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3. Tim's utility function labor and consumption is given by the utility function UTUJ) = 10(12 L) + C, where L is the number of hours of labor supplied in a day and C is the dollar amount of consumption goods purchased with wage income (pc = $1). R = 12 L is the number of hours of leisure that the worker has during the day. Tim's friend Sophia has the following preferences for labor and consumption: U5(L) = 5(12 L) + C, a) Derive Tim and Sophia's labor supply functions. b) Draw Tim and Sophia's labor supply functions on the same graph and compare. The current wage rate is wow\" = $8. A minimum wage increase is being proposed by the state legislature that would increase the hourly wage to wnew = $12. c) How does the labor supply change for Tim and Sophia from the old wage rate to the new minimum wage? Suppose the state government were to impose a lump-sum tax (T) on their wage income. d) What is the highest tax that can be imposed so that Tim is no worse-off than before the minimum wage was increased? e) Redo part d) for Sophia and compare your answers for Tim and Sophia