Week 7: The Consumer (Malinvaud, Chapter 2 and 4) / Consumer Theory (Jehle and Reny, Chapter 1) Tsun-Feng Chiang* *School of Economics, Henan University, Kaifeng, China November 1, 2015 Week 7: The Consumer (Malinvaud, Chapter 2 and 4) / Consumer November Theory 1, 2015 (Jehle and 1 / Reny, 32
1.3 The Consumer s Problem (Continued) Because the feasible set B is compact (closed and bounded), and the utilty function u(x) is a continuous real-valued function, by Theorem A1.10, there must exists a x that maximizes u(x). Moreover, because B is convex and the u(x) is assumed to be strictly quasiconcave, the x is unique. Because the preferences are strictly monotonic, the solution x will satisfy the budget constraint with equality, lying on, rather than inside, the boundary of the budget set (if a part of income y is not used, then he can improve his utility by using up to buy more commodity). A typical solution to this problem in the two-good case is illustrated in Figure 1.10. (see the next slide). Clearly, the solution vector x depends on the parameters to the consumer s problem. We can view the solution of the consumer s problem x as a function for prices and income. Therefore, we will often write xi = x i (p, y), i = 1,, n, or in vector notation, x = x(p, y). Week 7: The Consumer (Malinvaud, Chapter 2 and 4) / Consumer November Theory 1, 2015 (Jehle and 2 / Reny, 32
Figure 1.10 The solution to the consumer s utility-maximization problem. When viewed as functions of p and y, the solutions to the utility-maximization problem, x = x(p, y), are known as ordinary, or Marshallian demand functions. Week 7: The Consumer (Malinvaud, Chapter 2 and 4) / Consumer November Theory 1, 2015 (Jehle and 3 / Reny, 32
The relationship between the consumer s problem and consumer demand behavior is illustrated in Figure 1.11. (see the next slide) In Figure 1.11(a), initially the consumer is endowed with income y and the prices for x 1 and x 2 are p1 0 and p0 2, respectively. Suppose the quantities x 1 (p1 0, p0 2, y) and x 2(p1 0, p0 2, y) solve the consumer s problem. Figure 1.11(b) illustrates the price of good 1, p 1 against the quantity of good 1, x 1. x 1 (p1 0, p0 2, y) is a point in Figure 1.11(b). Let the price of good 1 change from p1 0 to p1 1, where p1 1 < p0 1, now the solution becomes x 1 (p1 1, p0 2, y) and x 2(p1 1, p0 2, y). We can get another point x 1 (p1 1, p0 2, y) in the Figure 1.11(b). By considering all possible values for p 1, we trace out the consumer s entire demand curve x 1 (p 1, p2 0, y) for good 1 shown in Figure 1.11(b) given p 2 and y unchanged. As you can easily verify, different levels of income and different prices of good 2 will cause the position and shape of the demand curve for good 1 to change. Week 7: The Consumer (Malinvaud, Chapter 2 and 4) / Consumer November Theory 1, 2015 (Jehle and 4 / Reny, 32
Figure 1.11 The consumer s problem and consumer demand function Week 7: The Consumer (Malinvaud, Chapter 2 and 4) / Consumer November Theory 1, 2015 (Jehle and 5 / Reny, 32
If it is further assumed that u(x) is differentiable, we can use calculus methods to further explore demand behavior. Recall that the consumer s problem is, max x R n + u(x), s.t. p x y As we ve noted, a solution x exists and is unique. To find x, first we form the Lagrangian, L(x, λ) = u(x) + λ[y p x] Assuming that the solution x is strictly positive, we can apply Kuhn-Tucker methods to characterize it. If x 0 solve this consumer s problem, then by Theorem A2.20, there exists a λ 0 such that (x, λ ) satisfy the following Kuhn-Tucker conditions: L = u(x ) λ p i = 0, i = 1,, n, x i x i y p x 0, λ [y p x ] = 0. Week 7: The Consumer (Malinvaud, Chapter 2 and 4) / Consumer November Theory 1, 2015 (Jehle and 6 / Reny, 32
