Consider an electricity market where the base of the demand curve is vertical and varieslinearly between 4000MW and 8000MW (so is equally likely to be found anywhere inthat range). Above 7500MW demand can be curtailed at a price of $4000/MWh. Themarket is competitive with three types of plants as below.Technology FC per MWh Variable costs perMWhCarbon Emissionsin tonnes perMWhPeak (diesel peaker) $18 $160 25Mid (gas) $35 $50 10Baseload (hydro) $55 $0 0(a)What is the optimal amount of baseload plant that should be built? (2 marks)(b) What is the duration of the price spike so that the peaker plant just recovers itsfixed costs? (2 marks)(c) What is the optimum mid-load capacity? (2 marks)(d) What is the total capacity and hence peak capacity? (2 marks)(e) Sketch the load duration curve and the offer stack (2 marks)(f) Show all plants just cover costs (4 marks)(g) Suppose now the government imposes a tax of $30 a tonne. Suppose there is nochange in capacity (so capacity doesn’t adjust). Calculate the profit that each type ofplant makes. Why do some plants make a profit even though the market iscompetitive? (4 marks)

This question is quite complex and requires a deep understanding of energy economics. Here's how you might approach each part:

(a) The optimal amount of baseload plant that should be built depends on the demand and the costs of the other types of plants. Since the baseload plant has the lowest variable cost but the highest fixed cost, it would be optimal to build as much baseload plant as possible until the demand is met. However, the exact amount would require more specific information about the demand curve and the costs of the other plants.

(b) The duration of the price spike that allows the peaker plant to just recover its fixed costs can be calculated by dividing the fixed costs by the difference between the price spike and the variable costs.

(c) The optimum mid-load capacity would be the amount that can be produced at a lower cost than the peaker plant but at a higher cost than the baseload plant. This would require more specific information about the demand curve and the costs of the other plants.

(d) The total capacity would be the sum of the capacities of the baseload, mid-load, and peaker plants. The peak capacity would be the maximum amount that can be produced by the peaker plant.

(e) The load duration curve would be a graph showing the amount of load at different times, while the offer stack would be a graph showing the amount of energy offered at different prices.

(f) To show that all plants just cover costs, you would need to demonstrate that the revenue from selling energy equals the total costs (fixed and variable) for each type of plant.

(g) If the government imposes a tax of $30 a tonne, the profit that each type of plant makes would decrease by the amount of the tax times the carbon emissions per MWh. Some plants might still make a profit because they have lower carbon emissions or because the price of energy is high enough to cover the tax.

Please note that these are general approaches and the exact answers would require more specific information about the demand curve and the costs of the other plants.