Dick, Jane, and their dog, Spot, share an apartment. Their utilities depend upon the cleanliness of the apartment and the hours spent doing housework. Dick and Jane are both averse to housework, though Dick finds it especially troublesome
Dick, Jane, and their dog, Spot, share an apartment. Their utilities depend upon the cleanliness of the apartment and the hours spent doing housework. Dick and Jane are both averse to housework, though Dick finds it especially troublesome. Their utility functions are
where c is an index of cleanliness and h represents hours of housework. The degree
of cleanliness is determined by the “production function”
a) Define the marginal rate of substitution MRSi as the increase in cleanliness that exactly compensates person i for one more hour of housework:
and define the marginal rate of transformation as the number of hours of leisure that must must be given up to generate a one unit increase in cleanliness. Find the Samuelson condition for this economy.
b) An allocation is a triplet (h D, hJ, c). What two conditions must a Pareto optimal allocation satisfy? Show that there are Pareto optimal allocations in which Dick, despite his aversion to housework, does more housework than Jane.
c) Assume that Dick and Jane independently decide how much housework they will do. Find the hours of work done by each of them in the Nash equilibrium, and show that both people would be better off if each of them did a little extra housework.
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