Now, by strict monotonicity, the constraint reduce to have an equality form p x = y. Consequently, these conditions reduce to (Cons. 1) L = u(x ) λ p i = 0, i = 1,, n, x i x i y p x = 0. It is very likely u(x ) 0.Thus by strict monotonicity, u(x )/ x i > 0, for some i = 1,, n. From the first condition, we can see u(x ) x i = λ p i > 0 This says marginal utility is proportional to price for all goods at the optimum. Alternatively, for any two goods j and k, we can combine the first condition to conclude that MRS jk = u(x )/ x j u(x )/ x k = p j p k Week 7: The Consumer (Malinvaud, Chapter 2 and 4) / Consumer November Theory 1, 2015 (Jehle and 7 / Reny, 32
This says at the optimum, the MRS between any two goods must be equal to the ratio of the goods prices. In a two-good case, this requires that the slope of the indifference curve through x be equal to the slope of the budget constraint, and that x lie on, rather than inside the budget line. In general, (Cons. 1) are merely necessary conditions for a local optimum. However, for the particular problem at hand, these necessary first-order conditions are in fact sufficient for a global optimum by the following theorem. Theorem 1.4 Sufficiency of Consumer s First-Order Conditions Suppose that u(x) is continuous and quasiconcave on R n +, and that (p, y) 0. If u is differentiable at x, and (x, λ ) 0 solve the (Cons. 1), then x solves the consumer s maximization problem at prices p and income y. Proof: For all x, x 1 0, because u is quasiconcave, u(x)(x 1 x) 0 whenever u(x 1 ) u(x) and u is differentiable at x. Week 7: The Consumer (Malinvaud, Chapter 2 and 4) / Consumer November Theory 1, 2015 (Jehle and 8 / Reny, 32
Theorem 1.4 (Continued) Suppose that u(x ) exists and (x, λ ) 0 solves (Cons. 1). Then u(x ) = λ p, (P.1) p x = y. (P.2) If x can t maximize u, there must be some x 0 0 such that u(x 0 ) > u(x ), p x 0 y. Multiply x 0 by t which is between 0 and 1, but also close enough to 1, the preceding inequalities imply that Let tx 0 = x 1, we then have u(tx 0 ) > u(x ), (P.3) p tx 0 <y. (P.4) Week 7: The Consumer (Malinvaud, Chapter 2 and 4) / Consumer November Theory 1, 2015 (Jehle and 9 / Reny, 32
Theorem 1.4 (Continued) u(x )(x 1 x ) = (λ p) (x 1 x ) by (P.1) = λ (p x 1 p x ) < λ (y y) by (P.2) and (P.4) = 0 However, by (P.3), u(x 1 ) u(x ), u(x )(x 1 x ) < 0 contradicts the fact set forth at the beginning of the proof. Therefore, x is the utiiity-maximizer. With the sufficiency result in hand, it is enough to find a solution (x, λ ) 0 to (Cons. 1). Example 1.1 Given p 1, p 2 and y, find the solution for the consumer with the CES utility function u(x 1, x 2 ) = (x ρ 1 + x ρ 2 )1/ρ, where 0 ρ < 1. Week 7: The Consumer (Malinvaud, Chapter 2 and 4) / Consumer November Theory 1, 2015 (Jehle and 10 / Reny, 32
Example 1.1 (Continued) The consumer s problem is to find a nonnegative consumption bundle x 1, x 2 solving First form the Lagrangian: max x1,x 2 (x ρ 1 + x ρ 2 )1/ρ, s.t. p 1 x 1 + p 2 x 2 y L(x 1, x 2, λ) (x ρ 1 + x ρ 2 )1/ρ + λ(y p 1 x 1 p 2 x 2 ) Because preferences are monotonic, the budget constraint will hold with equality at the solution. The we have the following conditions which hold at the solution x 1, x 2 and λ: L x 1 = (x ρ 1 + x ρ 2 )(1/ρ) 1 x ρ 1 1 λp 1 = 0 Week 7: The Consumer (Malinvaud, Chapter 2 and 4) / Consumer November Theory 1, 2015 (Jehle and 11 / Reny, 32
Example 1.1 (Continued) L x 2 = (x ρ 1 + x ρ 2 )(1/ρ) 1 x ρ 1 2 λp 2 = 0 L λ = y p 1x 1 p 2 x 2 = 0 There are three euqations and three unknowns (x 1, x 2, λ ), so we can solve the equation systems. For x 1 and x 2, x 1 = p 1/(ρ 1) 1 y p ρ/(ρ 1) 1 + p ρ/(ρ 1) 2 x2 p 1/(ρ 1) 2 y = p ρ/(ρ 1) 1 + p ρ/(ρ 1) 2 x1 and x 2, the solutions to the consumer s problem, are the consumer s Marshallian demand functions. If we define the parameter r = ρ/(ρ 1), we can simply x1 and x 2 as Week 7: The Consumer (Malinvaud, Chapter 2 and 4) / Consumer November Theory 1, 2015 (Jehle and 12 / Reny, 32
Example 1.1 (Continued) x 1 (p, y) = pr 1 1 y p r 1 + pr 2 x 2 (p, y) = pr 1 2 y p r 1 + pr 2 Notice that the solutions to the consumers s problems depend only on its parameters, p 1, p 2 and y. Different prices and income will give different quantities of each good demanded. Usually we d like to be able to consider the slopes of demand curves and hence we d like the x(p, y) to be differentiable whenever we need it to be. The following theorem says under what conditions, the demand curve would be differentiable. Week 7: The Consumer (Malinvaud, Chapter 2 and 4) / Consumer November Theory 1, 2015 (Jehle and 13 / Reny, 32
