These are 3 questions. Please help me solve them.

1. Suppose consumer's utility function for the one-period model we considered in class is given by with w > ; > 0, * = 0 and T = 0. (a) What assumptions about the utility function we made in class are violated by this utility function? (b) Write down the consumer's maximization problem given the other features of the econ- omy as given above. (c) Solve for equilibrium c' and (* given w. Graphically illustrate the equilibrium. 2. For the consumer's problem discussed in class, suppose Utility function is given by U(c !) = 25 . 10.5 and instead of a lump-sum tax T we had a proportional tax f on wage income. Then the consumer's after-tax wage income is given by w(h - 1)(1 - t). Assume # = 0, w = 4, h = 1. Show that if both taxes yield the same revenue for the government the consumer would prefer a lump-sum tax to a proportional tax. 3. For the closed-economy one-period model discussed in class, suppose that consumer's prefer- ences are given by U(c, !) = me+ Ind and the aggregate production function is given by Y = F(K, Na) = KIN (1) There is no government in the economy; and K = 1 and h = 1 are exogenous and given. (a) Do all assumptions that we made for the aggregate production function hold for the function given by equation (1)? Explain. (b) Define the competitive equilibrium for this economy. (c) Characterize the competitive equilibrium. Derive the equilibrium profit for the firm and the equation that would be used to solve for wages for this economy. (d) Draw a diagram with consumer's indifference curve (IC) and the production possibilities frontier (PPF), and show the competitive equilibrium in this diagram. (Note: you need to derive the equations for the indifference curve and the PPF) (e) Suppose the economy is at this competitive equilibrium and capital, K, increases. Using a diagram illustrate the change in equilibrium. Provide and explain the changes (increase or decrease) in equilibrium values for c, l, a and w