Plot capital as a function of the herring stock and entitle

Question

Read the article, then do the problem set.

Article: The Dynamics of an Open Access Fishery by Trond Bjorndal and Jon M. Conrad.

Problem Set 6 Renewable Resources

Part 1. Article Review

Bjorndal and Conrad wrote an article entitled "The dynamics of an open access fishery" in the Canadian Journal of Economics vol 20 issue1, pages 74-85. The article describes the dynamics of the North Sea herring fishery during the 1960s and 1970s. The fishery was in open access. The boats used are called "purse seiners" (their nets look like a big purse). Please refer to this article in answering the questions in Part 1. You do not need to read everything in the article to answer the questions below but you will find it helpful. You can download the article on the class webpage.

1. Set up an Excel spreadsheet to plot capital (K - the number of vessels) as a function of the size of herring stock (S) for the years 1963-1977 using Table 1. Entitle sheet 1 and your graph, "real data". Explain the biological dynamics and the vessel dynamics in detail.

2. We can write the dynamic system representing the fishery with the two difference equations:

K_t+1=K_t+nπ_t/(p_tK_t) where π_t = p_tY_t - c_tK_t
S_t+1=S_t+rS_t(1-S_t/L)-Y_t.

Let us assume the same parameters as in the text:
L = 3,200,000; n = 0.1; r = 0.8 and the starting values S0 = 2,325,000 and K0=120.

Use the production function, Y_t, specified in Table 2 equation (a)(ii). On sheet 2 of your excel spreadsheet, simulate stock size (S) , fleet size (K) and harvest (Y) using the dynamic system above for the years 1963-1977 along with cost per vessel and price of herring from Table 3. Plot capital as a function of the herring stock. Entitle the sheet 2 and your graph "simulated a".

Copy this sheet on sheet 3 but change the production function to the specification in (b)(ii). Call this sheet and your graph "simulated b". Do the same thing in two more sheets using the two remaining production functions: (c)(ii) and (d)(ii) and call them "simulated c" and "simulated d," respectively. Identify the production function that gives a simulated result that most closely follows the real data by writing "Best Fit" beside the graph title.

3. Let us assume that starting from 1978 onwards the price of herring and cost of vessels are fixed at p=2000 and c = 857000, respectively. On sheet 6 of your spreadsheet, simulate stock size, fleet size and harvest for years 1963-2000 using the production function that best fits the data and graph your results. Plot capital as a function of the herring stock and entitle sheet 6 and your graph as "forecasted data." Describe what happens to stock and capital. Write your response in sheet 6.

Part 2. Solver analysis

Your uncle plans to invest in the tuna fishery for the coming 15 years. The natural growth function of the tuna population (X_t) is rX_tln(K/X_t). Your uncle could use up to 2 fishing boat the first 5 years, up to 4 boats the following 5 years and up to 6 boats the final 5 years. The amount of tuna caught during a season (Y_t) is a function of the number of boats he decides to use (E_t) and X_t according to Y_t=qE_tX_t. The season market price (p) depends on the quantity harvested: p_t=a/(1+Y_t). The seasonal fishing cost per boat increase with, Et, the total number of boats used this season: c=b ln(1+E_t). Finally, no matter how many tunas your uncle catches, he has to leave some fish behind when he withdraws from the fishery after 15 years. Note, however, that he does not value any of the remaining stock. Assume that the discount rate, δ, is equal to 0.10 and the following parameter values: a = 200, b = 10, q = 0.2, X₀ = 5, r = 1.8 and K = 50.

Setup the problem in sheet 7 using an initial guess of 0 for all Et. How many boats should he use each season? To answer this, copy everything into sheet 8 and use solver to derive the optimal number of boats. In the same sheet, plot Xt and Et as a function of time on separate graphs.

Entitle your graphs, stock over time and effort over time respectively. How much should your uncle invest? (write your answer in the same sheet).

Hint: You cannot use ½ or 0.667 boats. Think integers!