Theorem 1.5 Differentiable Demand Let x 0 solve the consumer s maximization problem at prices p 0 0 and income y 0 > 0. If u is twice continuously differentiable on R n ++, u(x )/ x i > 0 for some i = 1,, n, and the bordered Hessian of u has a nonzero determinant at x, then x(p, y) is differentiable at (p 0, y 0 ). Week 7: The Consumer (Malinvaud, Chapter 2 and 4) / Consumer November Theory 1, 2015 (Jehle and 14 / Reny, 32
1.4 Indirect Utility Function The utilty function, u(x), defined over the consumption set X and represents the consumer s preferences directly, is called the direct utility function. Given prices p and income y, the consumer chooses a utility-maximizing bundle x(p, y) which give the maximum utility. Different prices or income, giving different budget constraints, will generally give rise to different choices by the consumer and so to different level of maximizing utility. The relation among prices, income, and the maximized value of utility can be summarized by a real-valued function v : R n + R + R defined as follows: v(p, y) = max x R n + u(x) s.t. p x y. The function v(p, y) is called the indirect utility function. The maximum level of utility can be achieved when facing prices p and income y therefore will be that which is realized when x(p, y) is chosen. Hence, Week 7: The Consumer (Malinvaud, Chapter 2 and 4) / Consumer November Theory 1, 2015 (Jehle and 15 / Reny, 32
v(p, y) = u(x(p, y)) (eq.1) Geometrically, we can think of v(p, y) as giving the utility level of the highest indifference curve the consumer can reach, given prices p and income y, as illustrated in Figure 1.13. Figure 1.13 Indirect utility at prices p and income y. Week 7: The Consumer (Malinvaud, Chapter 2 and 4) / Consumer November Theory 1, 2015 (Jehle and 16 / Reny, 32
There are several properties that the indirect utility function will possess. First, according to the Theorem of the Maximum, the continuity of the constraint function in prices p and income y is sufficient to guarantee that v(p, y) will be continuous in p and y. In the following theorem, we collect together a number of additional properties of v(p, y). Theorem 1.6 Properties of the Indirect Utility Function 1 Continuous on R n + R +, 2 Homogeneous of degree zero in (p, y), 3 Strictly increasing in y, 4 Decreasing in p. Moreover, it satisfies 5 Quasiconvex in (p, y). 6 Roy s identity: If v(p, y) is differentiable at (p 0, y 0 ) is differentiable and (p, y)/ y 0, then Week 7: The Consumer (Malinvaud, Chapter 2 and 4) / Consumer November Theory 1, 2015 (Jehle and 17 / Reny, 32
Theorem 1.6 (Continued) x i (p 0, y 0 ) = v(p, y)/ p i, i = 1,, n. v(p, y)/ y Proof: To prove the second property, we must show that v(p, y) = v(tp, ty) for all t > 0. By definition of the indirect utility function, v(tp, ty) = max x R n + u(x) s.t. tp x ty but the budget constraint does not change, which implies that v(tp, ty) = [max x R n + u(x) s.t. p x y] = v(p, y) Then we prove the third property. Remember that v(p, y) = max x R n + u(x) s.t. p x = y. (P.1) The Lagrangian for (P.1) is L(x, λ) = u(x) + λ(y p x) (P.2) Week 7: The Consumer (Malinvaud, Chapter 2 and 4) / Consumer November Theory 1, 2015 (Jehle and 18 / Reny, 32
Theorem 1.6 (Continued) For the solutions (x, λ ) that solves (P.1), they must satisfy the first-order condition: L(x, λ ) x i = u(x ) x i λ p i = 0, i = 1,, n. (P.3) Since u(x )/ x i > 0 and p i > 0, λ > 0. According to the Envelope theorem (Theorem A2.21), the partial derivative of the maximum value function v(p, y) with respect to y is equal to the partial derivative of the Lagrangian with respect to y evaluated at (x, λ ), v(p, y) y = L(x, λ ) y = λ > 0 (P.4) Thus, v(p, y) is strictly increasing in y. Also, by the Envelope theorem, v(p, y) = L(x, λ ) = λ xi < 0 (P.5) p i p i Week 7: The Consumer (Malinvaud, Chapter 2 and 4) / Consumer November Theory 1, 2015 (Jehle and 19 / Reny, 32
