Chapter 3. Constructing a Model of consumer behavior Part C, Copyright Kwan Choi, The consumer s utility U = XY, his income M = 100, and p

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1 Chater 3. Constructing a Model of consumer behavior Part C, Coright Kwan Choi, 009 Practice Problems The consumer s utilit =, his income M = 00, and =, and = 5. The consumer s roblem is to choose and to maimize =. () subject to the budget line, 00 = + 5. () Marginal utilities: M =, M =. Marginal Rate of Substitution: M MRS =. M = (3) Equilibrium condition: Equate MRS to rice ratio. M = =. (4) M Thus, =. (5)

2 An otimal consumtion bundle is obtained b substituting the equilibrium condition in (5) into the budget line: M M =, =. (6) The solution in (6) defines the demand functions for the two goods. Thus, 00 = = 5, 00 = = 0, and = = 5 0 = What is the rice elasticit of demand for? One. Because the eenditure on is M =. That is, eenditure on remains fied as rice changes. Constrained Otimization Problem An otimal solution can also be obtained using the Lagrangian method. First, state the consumer s roblem as a constrained otimization roblem. The consumer s roblem is to choose and to Maimize (, ) Subject to: M = +. The Lagrangian function is written as L = (, ) + λ( M ), (7) where λ is a Lagrange multilier. The first order conditions are derived b differentiating the Lagrangian function in (7).

3 L = λ = 0, (8) L = λ = 0, (9) L = M = λ 0, (0) where i = / iis marginal utilit of good i. Note that (0) reroduces the budget constraint. From (8) and (9), we get the equilibrium condition: λ =, = () which indicates that marginal utilit is roortional to rice. Water and Diamonds Parado Equation () elains the arado of water and diamonds, which states that consumers get much less utilit from diamonds than from water, and et as a higher rice for diamonds than water. Marginal utilit, not total utilit, is roortional to rice! Assuming additive utilit function (i.e., utilit of each good is indeendent of the other good), total utilit of diamonds is the area below the M of diamonds and that of water is the area below the M of water. Total utilit is not roortional to rice. 3

4 Equation () can also be written as =. () That is, marginal rate of substitution is equal to the rice ratio. This equilibrium conditions, together with the budget constraint in () ields the solutions, which are demand functions. The can be written as = (,, M), and = (,, M). Substituting the demand functions into the utilit function, we get [ ] [,, M] = (,, M), (,, M), (3) which is called an indirect utilit function, whereas is called a direct utilit function. Differentiating (3) with resect to mone income M, we get 4

5 = + M M M. (4) sing (8) and (9), marginal utilit of income rewritten as: = λ + M M M. (5) Differentiating the budget constraint in (0), wet get = +. M M (6) Thus, (5) can be written as M = λ. (7) That is, the Lagrange multilier is marginal utilit of income. In general, it is the derivative of the objective function with resect to the constant in the constraint. Proortionate Increases in Income and Prices Demonstrate that demand functions are homogeneous of degree zero in all rices and income. That is to sa that if all rices and income double, the otimal consumtion bundle remains the same. Suose mone income and rices all rise roortionatel due to inflation. (,, M ) ( k, k, km ). Then the budget constraint can be written as km k k = 0. Note that since each term contains k, it can be factored out. Thus, the budget constraint can be written as: km ( ) = 0. (8) Note that since k > 0, the new budget constraint holds, iff (if and onl if) M = 0, 5

6 which is the old budget constraint. In other words, if all rices and mone income rise roortionatel, it does not shift the budget line. Accordingl, consumers will derive the same otimal solution. In other words, consumer demands for goods are unaffected if all rices and income rise roortionatel. Monotonic Transformations and Ordinal tilit Consider another utilit function: / = /. (9) Marginal utilities are: M =, M =, Thus, marginal rate of substitution is: MRS =. (0) Comaring (3) and (0), we see that MRS is indeendent of a monotonic transformation. That is, a monotonicall increasing transformation of does not change MRS. Definition: is a monotonic transformation of if (, ) = f[ (, )], f ' 0. () In the above eamle, = /. 6

7 Since MRS s of the two utilit functions are the same and demand functions are derived from the condition that MRS = rice ratio, it follows that the two utilit functions will have the same demand functions. In general, differentiating () with resect to and, we get = f ', = f '. () Thus, MRS = f ' MRS. = f ' = (3) Numerical Eamle Solve the following maimization roblem. Choose and to /3 Maimize = /3, Subject to: M = 0. 3 Note that = = is a monotone increasing transformation of, and hence will ield identical demand functions. One can solve the above utilit maimization roblem directl, or maimize. Both utilit functions and will ield the same demand functions. The Lagrangian function associated with is written as: L = + [ ]. (4) λ M The first order conditions (FOCs) are: λ = 0, (5) λ = 0, (6) 7

8 M = 0. (7) From the FOCs in (5) and (6), we get λ =. = (8) It follows that =, or =. (9) Substituting (9) into the budget constraint in (7), we get M = 0, or M =, (30) 3 From M = 0, we get M =. (3) 3 The otimal utilit is now given b 3 M M 4 M = = (3) Marginal utilit of income can be obtained b differentiating the Lagrangian function in (3) with resect to M, L 4 M = λ = = > 0. M M 9 (33) Intuitivel, the Lagrange multilier is marginal utilit of income. It deends on which utilit function is chosen. If the original utilit function were used, otimal utilit is 8

9 /3 /3 /3 /3 M M = = M. (34) And the marginal utilit of income or the Lagrange multilier associated with is /3 u λ = = M 3 3 /3, (35) which is constant. That is, with, marginal utilit of income does not diminish as income increases. 9