Please help me with detailed calculations and explanations if necessary.

Consider an economy populated by a representative consumer, that has preferences over leisure () and consumption (c). The consumer's utility function in this one-period model is: u(c, 1 ) = [In(c) + - In(1) The consumer has 1 unit of time (h = 1) at his disposal to spend between leisure and work. With T being a fixed lump-sum tax, # being profits from dividends, and w being wage, the consumer's budget constraint can be written as: ct w= wth-T The representative firm's production function is Y = zK1/2 N1/2 Assume that K = 1, z = 10. (a) Define and Calculate the marginal rate of substitution of leisure for consumption (MRSlc). (b) Show why it is not optimal for the consumer to choose the allocation c = 5 and 1 = 0.64 if the wage is w = 10. (c) Show why this firm does not maximize profits , if it chooses N = 0.64 when w = 10. (d) Explain intuitively, what should this firm do