Theorem 1.6 (Continued) Thus, the fourth property is then proved. To prove the sixth property, we divide (P.5) by (P.4) and multiply the resulting equation by 1, which proves the Roy s identity. Example 1.2 v(p, y)/ p i v(p, y)/ y = x i = x i (p, y) In Example 1.1, the CES utility function u(x 1, x 2 ) = (x ρ 1 + x ρ 2 )1/ρ is a directly utility function. The corresponding demand functions are x 1 pr 1 1 y (p, y) = p1 r +, x pr 1 2 y pr 2 (p, y) = 2 p1 r +. pr 2 Week 7: The Consumer (Malinvaud, Chapter 2 and 4) / Consumer November Theory 1, 2015 (Jehle and 20 / Reny, 32
Example 1.2 (Continued) By (eq.1), we can form the indirect utility function by substituting these back into the direct utility function. v(p, y) = [x 1 (p, y) ρ + x 2 (p, y) ρ ] 1/ρ = y(p r 1 + pr 2 ) 1/r It is easy to see that v(p, y) is homogeneous of degree zero in prices and income, because for any t > 0, v(tp, ty) = ty((tp 1 ) r + (tp 2 ) r ) 1/r = tyt 1 (p r 1 + pr 2 ) 1/r = v(p, y) To see that its increasing in y and decreasing in p. differentiate the indirect utility function with respect to income and any price to obtain v(p, y) y = (p r 1+p r 2) 1/r > 0, v(p, y) p i = (p r 1+p r 2) ( 1/r) 1 yp r 1 i < 0, i = 1, 2 To verify the Roy s identity, just form the required ratio and multiply it by 1, Week 7: The Consumer (Malinvaud, Chapter 2 and 4) / Consumer November Theory 1, 2015 (Jehle and 21 / Reny, 32
Example 1.2 (Continued) [ ] v(p, y)/ pi ( 1) = ypr 1 i v(p, y)/ y p1 r + pr 2 = x i (p, y), i = 1, 2 Week 7: The Consumer (Malinvaud, Chapter 2 and 4) / Consumer November Theory 1, 2015 (Jehle and 22 / Reny, 32
The Expenditure Function To construct the indirect utility function, we fixed market prices and income, and sought the maximum level of utility the consumer could achieve. To construct the expenditure function, we again fixed the prices, but we ask a different sort of question: What is the minimum level of money expenditure the consumer must make facing a given set of prices to achieve a given level of utility? To better understand this question, consider Figure 1.15 (see the next slide), a two-good case. The consumer faces the fixed utility u, now he has to choose a level of expenditure e that exactly achieve that much level of utility u. Since the price is fixed, the slope of each expenditure curve is the same. In constructing the expenditure function, we seek the minimum expenditure the consumer requires to achieve utility u. Clearly, that will be level e, and the least cost bundle that achieves utility u at prices p will be the bundle x h = (x1 h(p, u), x 2 h (p, u)), and the level of expenditure, or e(p, u) = p 1 x1 h(p, u) + p 2x2 h(p, u) = e. Week 7: The Consumer (Malinvaud, Chapter 2 and 4) / Consumer November Theory 1, 2015 (Jehle and 23 / Reny, 32
Figure 1.15 Finding the lowest level of expenditure to achieve utility level u More generally, we define the expenditure function as minimum-value function, e(p, u) min x R n + p x s.t. u(x) u (Prob.1) for all p 0 and all attainable utility levels u. Week 7: The Consumer (Malinvaud, Chapter 2 and 4) / Consumer November Theory 1, 2015 (Jehle and 24 / Reny, 32
Because p and x are positive, the set of numbers {e e = p x for some x with u(x) u} is bounded below by zero. Moreover, because p 0, the set is closed. Hence it contains a smallest number. The value e(p, u) is precisely this smallest number. We can denote the solution if the minimization problem as the function x h (p, u) 0. Notice also that if u(x) is continuous and strictly quasiconcave, the solution will be unique. As we ve seen, if x h (p, u) solves this problem, the lowest expenditure necessary to achieve utility u at prices p will be exactly equal to the cost of the bundle x h (p, u), or e(p, u) = p x h (p, u). x h (p, u) is also a demand function, but unobservable. Imagine a situation when the prices change. It could not keep the original consumption bundle given the original amount of income, so the consumer s utility level changes. To have the original, fixed level of utility, there should be income compensated to him to rebuild a new consumption bundle for this purpose. Therefore, x h (p, u) is called the compensated demand function which describes how quantity demands change with prices by keeping utility fixed. x h (p, u) is also Week 7: The Consumer (Malinvaud, Chapter 2 and 4) / Consumer November Theory 1, 2015 (Jehle and 25 / Reny, 32
