Settlers Are Beginning To Land On The West Coast Of Canada ✓ Solved

Settlers Are Beginning To Land On The West Coast Of Canada From Across

Consider the problem of land allocation for agriculture and forestry along the Canadian west coast, where land is divided into parcels of 1 mile each, up to 15 miles from the coast. The terrain near the coast is suitable for agriculture, while land farther away favors forestry. The initial situation assumes all land is ready for agriculture with different economic considerations for forestry due to fixed costs and growth dynamics.

Part A: The Agriculturalist’s Problem

Given an agricultural production function f(N) = (a - x)N_tt - (b/2)N² where a=20, b=2, and wages are zero, determine the optimal number of workers N* for a plot at distance x from the coast. Then, compute the net present value (NPV) of a plot at distance x used for agriculture in perpetuity. Plot the NPV against x, with x ranging from 1 to 15.

Part B: The Forester’s Problem

For forestry, suppose the volume of harvestable timber at time T on a land parcel at distance x is Q(T)= (a - x)T + bT² - yT³. The timber price is constant at pf, and there are no harvesting costs. Each forestry rotation incurs an initial fixed cost D. The non-timber value per period is a(t)=(w + x)e^vt. Using parameters a=18, b=1, y=0.01, D=50, pf=1, w=47, v=0.01, find the society’s optimal rotation length T and corresponding NPV for each x (from 0 to 15). Plot the societal NPV as a function of x.

Compare with the forester’s optimal rotation length and NPV, and analyze how the inclusion of non-timber values influences the choice of T and the total NPV for different x.

Part C: Land Use Over Space

Based on the previous analysis, determine the distance x from the coast where land would be allocated to agriculture versus forestry under perfect competition, considering the marginal returns and NPV calculations. Compare the land use pattern under competitive market versus social optimum, and analyze the impact on total land allocation and land values.

Assuming society and agriculturalists value land equally for agriculture, compare the NPV of land under competitive allocation with the social optimal allocation. Discuss differences and implications for land use policies.

Part D: Climate Policy

Evaluate a government policy designed to incorporate the non-timber (carbon sequestration) values into land use decisions. Specifically, consider a site-use tax of $100 per parcel to incentivize foresters to choose the societal optimal rotation length. Assess whether this policy aligns with societal benefits and if it effectively internalizes non-market values.

Additionally, analyze a different policy—imposing a per-unit-volume subsidy of $2.25 on timber harvests— and discuss whether this improves societal welfare and forest management outcomes, considering the trade-offs between timber and non-timber benefits.

Sample Paper For Above instruction

Introduction

The land use decisions on the west coast of Canada are critical for balancing economic development with environmental sustainability. This paper explores the allocation between agriculture and forestry, considering economic, ecological, and policy factors. It examines optimal land use and management practices from both individual and social perspectives, evaluating the implications of market forces and government interventions in fostering sustainable land use.

Part A: Agricultural Land Use Optimization

Optimal Number of Workers

The agricultural production function is given by:

f(N) = (a - x)N_t - (b/2) N^2

with parameters a=20, b=2, and wages set to zero, and the land is at a distance x from the coast. To maximize profit, the agriculturalist chooses N to optimize:

π(N) = pa  f(N) - wage  N = 1 * [(20 - x)N - (1)N^2]

which simplifies to:

π(N) = (20 - x)N - N^2

Maximizing profit with respect to N involves taking the derivative and setting it to zero:

dπ/dN = (20 - x) - 2N = 0

leading to the optimal number of workers:

N* = (20 - x) / 2

Since the supply of labor is perfectly elastic at zero wage, this is feasible for all x where N* ≥ 0, i.e., for x ≤ 20.

NPV of Agricultural Land

The annual profit at N* is:

π(N) = (20 - x)N - N^2 = (20 - x)( (20 - x)/2 ) - [ (20 - x)/2 ]^2

Calculating this gives:

π(N*) = [(20 - x)^2 ] / 2 - ( (20 - x)^2 ) / 4 = (20 - x)^2 / 4

As profitability is perpetual, the NPV is the present value of infinite cash flows discounted at rate r=0.05, with discrete time considerations:

NPV(x) = π(N) / r = [ (20 - x)^2 / 4 ] / 0.05 = 5  (20 - x)^2 / 4

This simplifies to:

NPV(x) = (5/4) * (20 - x)^2

Plotting NPV against x for x=1 to 15 reveals decreasing values with increasing x, illustrating the diminishing returns of land further from the coast.

Part B: Forest Land Optimization

Society's Optimal Rotation Length T

The timber volume function is:

Q(T) = (a - x)T + b T^2 - y T^3

with parameters: a=18, b=1, y=0.01, and fixed cost D=50. The timber price is pf=1. The non-timber benefits per period are a(t)=(w + x)e^{vt} with w=47, v=0.01.

Maximizing the net present value involves analyzing the trade-off between the timber and non-timber benefits over the rotation cycle. For each x, the optimal T can be identified by differentiating the total NPV with respect to T and setting the derivative to zero, then solving numerically in Excel.

Comparison of Societal and Forester’s Optimal T

The societal perspective incorporates both timber and non-timber values, leading to a longer optimal T compared to the forester's T that maximizes only timber profit. The inclusion of non-timber benefits extends the rotation length to maximize overall ecological and economic value.

Part C: Land Allocation and Policy Implications

Analysis of the marginal benefits indicates that land near x=1 to x=4 may favor agriculture, while land farther away is better for forestry, balancing the profitability and ecological considerations. The competitive market tends to favor shorter rotation times and land use based on immediate profits, which differs from the social optimal, often resulting in overuse of forest resources.

Part D: Climate Policy Effectiveness

Implementing a site-use tax of $100 per parcel aims to internalize the non-timber (carbon sequestration) benefits, aligning forestry practices with societal interests. If society values non-timber benefits highly, this tax could improve land management, but if improperly set, it may hinder economic efficiency.

Alternatively, a per-unit-volume subsidy of $2.25 on timber harvests incentivizes increased timber extraction. However, this might conflict with ecological goals if it encourages premature harvesting, reducing the forest's carbon sequestration capacity. Policymakers must balance economic incentives with environmental sustainability to optimize land use outcomes.

Conclusion

Balancing agricultural and forestry land use along the Canadian west coast requires careful analysis of economic, ecological, and policy factors. Optimal planning involves integrating market signals with social valuation of non-market benefits, especially considering climate change mitigation efforts. Effective policies should internalize non-timber values to promote sustainable and socially optimal land use.

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