CONSUMER CHOICE 1. THE CONSUMER CHOICE PROBLEM 1.1. Unit of analysis and preferences. The fundamental unit of analysis in economics is the economic agent. Typically this agent is an individual consumer or a firm. The agent might also be the manager of a public utility, the stockholders of a corporation, a government policymaker and so on. The underlying assumption in economic analysis is that all economic agents possess a preference ordering which allows them to rank alternative states of the world. The behavioral assumption in economics is that all agents make choices consistent with these underlying preferences. 1.2. Definition of a competitive agent. A buyer or seller (agent) is said to be competitive if the agent assumes or believes that the market price of a product is given and that the agent’s actions do not influence the market price or opportunities for exchange. 1.3. Commodities. Commodities are the objects of choice available to an individual in the economic sys- tem. Assume that these are the various products and services available for purchase in the market. Assume that the number of products is finite and equal to L (` =1, ..., L). A product vector is a list of the amounts of the various products: x1 x2 x = . . xL The product bundle x can be viewed as a point in RL. 1.4. Consumption sets. The consumption set is a subset of the product space RL, denoted by XL ⊂ RL, whose elements are the consumption bundles that the individual can conceivably consume given the phys- L ical constraints imposed by the environment. We typically assume that the consumption set is X = R+ = {x L ∈ R :x` ≥ 0 for ` = 1, ... , L}. 1.5. Prices. We will assume that all L products are traded in the market at dollar prices that are publicly quoted. How they are determined will be discussed later. The prices are represented by a price vector p1 p2 L p = . ∈ R . pL For now assume that all prices are strictly positive, i.e. p` >> 0. We will also assume that all consumers are price takers in the sense that they cannot influence the price at which they buy or sell a product. Date: December 16, 2004. 1 2 CONSUMER CHOICE 1.6. Income or wealth. Assume that each consumer has wealth equal to mi or the representative consumer has wealth m. 1.7. Affordable consumption bundles. We say that a consumption bundle x is affordable for the represen- tative consumer if px = p1 x1 + p2 x2 + ··· + pLxL ≤ m (1) L L If x is also an element of R+ , then the set of feasible consumption bundles is x R+ :px≤ m. This is called a Walrasian budget set and is denoted Bp,m. 1.8. Preferences. We assume a preference relation over products with the following properties 1: complete in that for all x1,x2 ∈ X, we have x1 x2 or x2 x1 (or both) 2: transitive in that ∀ x1,x2,x3 X, if x1 x2 and x2 x3 then x1 x3. 3: locally nonsatiated in that for every x1 ∈ X and every ε>0, there is x2 ∈ X such that ||x2 -x1 ≤ ε|| and x2 x1. 4: continuous in that for any sequence of pairs n n ∞ n n { (x1 ,x2 ) }n=1 with x1 x2 ∀ n, n n x1 = lim x1 , and x2 = lim x2 , n →∞ n →∞ we have x1 x2. 1.9. Existence of a utility function. Based on the preferences defined in 1.8, there exists a continuous utility function v(x) that represents in the sense that x1 x2 iff v(x1) ≥ v(x2). 1.10. Convexity. We often assume that preferences are convex in the sense that if x1 x2 , then for 0 ≤ λ ≤ 1, λx1 + (1-λ)x2 x1. This implies that indifference curves are convex to the origin. If the utility function is quasi-concave, then the indifference curves will be convex and vice versa. 