Find the partnership equilibrium in a two-person firm with the following features: T = 24, and each individual has the utility function U =5 ln(x + 1) + y. Each dollar of income generated by the firm requires a total of two hours of effort per day as input.

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Consider a 2-person firm where each person i, i = 1,2, has a utility function defined over income y, and leisure, x;e U,(X, yi) = 5in(x; + 1) + y;" Each person has 24 hours a day to divide between work and leisure. The production function for the firm generates $1 of income for every two hours of work per day. a) What is the equation for this firm's production function? b) What levels of leisure for each individual maximizes joint welfare in this case? c) If the individuals cooperated to maximize the sum of their utilities while sharing the joint income equally, what income would each receive? d) If each individual chose their leisure independently, assuming that the other would choose the leisure you found in (b), would each person's income be equal to that in (b)? Briefly explain why or why not. e) Derive the partnership outcome for this firm. Suppose now that another person joins the firm, so there are now 3 individuals producing income. This person is a clone of the other two: same utility function and same productivity. f) Would the difference between the answers to parts (c) and (d) for this 3 person firm be greater or smaller than those you derived above for the two person firm? What is the economic intuition behind this result? Consider a 2-person firm where each person i, i = 1,2, has a utility function defined over income y, and leisure, x;e U,(X, yi) = 5in(x; + 1) + y;" Each person has 24 hours a day to divide between work and leisure. The production function for the firm generates $1 of income for every two hours of work per day. a) What is the equation for this firm's production function? b) What levels of leisure for each individual maximizes joint welfare in this case? c) If the individuals cooperated to maximize the sum of their utilities while sharing the joint income equally, what income would each receive? d) If each individual chose their leisure independently, assuming that the other would choose the leisure you found in (b), would each person's income be equal to that in (b)? Briefly explain why or why not. e) Derive the partnership outcome for this firm. Suppose now that another person joins the firm, so there are now 3 individuals producing income. This person is a clone of the other two: same utility function and same productivity. f) Would the difference between the answers to parts (c) and (d) for this 3 person firm be greater or smaller than those you derived above for the two person firm? What is the economic intuition behind this result?

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