Albert and Bob consume video games and books. Albert has an initial endowment of 60 books and 20 video games. Bob has an initial endowment of 20 books and 30 video games. They have no other assets and make no trades with anyone other than each other. For Albert, two books and one video game are perfect substitutes. His utility function is U(b,g)=b+2g, where b is the number of books he consumers and g is the number of games he consumes. Bob's preferences are more subtle and convex. He has a Cobb-Douglas utility function, U(b,g)=b*g. in the Edgeworth box below, Albert's consumption is measured from the lower left, and Bob's is measured from the upper right corner of box.
a. On this diagram, write the total number of books and games at the two question marks.
b. Mark the initial endowment and label it E. Use red ink to draw Albert's indifference curve that passes through his initial endowment. Use blue ink to draw Bob's indifference curve that passes through his initial endowment. (Remember that quantities for Bob are measured from the upper right corner, so his indifference curves are flipped over.
c. At any Pareto optimum, where both people consume some of each good, it must be that their marginal rates of substitution are equal. No matter what he consumes, what is Albert's marginal rate of substitution?
When Bob consumes the bundle, (bb, gb), what is his MRS? (hint: should be a function of bb and gb).
What equation would you get if you equate the two MRS?
d. What should the price ratio be at a competitive equilibrium if Albert consumes some books and some wine? (hint: the price ratio should be equal to Albert's marginal rate of substitution.)
e. At the equilibrium price you found in the last part of the question, how many books and games will Bob consume? How many books and games will Albert consume? Mark this point as F on the diagram.
f. At the competitive equilibrium you found above, how much more utility does Albert get from the trade (as compared to the endowment)? How much more utility does Bob get from the trade?
g. We know the equilibrium allocation at F represents a Pareto optimum. On the diagram, please find another two Pareto optimal points and use black ink to draw a contract curve that goes through them and F.