# What does this imply for the preferences of Bill in terms of consumption this year and next year relative to Janeís preferences?

) Bill and Jane grow wheat on their farms this year (period 1) and next year (period 2). Each farm generates an endowment of one unit wheat in each period. Both farmers like to consume wheat, but they like wheat this year better than wheat next year. The utility function for each farmer is: u B x B 1 ; xB 2 = ln x B 1 + B ln x B 2 , and u J x J 1 ; xJ 2 = ln x J 1 + J ln x J 2 , where x B 1 is the consumption of wheat this year and x B 2 is the consumption of wheat next year, by Bill, and similarly for Jane. The parameter B (and J ) is the discount factor, which diminishes the value of consumption next year relative to this year and we assume that 0 < B; J < 1. Farmer Bill and Jane can trade wheat in period 1 against wheat in period 2 and the price of wheat in period 1 is normalized to p1 = 1. Thus each farmer i faces a single (intertemporal) budget constraint: p1x i 1 + p2x i 2 = p1e i 1 + p2e i 2 , and given the information can simplify it to 1 x i 1 + p2x i 2 = 1 1 + p2 1. 1. Describe graphically the consumption problem for Farmer Bill with his endowment and prices p1 = 1 and arbitrary p2 (in a single agent diagram). 2. Determine graphically how the solution to Farmer Billís consumption problem changes with changes in his discount factor B and the price p2. Brieáy describe the intuition behind the comparative static results (in a single agent diagram). 3. Determine the optimal demand for wheat by Farmer Bill analytically as a function of his discount factor and the price p2. For which values of B and p2 does Farmer Bill wish to consume identical amounts of wheat in period 1 and period 2. 4. Graphically describe the trading environment for Farmer Bill and Jane in the Edgeworth box, identify the endowment point and label all axis and other objects you wish to introduce. 5. Suppose now that the discount factor for Bill and Jane are identical, or B = J = 2 (0; 1). Find the competitive equilibrium and relate it to the endowment of the farmers and the discount factor. 6. Suppose now that the discount factor of Bill is lower than the discount factor of Jane, or B < F . What does this imply for the preferences of Bill in terms of consumption this year and next year relative to Janeís preferences? Intuitively, how would you think that the competitive equilibrium in this new environment changes to the earlier environment? 7. For B < F derive the competitive equilibrium. What can you say about the trade pattern relative to the endowment and about the price of the second good, i.e. consumption of wheat tomorrow.

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