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. Decomposition Bundles.® Consider an individual with an utility function u(x, y) = i y, and facing prices py = $2 and py, = $4. (a) Assuming that his income is / = $800, find the optimal consumption of goods x and y that maximizes his utility. That is, solve his UMP. (b) Consider now that the price of good y decreases from p, = §4 to p)', = $3. Find this consumer’s new optimal consumption bundle. Then, identify the total effect of the price change, and decompose it into the substitution and income effects. (¢) Considering that the price of good y remains at p, = $4, assume that the consumer seeks to reach the same utility level as in part (a). Find the optimal consumption of goods x and y that minimizes his expenditure. That is, solve his EMP. (d) As in part (b), assume that the price of good y decreases from py, = $4 to p{, = $3. Find this consumer’s new optimal consumption bundle. Comparing your results from parts (c) and (d), argue that the total effect that we find when using the compensated demand (the result of the EMP) measures the substitution effect alone. Interpret. . Perfect Substitutes.® Peter’s preferences for tea and coffee are given by u(x,y) =2x+ y, where x denotes the units of tea and y the units of coffee. His income is $500, and the initial prices are Px= $36 and py - $22 (a) Find the utility-maximizing pair of tea and coffee. [Hint: Peter regards tea and coffee as perfectly substitutable, so you should anticipate that he consumes only one of the two goods. ] (b) Assume now that the price of tea increases, to py = $40. Is his consumption of tea and cof- fee affected by the price change? Your answer defines the total effect of the price change. Decompose it into substitution and income effects. (c) What if the price of tea further increases to p, = $83? What is the total effect of the price change? What are the substitution and income effects?