Quantitative Econ. Please help thanks!

(16 points) Household Production. Consider a static economy. A consumer has preferences defined over consumption C and leisure L, i.e. u(C, L). The equation L + N = H represents the consumer's time- allocation constraint. This consumer allocates time, N, to home production, where they generate goods according to the technology Y =f(N). The consumer does not have any external sources of income and consumes all the goods they produce. (a) (4 points) Write the consumer's constraint relating consumption to leisure and derive the relative cost of leisure in terms of consumption. (b) (4 points) Set up the consumer's problem and derive the optimality condition, a set of equations that will allow you to find the optimal amount labor N*, optimal consumption, C*, and optimal leisure, L*. (c) (4 points) Assume that the utility function is given by u(C, L) = log C + vlog L over consumption C and leisure L, where y > 0. The production function is f ( N) = No, 0 0. The production function is f(N) = No, 0