Slab Tariffs, Marginal Units and the Solar Payback Decision
A household in Lahore consumed 410 units in a 30-day billing month, up from 330 units in the previous month. The distribution company applies these slab rates on a progressive basis: first 100 units at PKR 12.00 each, next 100 at PKR 18.50, next 100 at PKR 24.00, next 100 at PKR 32.00, and all remaining units at PKR 40.00. A fixed charge of PKR 500 and general sales tax of 17 per cent on the energy charge plus fixed charge are added. The household's only new load is a 1.5-ton inverter air conditioner rated at 1.5 kW, run 8 hours a day; everything else is unchanged.
The owner is offered a 3 kW rooftop solar system for PKR 600,000, expected to generate about 12 units per day on average, net-metered so that exported units offset imported units one for one. He wants to know whether cutting air-conditioner use or buying the system is the better financial decision, and how much each option actually saves.
1. Compute the current monthly bill and the average cost per unit. Then compute the bill at 330 units and state the true marginal cost of the 80 extra units. Where is the most likely arithmetic trap?
Read the worked answer
Step 1 - price each slab separately. Progressive slabs mean 410 units are not billed at one rate:
- 1–100: 100 × 12.00 = 1,200.00
- 101–200: 100 × 18.50 = 1,850.00
- 201–300: 100 × 24.00 = 2,400.00
- 301–400: 100 × 32.00 = 3,200.00
- 401–410: 10 × 40.00 = 400.00
Energy charge = 9,050.00; fixed charge 500.00 → subtotal 9,550.00; GST 17% = 1,623.50.
Total ≈ PKR 11,173.50 (11,174). Average per unit = 11,173.50 ÷ 410 ≈ PKR 27.25.
Step 2 - previous month. 330 units: 1,200 + 1,850 + 2,400 + (30 × 32 = 960) = 6,410.00; plus 500 = 6,910.00; GST 1,174.70; total ≈ PKR 8,084.70.
Step 3 - marginal cost. 11,173.50 − 8,084.70 = 3,088.80 for 80 units ≈ PKR 38.61 per unit. The extra units sit mostly in the 32-rupee slab, the last ten in the 40-rupee slab, and GST is charged on all of it.
The trap is multiplying total units by the top or an assumed rate (410 × 32, or 410 × 27), which hides the decision-relevant number. The average, 27.25, understates what the next unit costs by about 11 rupees; every unit avoided is worth roughly 38.6, not 27.25.
2. Estimate the monthly saving and simple payback of the solar system, then compare it with halving the air conditioner's run-time. Which assumptions is the payback most sensitive to?
Read the worked answer
Load first. 1.5 kW × 8 h = 12 units/day ≈ 360 units/month - about 88% of the 410 units consumed. The bill is largely an air-conditioner bill.
Solar saving. 360 units/month offset leaves 410 − 360 = 50 imported units. Bill at 50 units = (50 × 12.00) 600 + 500 fixed = 1,100; GST 187; ≈ PKR 1,287. Saving ≈ 11,173.50 − 1,287 = PKR 9,886.50/month. Simple payback = 600,000 ÷ 9,886.50 ≈ 61 months, about five years.
Halving the run-time. 4 h/day saves 180 units → 230 units. Bill = 1,200 + 1,850 + (30 × 24 = 720) = 3,770 + 500 = 4,270; GST 725.90; ≈ PKR 4,995.90. Saving ≈ PKR 6,177.60/month at zero capital cost - about 62% of the solar saving for nothing.
Sensitivities. (i) Tariff slabs: five-year payback assumes today's rates; repeated upward revisions shorten it, which is the strongest argument for solar. (ii) Net metering: one-for-one crediting is a policy, not a physics fact, and could be revised. (iii) Timing: an air conditioner run at night imports rather than offsets, so generation must match load without a battery. (iv) Degradation, inverter replacement and panel cleaning are unaccounted. Solar is a five-year bet on tariffs, not a five-year certainty.
3. If only one action can be taken this month, which should it be, and how should the solar system be sized? Give your recommendation with the comparison that drives it.
Read the worked answer
Recommendation: cut the run-time first, then size solar against the reduced load.
- The AC cut costs nothing and returns about 6,178 rupees a month immediately; solar returns about 9,887 a month but only after a 600,000-rupee outlay. The cut therefore dominates on both cash-flow and risk.
- Sequencing matters because the two are not alternatives: a system sized for 410 units would be oversized once the AC is halved, and oversized systems are exactly where net-metering disputes and wasted capital appear.
The decisive comparison is cost per unit avoided versus cost per unit generated:
- Grid unit avoided ≈ PKR 38.61 (the marginal rate computed above, not the 27.25 average).
- Solar unit generated = 600,000 ÷ (360 units × 60 months) = 600,000 ÷ 21,600 ≈ PKR 27.78 per unit, excluding battery and inverter replacement.
So solar is cheaper than the marginal grid unit, but only by about 11 rupees a unit over five years - a narrow gap that a single adverse change in net-metering rules or a fall in the household's consumption can erase. That gap, not the size of the gross bill, is what the decision turns on. Size the array to the post-cut load, keep the AC on a timer, and revisit once two or three bills confirm the new baseline.