Omar consumes only two goods, whose quantities are measured by x and y. His preferences are described by the utility function U(x, y) = xy + 10(x + y). His marginal utilities are MUx = y + 10 and MUy = x + 10. The prices of the goods are PX = $9 and Py = $3. He has a daily income of $30. (10 Marks) a) Show that Omar has a diminishing marginal rate of substitution of x for y. Why is this important in applying the method of Lagrange to the consumer choice problem? b) With Omar’s utility function, is it possible that the utility-maximizing basket (x, y) will be at a corner point, with either x = 0 or y = 0? Do you expect the budget constraint to be binding? c) Using the method of Lagrange, find Omar’s optimal consumption basket (x, y) when his income is $30. Also find the numerical value of the Lagrange multiplier, λ, which measures Omar’s marginal utility of income (the rate of change) when his income is $30. d) Find Omar’s optimal consumption basket (x, y) and the value of λ if his income is $31. e) Show that the increase in Omar’s utility when his income rises from $30 to $31 is close to the values of λ you found in parts (c) and (d).
Answer: Diminishing Marginal Rate of Substitution The marginal rate of substitution (MRS) is the rate at which one good must be substituted for another to maintain the same level of utility. As the quantity of one good increases while the