Suppose your utility over donuts (D) and sodas (S) is given by U(D, S) = D^0.5S^0.5 + 10.

a. Calculate the marginal rate of substitution (MRS) between donuts and sodas.

b. Suppose you have income of Y dollars to spend on donuts and sodas each week. Denoting the price of donuts PD and the price of sodas Ps, derive your demand functions for donuts and for sodas.

c. Do you view donuts as a normal or inferior good? Explain both mathematically and in words how you arrived at your answer.

d. Suppose that Y = $20 and that Ps = $2. Using the demand function you derived for donuts in part b, draw your demand curve for donuts (where the price per donut is on the y-axis and the quantity of donuts is on the x-axis).

e. Suppose now that Y increases to $24. Still assuming Ps = $2, use the demand function you derived for donuts in part b to draw your new demand curve for donuts. Explain in words why the demand curve is (or is not) different given the change in income.

f. If Y = $20, Ps = $2, and PD = $1, how many donuts and sodas would you consume? What is the utility level associated with this consumption choice?

g. If Y = $24, Ps = $2, and P_D = $1, how many donuts and sodas would you consume? What is the utility level associated with this consumption choice? How does your utility level in this case compare with that which you found in part f?

h. Without doing any further math, draw the budget constraints and indifference curves associated with your solutions in parts f and g (the figure does not need to be perfectly to scale, but be sure to label all axes and lines).

i. Now suppose that the government wants to discourage consumption of donuts. It places a tax on donuts such that PD = $4 (instead of $1). Assuming the price of sodas remains the same at $2 and your income is $24, determine the new optimal consumption of donuts and sodas. What is the utility level associated with this new consumption choice? How does it compare to that which you found in part g?

j. Suppose now that your utility over donuts (D) and sodas (S) is given by U(D, S) = D^0.5S^0.5. Without doing any math, discuss how and why this would (or would not) affect the optimal consumption bundles you derived in parts f, g, and i of this problem.