Utility Function, Deriving MRSBudget ConstraintOptimization: Interior SolutionCite 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 Utility Function, Deriving MRS 14.01 Principles of Microeconomics, Fall 200 7 Chia-Hui Chen September 14, 2007 Lecture 5 Deriving MRS from Utility Function, Budget Constraints, and Interior Solution of Optimization Outline 1. Chap 3: Utility Function, Deriving MRS 2. Chap 3: Budget Constraint 3. Chap 3: Optimization: Interior Solution 1 Utility Function, Deriving MRS Examples of utility: Example (Perfect substitutes). U(x, y) = ax + by. Example (Perfect complements). U(x, y) = min{ax, by}. Example (Cobb-Douglas Function). cU(x, y) = Axb y . Example (One good is bad). U(x, y) = −ax + by. An important thing is to derive MRS. dy MRS = = Slope of Indifference Curve .−dx | |10 9 8 7 6 U(x,y)=ax+by=Const y5 4 3 2 1 0 0 2 4 6 8 10 x 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 Utility Function, Deriving MRS Figure 1: Utility Function o f Perfect Substitutes y 10 9 8 7 6 5 4 3 U(x,y)=min{ax,by}=Const 2 1 0 0 2 4 6 8 10 x Figure 2: Utility Function of Perfect Complements2 4 6 8 10 y U(x,y)=Axa y b=Const 0 0 2 4 6 8 10 x 10 9 8 7 6 U(x,y)=−ax+by=Const y5 4 3 2 1 0 0 2 4 6 8 10 x 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 1 Utility Function, Deriving MRS Figure 3: Cobb-Douglas Utility Function Figure 4: Utility Function of the Situation That O ne Go od Is BadCite 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]. r 4 2 Budget Constraint Because utility is constant along the indifference cur ve, u = (x, y(x)) = C, = ⇒ ∂u ∂u dy ∂x + ∂y dx = 0, = ⇒ ∂u dy ∂x −dx = .∂u ∂y Thus, ∂u MRS = ∂x .∂u ∂y Example (Sample utility function). u(x, y) = xy 2 . Two ways to derive MRS: • Along the indifference curve xy 2 = C. c y = . x Thus, dy √c yMRSd = = = .−dx 2x3/2 2x • Using the conclusion above ∂u 2 ∂x y yMRS = = = .∂u 2xy 2x ∂y 2 Budget Constraint The problem is ab out how much goods a person can buy with limited income. Assume: no saving, with income I, only spend money on goods x and y with the price Px and Py. Thus the budget constraint is Px x + Py y � I. · · Suppose Px = 2, Py = 1, I = 8, then 2x + y � 8. The slope of budget line is dy Px −dx = Py . Bundles b e low the line are affordable. Budget line can shift: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 2 Budget Constraint 10 8 6 2x+y≤8 y 4 2 0 0 2 4 6 8 10 x Figure 5: Budget Constr aint 10 9 0 2 4 6 8 10 x 0 1 2 3 4 5 6 7 8 y 2x+y≤8 2x+y≤6 Figure 6: B udget Line Shifts Because of Change in IncomeCite 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 Optimization: Interior Solution 10 0 2 4 6 8 10 x 0 1 2 3 4 5 6 7 8 9 y 2x+y≤8 x+y≤4 Figure 7: B udget Line Rotates Because of Change in Price ′ • Change in Income Assume I = 6, then 2x + y = 6. The budget line shifts right which means more income makes the affordable region larger. ′ • Change in Price Assume Px = 2, then 2x + 2y = 8. The budget line changes which means lower price makes the affordable region larger . 3 Optimization: Interior Solution Now the consumer’s problem is: how to be as happy as possible with limited income. We can simplify the problem into language of mathematics:    xPx + yPy � I  max U (x, y) subject to x � 0 . x,y   y � 0 Since the preference has non-satiation property, only (x, y) on the budget line can be the s olution. T herefore, we can simplify the inequality to an equality: xPx + yPy = I. First, consider the case where the solution is interior, that is, x > 0 and y > 0. Example solutions: Method 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]. 7 3 Optimization: Interior Solution 0 1 2 3 4 5 6 7 8 9 10 y P x x+P yy=I U(x,y)=Const 2 4 6 8 10 x Figure 8: Interior Solution to Consumer’s Problem From Figure 8 , the utility function reaches its maximum when the indif-ferent curve and cons traint line are tangent, namely: Px ∂u/∂x ux = MRS = = . Py ∂u/∂y uy – If Px ux > ,Py uy then one should consume more y, less x. – If Px ux < ,Py uy xthen one should consume mor e x, less y. Intuition behind PPy = MRS: Px is the market price of x in terms of y, and MRS is the pr ic e of x in Py terms of y valued by the individual. If Px/Py > MRS, x is relatively exp ensive for the individual, and hence he should consume more y. On the other hand, if Px/Py < MRS, x is relatively cheap for the individual, and hence he should consume more x. Method 2: Use L agrange Multipliers • L(x, y, λ) = u(x, y) − λ(xPx + yPy − 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]. 8 3 Optimization: Interior Solution In or der to maximize u, the following fir st order conditions must be satis-fied: ∂L ux = 0 = = λ, ∂x ⇒ Px ∂L uy ∂y = 0 = ⇒ Py = λ, ∂L ∂λ = 0 = ⇒