Question 1 Consider an economy consisting of one consumer, one firm, and two commodities: consumption (3 and leisure/labour E. The consumer is endowed with one unit of leisure time and zero units of consumption. Given 6 units of consumption and 2 units of leisure, their utility is given by u(c, 1?) : lnc+ alnf, for some parameter or > 0. Suppose that the price of consumption is normalised to 1. In addition, we assume that any time that is not devoted to leisure can be sold as labour at price it). Finally, as the sole shareholder, the consumer has rights to all the profits 7r generated by the firm. The technology ofthe firm is given by FUD) : x/E, where L units of labour is transformed into FUD) units of consumption. The objective ofthe price-taking firm is to maximise profits 71' = F(L) wL, given the price in. As stated above, all profits are transferred to the consumer as a lump sum. (i) Write down the optimisation problem of the consumer. Show that the demand for consumption and leisure is given by C = LIUTT and E = % by the consumer to the labour market? , respectively. How much labour time is supplied [7 marks] (ii) Solve the optimisation problem ofthe firm. Show that the demand for labour is given by L = 1/?112. How much consumption is being produced given the price of labour in? How high are the profits generated by the firm? [7 marks]