Let's solve each part step by step: --- ### **Given Data** - Total stock \( S_0 = 15 \) units - Demand each year: \( Q_t = 10 - P_t \) (so \( P_t = 10 - Q_t \)) - Marginal cost of extraction = 0 - Discount rate \( r = 10\% \) --- ## **a) Resource Only Useful in Year 0** #### **i) Efficient Quantity to Extract This Year** Since the resource is only useful this year, extract as much as is demanded at a positive price, but not more than the stock. - **Demand:** \( Q_0 = 10 - P_0 \) - **Supply available:** 15 units If we set price so all 15 units are used, \( Q_0 = 15 \): - \( P_0 = 10 - Q_0 = 10 - 15 = -5 \), which is negative → **Not possible**, as negative prices make no sense. **Set \( P_0 = 0 \) (i.e., give it away for free):** - \( Q_0 = 10 - 0 = 10 \) So, **efficient extraction** is 10 units at price \( P_0 = 0 \). \[ \boxed{Q_0^* = 10} \] --- #### **ii) Net Benefit to Consumers (Consumer Surplus)** Consumer surplus is the area under the demand curve above the price, up to the quantity consumed. - Demand curve: \( P = 10 - Q \) - Price paid: \( P_0 = 0 \) - Quantity: \( Q_0 = 10 \) Consumer surplus is the area of the triangle under the demand curve from \( Q = 0 \) to \( Q = 10 \): \[ CS = \frac{1}{2} \times \text{base} \times \text{height} \] Base = 10 (from 0 to 10 units) Height = 10 (from price 0 up to 10) \[ CS = \frac{1}{2} \times 10 \times 10 = 50 \] \[ \boxed{CS = 50} \] --- ## **b) Profit-Maximizing Monopolist** #### **i) Monopolist Extraction Quantity** Monopolist chooses \( Q \) to maximize profit. With MC = 0, maximize revenue: - Demand: \( P = 10 - Q \) - Revenue: \( R = P \times Q = (10 - Q)Q = 10Q - Q^2 \) - Marginal revenue: \( MR = \frac{dR}{dQ} = 10 - 2Q \) Set MR = MC = 0: \[ 10 - 2Q = 0 \implies Q = 5 \] Check if \( Q = 5 \leq S_0 = 15 \): Yes. So, \[ \boxed{Q_{mon}^* = 5} \] - \( P_{mon} = 10 - Q_{mon} = 10 - 5 = 5 \) --- #### **ii) Net Benefit to Consumers, Monopolist's Revenue, Other Term** **Consumer Surplus:** Area under demand curve above price, up to \( Q = 5 \): - At \( Q = 5 \), \( P = 5 \) - CS = area of triangle from \( Q = 0 \) to \( Q = 5 \), from \( P = 5 \) to \( P = 10 \): \[ CS = \frac{1}{2} \times 5 \times (10 - 5) = \frac{1}{2} \times 5 \times 5 = 12.5 \] \[ \boxed{CS_{mon} = 12.5} \] **Monopolist's Revenue:** \[ R = P \times Q = 5 \times 5 = 25 \] \[ \boxed{R_{mon} = 25} \] **Other term for this revenue:** Since MC = 0, all revenue is **producer surplus**. In nonrenewable resource economics, this is also called **scarcity rent** or **resource rent**. \[ \boxed{\text{Scarcity rent (resource rent)}} \] --- ## **c) Resource Useful in Years 0 and 1** Total stock: \( S_0 = 15 \) units. Objective: Allocate between \( t=0 \) and \( t=1 \) to maximize present value of net benefits. Let \( Q_0 \) be extracted in year 0, \( Q_1 = 15 - Q_0 \) in year 1. - In each year: \( P_t = 10 - Q_t \) - MC = 0 - Discount rate \( r = 0.1 \) Maximize present value of consumer surplus: \[ \text{Maximize: } CS_0 + \frac{CS_1}{1.1} \] Consumer surplus in year \( t \): \[ CS_t = \frac{1}{2} Q_t (10 - P_t) = \frac{1}{2} Q_t^2 \] (because \( P_t = 10 - Q_t \), so vertical distance from highest price to market price is \( Q_t \)) But let's check this: - Height of triangle at \( Q_t \): \( 10 - P_t = Q_t \) - Base: \( Q_t \) - So area = \( \frac{1}{2} Q_t Q_t = \frac{1}{2} Q_t^2 \) So, \[ \text{Maximize: } \frac{1}{2} Q_0^2 + \frac{1}{2} \frac{(15 - Q_0)^2}{1.1} \] Set up the Lagrangian (but since all must be used, just substitute \( Q_1 = 15 - Q_0 \)): \[ \text{Maximize: } V(Q_0) = \frac{1}{2} Q_0^2 + \frac{1}{2} \frac{(15 - Q_0)^2}{1.1} \] Take derivative w.r.t. \( Q_0 \): \[ \frac{dV}{dQ_0} = Q_0 - \frac{1}{1.1}(15 - Q_0) \] Set to zero: \[ Q_0 - \frac{1}{1.1}(15 - Q_0) = 0 \] \[ Q_0 - \frac{15}{1.1} + \frac{Q_0}{1.1} = 0 \] \[ Q_0 + \frac{Q_0}{1.1} = \frac{15}{1.1} \] \[ Q_0 \left(1 + \frac{1}{1.1}\right) = \frac{15}{1.1} \] \[ 1 + \frac{1}{1.1} = \frac{1.1 + 1}{1.1} = \frac{2.1}{1.1} \] So \[ Q_0 \cdot \frac{2.1}{1.1} = \frac{15}{1.1} \] \[ Q_0 = \frac{15}{2.1} = \frac{150}{21} \approx 7.14 \] So, \[ \boxed{Q_0^* \approx 7.14} \] \[ Q_1^* = 15 - 7.14 = 7.86 \] **Prices:** - \( P_0 = 10 - Q_0 = 10 - 7.14 = 2.86 \) - \( P_1 = 10 - Q_1 = 10 - 7.86 = 2.14 \) --- #### **i) Efficient price and quantity in year 0** \[ \boxed{Q_0^* \approx 7.14,\quad P_0^* \approx 2.86} \] --- #### **ii) Efficient price and quantity in year 1** \[ \boxed{Q_1^* \approx 7.86,\quad P_1^* \approx 2.14} \] --- #### **iii) Consumer Surplus in year 0 and 1? Comparison with (a) ii** - \( CS_0 = \frac{1}{2} (7.14^2) = 0.5 \times 51.0 = 25.5 \) - \( CS_1 = \frac{1}{2} (7.86^2) = 0.5 \times 61.78 = 30.89 \) - Present value of \( CS_1 \): \( \frac{30.89}{1.1} = 28.08 \) - Total present value: \( 25.5 + 28.08 = 53.58 \) In (a) ii, consumer surplus was 50. \[ \boxed{ \text{Year 0 CS: } 25.5,\quad \text{Year 1 CS: } 30.89,\quad \text{Total PV: } 53.58 } \] So, splitting the resource between two years **increases total consumer surplus** (from 50 to 53.58), due to discounting and more efficient allocation. --- ## **d) New Demand: \( Q_t = 5 - \frac{P_t}{2} \) (or \( P_t = 10 - 2Q_t \))** Total stock: 15 Let \( Q_0 \) in year 0, \( Q_1 = 15 - Q_0 \) in year 1. --- ### **i) Efficient price and quantity in year 0 and 1** Consumer surplus in each year: - Invert demand: \( P = 10 - 2Q \) - For given \( Q \), highest price willing to pay is \( P_{max} = 10 \), actual price is \( P \). - Height = \( 10 - P \) - So, for \( Q_t \), \( P_t = 10 - 2Q_t \), so \( 10 - P_t = 2Q_t \) - Consumer surplus per year: \( CS_t = \frac{1}{2} Q_t (10 - P_t) = Q_t^2 \) Maximize present value: \[ \text{Maximize: } Q_0^2 + \frac{(15 - Q_0)^2}{1.1} \] Take derivative w.r.t. \( Q_0 \): \[ \frac{dV}{dQ_0} = 2Q_0 - \frac{2}{1.1}(15 - Q_0) \] Set