please can expert help with explanation on the following utility functions?

1. Suppose Michael purchases only two goods, hamburgers (H) and Cokes (C). (a) What is the relationship between MRSH,C and the marginal utilities MUH and MUc ? (b) Draw a typical indifference curve for the case in which the marginal utilities of both goods are positive and the marginal rate of substitution of hamburgers for Cokes is diminishing. Using your graph, explain the relationship between the indifference curve and the marginal rate of substitution of hamburgers for Cokes. (c) Suppose the marginal rate of substitution of hamburgers for Cokes is constant. In this case, are hamburgers and Cokes perfect substitutes or perfect complements? (d) Suppose that Michael always wants two hamburgers along with every Coke. Draw a typi- cal indifference curve. In this case, are hamburgers and Cokes perfect substitutes or perfect complements? 2. Consider the following utility functions: (a) U(x, y) = Ty. (b) U(x, y) = x2y. (c) U(x, y) = Inc + Iny. Show that each of these has a diminishing MRS but that they exhibit constant, increasing, and decreasing marginal utility, respectively. What do you conclude? 3. (a) A consumer is willing to trade 3 units of x for 1 unit of y when she has 6 units of x and 5 units of y. She is also willing to trade in 6 units of x for 2 units of y when she has 12 units of x and 3 units of y. She is indifferent between bundle (6, 5) and bundle (12, 3). What is the utility function for goods x and y? Hint: What is the shape of the indifference curve? (b) A consumer is willing to trade 4 units of x for 1 unit of y when she is consuming bundle (8, 1). She is also willing to trade in 1 unit of x for 2 units of y when she is consuming bundle (4, 4). She is indifferent between these two bundles. Assuming that the utility function is Cobb-Douglas of the form U(x, y) = roy, where o and B are positive constants, what is the utility function for this consumer? (c) Was there a redundancy of information in part (b)? If yes, how much is the minimum amount of information required in that question to derive the utility function? 4. Consider the function U(x, y) = x + Iny. This is a function that is used relatively frequently in economic modeling as it has some useful properties. (a) Find the MRS of the function. (b) Find the equation for an indifference curve for this function. (c) Compare the marginal utility of x and y. How do you interpret these functions? How might consumers choose between x and y as they try to increase their utility by, for example, con- suming more when their income increases? (d) Considering how the utility changes as the quantities of the two goods increase, describe some situations where this function might be useful