Consider a couple consisting of a male m and a female f who live together. Each partner i has one unit of time to alloca

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Consider A Couple Consisting Of A Male M And A Female F Who Live Together Each Partner I Has One Unit Of Time To Alloca 1 (150.18 KiB) Viewed 26 times

Consider a couple consisting of a male m and a female f who live together. Each partner i has one unit of time to allocate to either household production or market work. Let t; be the time that partner i spends in household production and let 1 – ti be the time that he/she spends doing market work. Let w; be the wage rate of partner i and assume that 0 < w; <Wm. The couple use their total joint earnings to buy a household public good, denoted Q. Hence Q=Wm (1 – tm) + wf (1 – t) (1) The household produced goods is denoted 2 and is given by the sum of their time inputs z = tm +ts (2) and z is also a household public good. Specifically, each partner has preferences given by u; = { log (Q)+{ log (2). For simplicity, assume that they choose their time allocation {tm;tp} to maximize the sum of their utilities, U = Um + up = log (Q) + log (2) (3) (a) Argue that, at the optimum, at least one partner spends some time in household production, that is t; > 0 for at least one partner, i = m, f. (6 Marks) (b) Argue that, at the optimum, at least one partner spends some time in market work, that is t; < 1 for at least one partner, i = m, f. (6 Marks) (c) Argue that, at the optimum, they cannot both be splitting their time between household production and market work, that it it cannot be that 0 <t; < 1 for both i = m, f. (10 Marks) (d) Argue in particular that, given the assumption that Wm > wf, it must be that either (i) tm = 0 and 0 < t < 1, or (i) 0 <tm < 1 and t; = 1. (11 Marks)