to zero: \[ 2Q_0 - \frac{2}{1.1}(15 - Q_0) = 0 \] \[ 2Q_0 - \frac{30}{1.1} + \frac{2Q_0}{1.1} = 0 \] \[ 2Q_0 + \frac{2Q_0}{1.1} = \frac{30}{1.1} \] \[ 2Q_0 (1 + \frac{1}{1.1}) = \frac{30}{1.1} \] \[ 1 + \frac{1}{1.1} = \frac{2.1}{1.1} \] \[ 2Q_0 \cdot \frac{2.1}{1.1} = \frac{30}{1.1} \] \[ Q_0 = \frac{30}{2 \times 2.1} = \frac{30}{4.2} = 7.14 \] \[ Q_1 = 15 - 7.14 = 7.86 \] **Prices:** - \( P_0 = 10 - 2Q_0 = 10 - 2 \times 7.14 = 10 - 14.28 = -4.28 \) (negative price: not possible) - At \( P_0 = 0 \), \( Q_0 = 5 \) - At \( Q_1 = 5 \), but with 15 units, we can't sell more than 5 in any year at non-negative price. So, check: At \( Q = 5 \), \( P = 10 - 2 \times 5 = 0 \) Therefore, **cannot sell more than 5 units in any one year at non-negative price**. Thus, in **each year, max possible is 5 units**. Over two years, can only sell 10 units at non-negative price; the other 5 units will not be sold. **So, efficient allocation:** - Year 0: \( Q_0 = 5 \), \( P_0 = 0 \) - Year 1: \( Q_1 = 5 \), \( P_1 = 0 \) - 5 units remain unused (or sold at negative price, which is not efficient). --- \[ \boxed{ \text{Year 0: } Q_0 = 5,\, P_0 = 0; \quad \text{Year 1: } Q_1 = 5,\, P_1 = 0 } \] --- ### **ii) Is this resource still scarce? Are there scarcity rents? Explain.** - **Scarcity:** The resource is **no longer scarce in the economic sense**,

Let's present the final answers clearly, summarizing each part: --- ## **a) Resource Only Useful in Year 0** ### **i) Efficient quantity to extract in year 0:** - Demand: \( Q_0 = 10 - P_0 \) - To maximize net benefit, extract as much as demanded at non-negative price, up to stock. - At \( P_0=0 \), \( Q_0=10 \) - Since total stock is 15, extracting 10 units at zero price yields maximum benefit. **Answer:** \(\boxed{Q_0^* = 10}\) --- ### **ii) Consumer surplus:** - Consumer surplus = \( \frac{1}{2} \times \text{quantity} \times \text{height} \) - Height = 10 (from demand curve at zero quantity to price zero) - CS = \( \frac{1}{2} \times 10 \times 10 = 50 \) **Answer:** \(\boxed{CS = 50}\) --- ## **b) Monopoly scenario** ### **i) Quantity monopolist extracts:** - Revenue: \( R = (10 - Q)Q = 10Q - Q^2 \) - Marginal revenue: \( MR = 10 - 2Q \) - Set \( MR=0 \): \( 10 - 2Q=0 \Rightarrow Q=5 \) - Price: \( P=10 - Q=5 \) **Answer:** \(\boxed{Q_{mon} = 5}\) --- ### **ii) Consumer surplus, monopolist revenue, and other terms:** - Consumer surplus: area of triangle with \( Q=5 \), \( P=5 \), maximum price 10: \[ CS = \frac{1}{2} \times 5 \times (10 - 5)= 12.5 \] - Revenue: \( R= P \times Q=5 \times 5=25 \) - This revenue is **scarcity rent** or **resource rent**. **Final answers:** \[ \boxed{ \text{Consumer Surplus} = 12.5,\quad \text{Revenue} = 25,\quad \text{Revenue term: scarcity rent} } \] --- ## **c) Resource useful in years 0 and 1** ### **i) Year 0 extraction:** Optimization of discounted consumer surplus: - \( CS_t= \frac{1}{2}Q_t^2 \) - Maximize \( V= \frac{1}{2}Q_0^2 + \frac{1}{2}\frac{(15 - Q_0)^2}{1.1} \) Derivative w.r.t. \( Q_0 \): \[ Q_0 - \frac{1}{1.1}(15 - Q_0) = 0 \] Solve: \[ Q_0 \left(1 + \frac{1}{1.1}\right) = \frac{15}{1.1} \] \[ Q_0 \approx 7.14 \] - Price: \( P_0 = 10 - Q_0 \approx 2.86 \) **Answer:** \(\boxed{ Q_0^* \approx 7.14,\quad P_0^* \approx 2.86 }\) ### **ii) Year 1 extraction:** - \( Q_1=15 - Q_0 \approx 7.86 \) - \( P_1=10 - Q_1 \approx 2.14 \) ### **iii) Consumer benefit in years 0 and 1:** - Year 0 CS: \( \frac{1}{2} \times 7.14^2 \approx 25.5 \) - Year 1 CS: \( \frac{1}{2} \times 7.86^2 \approx 30.89 \) - Present value of total CS: \[ 25.5 + \frac{30.89}{1.1} \approx 25.5 + 28.08 = 53.58 \] **Comparison:** In part (a) ii), consumer surplus was 50, now approximately 53.58—total consumer benefits increase when resource is allocated over two years. --- ## **d) New demand: \( Q_t=5 - \frac{P_t}{2} \)** Rearranged: \( P_t=10 - 2Q_t \) ### **i) Year 0 and Year 1 optimal extraction:** - Consumer surplus per year: \( CS = Q^2 \) (since \( 10 - P = 2Q \)) - Maximize total discounted surplus: \[ V= Q_0^2 + \frac{(15 - Q_0)^2}{1.1} \] Derivative: \[ 2Q_0 - \frac{2}{1.1}(15 - Q_0)=0 \] Solve: \[ Q_0 \left(1 + \frac{1}{1.1}\right) = \frac{15}{1.1} \Rightarrow Q_0 \approx 7.14 \] - Corresponding price: \[ P_0=10 - 2Q_0 \approx 10 - 14.28 = -4.28 \] Negative price is impossible; the maximum feasible quantity at \( P=0 \) is \( Q=5 \): \[ Q=5 \Rightarrow P=0 \] Similarly for year 1: \[ Q=5,\quad P=0 \] Remaining 5 units are effectively **not valuable at non-negative price**; the resource is **not scarce** anymore, as its demand at zero price is 5 units, less than total stock. ### **ii) Scarcity and scarcity rents:** - Because the maximum demand at \( P \geq 0 \) is 5 units per year, and total stock is 15, the resource **is no longer scarce**. - Scarcity rents **disappear** because the market demand is insufficient to generate scarcity rents; the resource is effectively abundant at zero price. --- **Final summary:** | Part | Question | Final Answer | |---------|------------------------------|------------------------------| | **a i** | Efficient extraction when only useful in year 0 | **10 units** | | **a ii** | Consumer surplus | **50** | | **b i** | Monopolist extract | **5 units** at \( P=5 \) | | **b ii** | Consumer surplus / revenue | **CS=12.5**, Revenue=25, revenue = scarcity rent | | **c i** | Optimal year 0 extraction | **7.14 units**, \( P \approx 2.86 \) | | **c ii** | Year 1 extraction | **7.86 units**, \( P \approx 2.14 \) | | **c iii** | Total discounted consumer surplus | **~53.58** (more than initial static case) | | **d i** | Year 0 & 1 extraction | **5 units each year**, \( P=0 \) | | **d ii** | Scarcity rents? | **No**, resource is not scarce; demand insufficient to generate scarcity rents. | This completes the step-by-step solution.