called Hicksian demand functions, named after John Hicks. Figure 1.16 (see the next slide). shows that when the price of good 1 changes, how do we sketch the Hicksian demand function. Week 7: The Consumer (Malinvaud, Chapter 2 and 4) / Consumer November Theory 1, 2015 (Jehle and 26 / Reny, 32
Figure 1.16 The Hicksian demand for good 1 Week 7: The Consumer (Malinvaud, Chapter 2 and 4) / Consumer November Theory 1, 2015 (Jehle and 27 / Reny, 32
The following theorem describes the important properties of the expenditure function. Theorem 1.7 Properties of the Expenditure Function If u( ) is continuous and strictly increasing, then e(p, u) defined in (Prob.1) is 1 Zero when u takes on the lowest level of utility. 2 Continuous on its domain. 3 For all p 0, strictly increasing and unbounded above in u. 4 Increasing in p. 5 Homogeneous of degree 1 in p. 6 Concave in p. If, in addition, u( ) is strictly quasiconcave, we have, 7 Shephard s lemma: e(p, u) is differentiable in p at (p 0, u 0 ) with p 0 0, and Week 7: The Consumer (Malinvaud, Chapter 2 and 4) / Consumer November Theory 1, 2015 (Jehle and 28 / Reny, 32
Theorem 1.7 (Continued) e(p 0, u 0 ) p i = x h i (p 0, u 0 ), i = 1,, n. Proof: To prove property 1, note that the lowest value of u is u(0). Consequently, e(p, u(0)) = 0 because x = 0 attains utility u(0) and requires an expenditure of p 0 = 0. Property 2 follows from the Theorem of Maximum. To prove property 3, we start from the consumer s minimization problem, e(p, u) min x R n + p x s.t. u(x)=u (P.1) Note the constraint is an equality. If the utility is not binding, there is always a cheaper consumption bundle achieving the same level of utility. Therefore, we can set up the Lagrangian for the problem Week 7: The Consumer (Malinvaud, Chapter 2 and 4) / Consumer November Theory 1, 2015 (Jehle and 29 / Reny, 32
Theorem 1.7 (Continued) L(x, λ) = p x + λ[u u(x)]. (P.2) Suppose we have x = x h (p, u) 0 and λ be the solutions, then the following conditions must hold, L(x, λ ) x i = p i λ u(x ) x i = 0, i = 1,, n. (P.3) Note that because p i and u(x )/x i are positive, so, too is λ. By the Envelop theorem, the partial derivative of the minimum-value function e(p, u) with respect to u is equal to the partial derivative of the Lagrangian with respect to u, evaluated at (x, λ ). Hence, e(p, u) u = L(x, λ ) u = λ > 0 (strictly increasing in u) Skip the proof of unbounded above in u, but imagine utility can be unbounded so does the expenditure achieving that unlimited level of utility. Week 7: The Consumer (Malinvaud, Chapter 2 and 4) / Consumer November Theory 1, 2015 (Jehle and 30 / Reny, 32
Theorem 1.7 (Continued) To prove property 4, we again appeal to the Envelope theorem but now differentiate with respect to p i, e(p, u) = L(x, λ ) = xi xi h (p, u) 0 p i p i This also proves property 7. For property 5, multiply p by t > 0, thus e(tp, u) = min x R n + (tp) x = t min x R n + p x = t e(p, u) We skip the proof of property 6 by just give an intuition. When the price of x i, say goes up from pi 0 to pi 1, the consumer would not buy the same consumption bundle where x i = xi 0 to keep the utility. Instead he will buy less x i and more other goods relatively cheaper. In that case, the expenditure need not to increase by xi 0 (pi 1 pi 0 ), the maximum income compensation, but some smaller amount. Week 7: The Consumer (Malinvaud, Chapter 2 and 4) / Consumer November Theory 1, 2015 (Jehle and 31 / Reny, 32
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