. Using a discount rate of 9%, we compare the NPV of a large generator costing $280,000 with operating costs of $200,000/year and a small generator costing $140,000 with operating costs of $220,000/year. This financial mathematics problem covers NPV, discounting, and cost analysis over a five-year period.

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