Corner Solution of OptimizationRevealed PreferenceDeriving Individual Demand, Engle CurveCite as: Chia-Hui Chen, course materials for 14.01 Principles of Microeconomics, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 1 1 Corner Solution of Optimization 14.01 Principles of Microeconomics, Fall 2007 Chia-Hui Chen September 17, 2007 Lecture 6 Optimization, Revealed Preference, and Deriving Individual Demand Outline 1. Chap 3: Corner Solution of Optimization 2. Chap 3: Revealed Preference 3. Chap 4: Deriving Individual Demand, Engle Curve 1 Corner Solution of Optimization When we have an interior solution, Px Ux = Py Uy must be satisfied. However, sometimes a consumer gets highest utility level when x = 0 or y = 0. If that’s the case, we have corner solutions, and Px Ux = ,Py �Uy as shown in Figure 1. In Figure 1, because people cannot consume negative amounts of goods (bundle A), their best choice is to consume bundle B, so the quantity o f y consumed is zero. Conditions for corner solutions: • Ux PxMRS = > when y = 0. Uy Py • Ux PxMRS = < when x = 0. Uy Py Example (An example of consumer’s problem). The parameters are Px = 1, Py = 1,Cite as: Chia-Hui Chen, course materials for 14.01 Principles of Microeconomics, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 2 1 Corner Solution of Optimization Figure 1: Corner Solution to Consumer’s Problem.Cite as: Chia-Hui Chen, course materials for 14.01 Principles of Microeconomics, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 3 2 Revealed Preference I = 2. The utility function is U(x, y) = x + 2√y. The budget constraint is x + y = 2 . According to the condition for an interior solution: Px Ux = . Py Uy = ⇒ 1 1 = . 1 √ 1 y = ⇒ y = 1 = x = 1.⇒If the price y changes to 1: Py = 1, then the solution is y = 4 = x = −3 < 0,⇒which is impossible. Then we have the corner solution: x = 0, y = 2. x = 0 since consumer wants to consume as little as possible. 2 Revealed Preference In the former chapters, we discussed how to decide optimal consumption from utility function and budget constraint: Utility Function = Optimal Consumption ⇒Budget Constraint And now we discus s how to know cons umer’s pre ference from budget constraint and consumption: Budget Constraint = Preference ⇒ConsumptionCite as: Chia-Hui Chen, course materials for 14.01 Principles of Microeconomics, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 4 3 Deriving Individual Demand, Engle Curve 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 X: 1.478 Y: 3.761 y X: 1.929 Y: 5.142 X: 4.751 Y: 2.124 X: 3.949 Y: 1.101 B C A D x Figure 2: A Contradiction o f Preference. A and B are the Choices. Example (Revealed preference). In Figure 2, two budget c onstraint lines inter-sect. Assume one person’s choices are A and B respectively. Then we have A � C, B � D. And Figure 2 obviously shows tha t C ≻ B, D ≻ A. Thus, A � C ≻ B � D ≻ A, which is a c ontradiction, which means utility does not optimized and the choice is not rational. 3 Deriving Individual Demand, Engle Curve Use the following utility function again: U(x, y) = x + 2√y, with a budget constraint: Pxx + Pyy = I.Cite as: Chia-Hui Chen, course materials for 14.01 Principles of Microeconomics, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 5 3 Deriving Individual Demand, Engle Curve When P 2 I � x ,Py we have an interior solution. MRS = Px/Py. Thus, I Px x = ,Px −Py � �2Px y = . Py When P 2 I � x ,Py we have a corner solution. x = 0, I y = . Py • Figure 3 shows a demand function of y and Py as an example. (Assume that I, x a nd Px are held constant.) • Engle Curve descr ibes the relation between quantity and income. Figure 4 shows the relation between x and income, and Figure 5 shows that between y and income. Normal good. Quantity demanded of good increases with income. Inferior good. Quantity demanded of good decreases with income. Substitutes. Increase in price of one leads to an incre ase in quantity demanded of the other. Complements. Increase in price of one leads to an decre ase in quantity demanded of the other. For this problem, P 2 x • if I < P, x and y are neither substitutes nor complements, and x is a y normal good. P 2 xif I � , x and y are substitutes, and y is a normal good. y • PCite as: Chia-Hui Chen, course materials for 14.01 Principles of Microeconomics, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 6 3 Deriving Individual Demand, Engle Curve Figure 3: Demand Function for Goods ‘y’.9 10 0 1 2 3 4 8 7 6 0 1 2 3 4 5x P x 2/Py 5 6 7 8 9 10 I 7 3 Deriving Individual Demand, Engle Curve Figure 4: The Relation between Income and Quantity Demanded o f ‘x’. Engle curve of x. 0 1 2 3 4 5 6 7 8 9 10 y P x 2/Py 0 1 2 3 4 5 6 7 8 9 10 I Figure 5: The Relation between Income and Quantity Demanded of ‘y’. E ngle curve of y. Cite as: Chia-Hui Chen, course materials for 14.01 Principles of Microeconomics, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month