# Deriving MRS from Utility Function, Budget Constraints, and Interior Solution of Optimization

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1 Utilit Function, Deriving MRS. Principles of Microeconomics, Fall Chia-Hui Chen September, Lecture Deriving MRS from Utilit Function, Budget Constraints, and Interior Solution of Optimization Outline. Chap : Utilit Function, Deriving MRS. Chap : Budget Constraint. Chap : Optimization: Interior Solution Utilit Function, Deriving MRS Eamples of utilit: Eample (Perfect substitutes). U(, ) = a + b. Eample (Perfect complements). U(, ) = min{a, b}. Eample (Cobb-Douglas Function). c U(, ) = A b. Eample (One good is bad). U(, ) = a + b. An important thing is to derive MRS. d MRS = = Slope of Indifference Curve. d Cite as: Chia-Hui Chen, course materials for. Principles of Microeconomics, Fall. MIT

2 Utilit Function, Deriving MRS U(,)=a+b=Const Figure : Utilit Function of Perfect Substitutes U(,)=min{a,b}=Const Figure : Utilit Function of Perfect Complements Cite as: Chia-Hui Chen, course materials for. Principles of Microeconomics, Fall. MIT

3 Utilit Function, Deriving MRS U(,)=A a b =Const Figure : Cobb-Douglas Utilit Function U(,)= a+b=const Figure : Utilit Function of the Situation That One Good Is Bad Cite as: Chia-Hui Chen, course materials for. Principles of Microeconomics, Fall. MIT

4 Budget Constraint Because utilit is constant along the indifference curve, u = (, ()) = C, = d + =, = d d =. d Thus, MRS =. Eample (Sample utilit function). Two was to derive MRS: Along the indifference curve Thus, Using the conclusion above Budget Constraint u(, ) =. = C. c =. d c MRSd = = =. d / MRS = = =. The problem is about how much goods a person can bu with limited income. Assume: no saving, with income I, onl spend mone on goods and with the price P and P. Thus the budget constraint is Suppose P =, P =, I =, then The slope of budget line is Bundles below the line are affordable. Budget line can shift: P + P I. +. d P =. d P Cite as: Chia-Hui Chen, course materials for. Principles of Microeconomics, Fall. MIT

5 Budget Constraint + Figure : Budget Constraint + + Figure : Budget Line Shifts Because of Change in Income Cite as: Chia-Hui Chen, course materials for. Principles of Microeconomics, Fall. MIT

6 Optimization: Interior Solution + + Figure : Budget Line Rotates Because of Change in Price Change in Income Assume I =, then + =. The budget line shifts right which means more income makes the affordable region larger. Change in Price Assume P =, then + =. The budget line changes which means lower price makes the affordable region larger. Optimization: Interior Solution Now the consumer s problem is: how to be as happ as possible with limited income. We can simplif the problem into language of mathematics: P + P I ma U(, )subject to., Since the preference has non-satiation propert, onl (, ) on the budget line can be the solution. Therefore, we can simplif the inequalit to an equalit: P + P = I. First, consider the case where the solution is interior, that is, > and >. Eample solutions: Method Cite as: Chia-Hui Chen, course materials for. Principles of Microeconomics, Fall. MIT

7 Optimization: Interior Solution U(,)=Const P +P =I Figure : Interior Solution to Consumer s Problem From Figure, the utilit function reaches its maimum when the indifferent curve and constraint line are tangent, namel: If If P / u = MRS = =. P / u P P u u >, then one should consume more, less. P P u u <, then one should consume more, less. Intuition behind P = MRS: P P is the market price of in terms of, and MRS is the price of in terms of valued b the individual. If P /P > MRS, is relativel epensive for the individual, and hence he should consume more. On the other hand, if P /P < MRS, is relativel cheap for the individual, and hence he should consume more. Method : Use Lagrange Multipliers L(,, λ) = u(, ) λ(p + P I). P Cite as: Chia-Hui Chen, course materials for. Principles of Microeconomics, Fall. MIT

8 Optimization: Interior Solution In order to maimize u, the following first order conditions must be satisfied: Thus we have Method Since P + P + I =, L u = = = λ, P L u = = = λ, P L = = P + P I =. λ P P Then the problem can be written as u u =. I P =. P I P ma u(, ) = u(, )., At the maimum, the following first order condition must be satisfied: P P u + u ( ) = u + u ( ) =. P = P P u u =. Cite as: Chia-Hui Chen, course materials for. Principles of Microeconomics, Fall. MIT

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