We are given the log-linear model:

$\log(\text{Costs}) = 4.77 - 0.002 \times \text{Total Production}$

Step 1: Current situation

We know the total production is 200 MW, and the current costs are $50 per Watt. Let's confirm this with the given equation.

$\log(\text{Costs}) = 4.77 - 0.002 \times 200 = 4.77 - 0.4 = 4.37$

Now, the cost in $/W can be found by exponentiating the result:

$\text{Costs} = e^{4.37}$

Let's calculate that:

$e^{4.37} \approx 79.04 \, \text{\$/W}$

However, the current cost is $50/W, which suggests that the coefficient of production (or intercept) might need recalibration. But we'll proceed with the model as given to find the relative change.

Step 2: New total production

Now, we increase production by 10 MW, making the new total production:

$\text{New Total Production} = 200 + 10 = 210 \, \text{MW}$

Substitute this into the log-linear model to find the new log cost:

$\log(\text{New Costs}) = 4.77 - 0.002 \times 210 = 4.77 - 0.42 = 4.35$

Step 3: New cost

Exponentiate to find the new cost:

$\text{New Costs} = e^{4.35}$

Calculating this:

$e^{4.35} \approx 77.56 \, \text{\$/W}$

Step 4: Fall in costs

The fall in costs is the difference between the current cost and the new cost:

$\text{Fall in Costs} = 79.04 - 77.56 = 1.48 \, \text{\$/W}$

Thus, the fall in costs when increasing production by 10 MW is approximately $1.48/W.

Would you like further details or clarifications?

Here are 5 related questions to explore further:

1. How would the fall in costs change if the production increase was 20 MW instead of 10 MW?
2. What does the coefficient of -0.002 represent in this context?
3. Can we calculate the elasticity of costs with respect to production from this model?
4. How accurate is a log-linear model for forecasting cost reductions in the solar industry?
5. What are the limitations of using such a regression model for cost predictions in real-world applications?

Tip: Log-linear models are often used to capture the learning effect, where cumulative production leads to cost reductions. This is known as the "experience curve."