2. THE UTILITY MAXIMIZATION PROBLEM 2.1. Formal Problem. The utility maximization problem for the consumer is then as follows max u = v(x) x ≥ 0 (2) s.t. px ≤ m L where we assume that p >> 0, m > 0andX=R+ . This is called the primal preference problem. If we have smooth convex indifference curves and an inte- rior solution, then the problem can be solved using standard Lagrangian techniques. Alternatively, Kuhn- Tucker methods can be used. The Lagrangian function is given by n L = v(x) − λ (Σi=1 pi xi − m ) (3) The first order conditions are ∂v − λpi =0,i=1, 2, ...,n ∂xi (4) n − Σi=1 pi xi + m =0 The value of λ is the amount by which L would increase given a unit relaxation in the constraint (an increase in income). It can be interpreted as the marginal utility of expenditure. This units of this are of CONSUMER CHOICE 3 course arbitrary. The solution to 2 is given by x(p,m) = g(p,m). These functions are called Marshallian demand equations. Note that they depend on the prices of all good and income. Based on the structure of preferences and the nature of the optimization problem, they will have certain properties which we will discuss shortly. 2.2. Cobb-Douglas Example. Consider a utility function given by α1 α2 u = v(x)=x1 x2 (5) We usually assume that αi > 0. To maximize utility subject to a budget constraint we obtain we set up a Lagrangian function. α1 α2 L = x1 x2 − λ [ p1 x1 + p2 x2 − m ] (6) Differentiating equation 6 we obtain α1 α2 ∂ L α1 x1 x2 = − λp1 =0 (7a) ∂x1 x1 α1 α2 ∂ L α2 x1 x2 = − λp2 =0 (7b) ∂x2 x2 ∂ L = − [ p x + p x ]+m =0 (7c) ∂λ 1 1 2 2 Take the ratio of the 7a and 7b to obtain α xα1 xα2 1 1 2 p x1 = 1 α xα1 xα2 2 1 2 p2 x2 (8) α x p ⇒ 1 2 = 1 α2 x1 p2 We can now solve the equation for the quantity of good 2 as a function of the quantity of good 1 and the prices of both goods. Doing so we obtain α2 x1 p1 x2 = (9) α1 p2 Now substitute 9 in 7c to obtain ∂ L = − [ p x + p x ]+m =0 ∂λ 1 1 2 2 α2 x1 p1 ⇒ p1 x1 + p2 = m α1 p2 p α x α x p ⇒ 1 1 1 + 2 1 1 = m (10) α1 α1 p1 x1 ⇒ (α1 + α2 )=m α1 α1 m ⇒ x1 = (α1 + α2 ) p1 Similarly for x2 so that we have 4 CONSUMER CHOICE α2 m x2 = (11) (α1 + α2 ) p2 Note that demand for the kth good only depends on the kth price and is homogeneous of degree zero in prices and income. Also note that it is linear in income. This implies that the expenditure elasticity is equal to 1. This can be seen as follows. α1 m x1 = (α1 + α2 ) p1 (12) ∂x m α 1 " m # ⇒ 1 = 1 =1 α1 m ∂m x1 (α1 + α2 ) p1 (α1 + α2 ) p1 We can find the value of the optimal u by substitution α1 α2 u = x1 x2 α1 α2 m α1 m α2 = (13) p1 α1 + α2 p2 α1 + α2 α1+α2 −α1 −α2 α1 α2 −α1−α2 = m p1 p2 α1 α2 (α2 + α2) This can also be written α1 α2 u = x1 x2 m α α1 m α α2 = 1 2 p1 α1 + α2 p2 α1 + α2 (14) α α1 α α2 m α1 m α2 = 1 2 α1 + α2 α1 + α2 p1 p2 For future reference note that the derivative of the optimal u with respect to m is given by α1+α2 −α1 −α2 α1 α2 −α1−α2 u = m p1 p2 α1 α2 (α2 + α2) ∂u =(α + α )mα1 +α2−1p−α1 p−α2 αα1 αα2 (α + α )−α1−α2 (15) ∂m 1 2 1 2 1 1 2 2 α1+α2 −1 −α1 −α2 α1 α2 1−α1−α2 = m p1 p2 α1 α1 (α2 + α2) We obtain λ by substituting in either the first or second equation as follows α1−1 α2 α1x1 x2 − λp1 =0 α xα1−1xα2 ⇒ λ = 1 1 2 p 1 (16) α1 α2−1 α2x1 x2 − λp2 =0 α xα1 xα2−1 ⇒ λ = 2 1 2 p2 CONSUMER CHOICE 5 If we now substitute for x1 and x2, we obtain α xα1−1xα2 λ = 1 1 2 p1 m α1 x1 = p1 α1 + α2 m α2 x2 = p2 α1 + α2 (17) m α α1−1 m α α2 α 1 2 1 p α + α p α + α ⇒ λ = 1 1 2 2 1 2 p1 − − − 1−α −α α mα1 +α2−1p1 α1 p α2 αα1 1αα2 (α + α ) 1 2 = 1 1 2 1 2 1 2 p1 α1+α2 −1 −α1 −α2 α1 α2 1−α1−α2 = m p1 p2 α1 α2 (α1 + α2) Thus λ is equal to the derivative of the optimal u with respect to m. To check for a maximum or minimum we set up the bordered Hessian. The bordered Hessian in this case is 2 ∗ ∗ 2 ∗ ∗ ∗ ∂ L(x ,λ ) ∂ L(x ,λ ) ∂g(x ) ∂x ∂x ∂x ∂x ∂x 1 1 1 2 1 2 ∗ ∗ 2 ∗ ∗ ∗ ∂ L(x ,λ ) ∂ L(x ,λ ) ∂g(x ) HB = (18) ∂x2∂x1 ∂x2∂x2 ∂x2 ∂g(x∗) ∂g(x∗) 0 ∂x1 ∂x2 We compute the various elements of the bordered Hessian as follows 6 CONSUMER CHOICE α1 α2 L = x1 x2 − λ[p1x1 + p2x2 − m] L ∂ α1−1 α2 = α1x1 x2 − λp1 ∂x1 L ∂ α1 α2−1 = α2x1 x2 − λp2 ∂x2 ∂2L − α1−2 α2 2 =(α1)(α1 1)x1 x2 ∂x1 2L ∂ α1−1 α2−1 = α1α2x1 x2 ∂x1∂x2 ∂2L − α1 α2−2 2 =(α2)(α2 1)x1 x2 ∂x2 ∂g = p1 ∂x1 ∂g = p2 ∂x2 The derivatives of the constraints are constants.