Andrew McLennan October 28, 2014 Economics 7250 Advanced Mathematical Techniques for Economics Second Semester 2014 Lecture 15 The Consumer, the Firm, and an Economy I. Introduction A. The material discussed here emphasizes how certain basic concepts are correctly understood as derivatives or partial derivatives. B. The focus is on the central behavioral models in economics. 1. The consumer s choice of a consumption bundle from the set of bundles satisfying a budget constraint. 2. A single-product firm s choice of an input bundle, given input wages and the price of output. C. In addition we consider Edgeworth box models of exchange (no production), considering notions of efficiency, perfectly competitive equilibrium, and fairness. II. The Theory of the Consumer A. We consider the problem of utility maximization in IR n + = {x IRn : x 1 0,...,x n 0}. 1. Let U : IR + n IR be the consumer s utility function. a. We will typically not fuss with assumptions about which partial derivatives exist, adopting the attitude that whenever we refer to a derivative, it has implicitly been assumed to exist. 1
b. The main example we consider is the Cobb-Douglas utility function U(x) = x 1 n 1... x 1 n n. 2. The budget constraint is p 1 x 1 +...+p n x n M, where p 1,...,p n > 0 are the prices of the goods and M > 0 is the consumer s total income. B. We express the economically relevant concepts in terms of partial derivatives. 1. At a particular bundle x, the marginal utility of good i is the partial derivative of U with respect to x i. We denote this by MU i (x), U i (x), or U x i (x). a. for the Cobb-Douglas example we have U (x) = 1 x i n x 1 n 1... x 1 n i 1 x 1 n 1 i x 1 n i+1... x 1 n. 2. Both in a commonsensical emotional sense, and also in the sense that people are typically risk averse, it is natural to suppose that the marginal utility of any good declines as the consumption of that good increases, holding the consumption of other goods fixed. This property is known as diminishing marginal utility. a. This can be exressed by the inequality 2 U x i 2(x) < 0. b. For the Cobb-Douglas utility function we have 2 U 2 x (x) = 1 i n (1 n 1) x 1 n 1... x 1 n i 1 x 1 n 2 i x 1 n i+1... x 1 n < 0. c. It is typically more readable to write U ij (x) in place of 2 U x i x j (x). d. If i j, and U ij (x) > 0, then we say that good i and good j are complements, meaning that an increase in the quantity of good i increases the marginal utility of good j, and they are substitutes if U ij (x) < 0. i. It is important to recognize that this notion refers to a specific utility function at a specific point. 2
ii. On the other hand Young s theorem (if all second partials existandarecontinuous, thenu ij = U ji for alli,j = 1,...,n) implies that it makes no difference whether we think of good i as a complement for good j or vice versa. 3. The consumer s willingness to accept decreased consumption of good j in return for increased consumption of good i, at a particular point x, is captured by the marginal rate of substitution between these two goods at x. a. Formally we write MRS ij (x) = dx j dx i U(x )=U(x), where dx j dx i U(x )=U(x) is the slope of the curve through x given by the conditions x k = x k, k i,j, and U(x ) = U(x). b. We project this curve onto the (x i,x j )-plane, regard it as the graph of a relation expressing x j as a function of x i, and apply Taylor s theorem, obtaining MRS ij (x) = U i(x) U j (x). c. For the Cobb-Douglas case we have MRS ij (x) = x j x i. 4. We now suppose that x maximizes utility subject to the budget constraint, and derive calculus-based necessary conditions. a. Consider the curve through x given by the conditions x k = x k, k i,j, and p 1 x 1 +...+p n x n = p 1 x 1 +...+p nx n. b. If we project this curve onto the (x i,x j )-plane, regard it as the graph of a relation expressing x j as a function of x i, and apply Taylor s theorem to the condition that utility cannot be increasing in either direction along this curve, we obtain MRS ij (x) = U i(x) U j (x) = p i p j, or U i(x) p i = U j(x) p j. c. In the Cobb-Douglas case this gives x j x i = p i p j or p 1 x 1 =... = p n x n. 3
The only point satisfying this condition and the budget constraint is x = ( M np 1,..., M np n ). 5. In the two good case we can write the indifference surface U(x 1,x 2 ) = Ū as the graph of a function x 2 = g(x 1 ). Differentiating the identity U(x 1,g(x 1 )) = Ū twice and isolating the second derivative of g, we find that g is convex at x 1 if, at x = (x 1,g(x 1 )), we have U 2 2U 11 2U 1 U 2 U 12 +U 2 1U 22 < 0. III. The Theory of the Firm A. We consider the problem of choosing a profit maximizing bundle of inputs in IR n + = {x IRn : x 1 0,...,x n 0}. 1. Let f : IR n + IR be the firm s production function. a. As above, the main example we consider is the Cobb-Douglas production function f(x) = x 1 n 1... x 1 n n. a. Note that f exhibits constant returns to scale: for any input bundle x and any α > 0, f(αx) = αf(x). 2. The profit function is pf(x) (w 1 x 1 +...+w n x n ), where p is the price of output and w 1,...,w n > 0 are the wages of the factors of production. B. We express the economically relevant concepts in terms of partial derivatives. 1. At a particular input bundle x, the marginal product of input i is the partial derivative of f with respect to x i. We denote this by MP i (x), f i (x), or f x i (x). a. For the Cobb-Douglas example we have f (x) = 1 x i n x 1 n 1... x 1 n i 1 x 1 n 1 i x 1 n i+1... x 1 n. 4
2. It is typically the case in most production processes that the marginal product of each factor declines as the amount of that input increases, holding the amounts of other inputs fixed. This property is know as diminishing marginal productivity or diminishing returns. a. This can be exressed by the inequality 2 f x i 2(x) < 0. b. For the Cobb-Douglas utility function we have 2 f 2 x (x) = 1 i n (1 n 1) x 1 n 1... x 1 n i 1 x 1 n 2 i x 1 n i+1... x 1 n < 0. c. If i j, and f ij (x) > 0, then we say that input i and input j are complements, meaning that an increase in the quantity of input i increases the marginal productivity of input j, and they are substitutes if f ij (x) < 0. 3. The production technology s ability to compensate for a decrease in the amount of input i with an increase in the amount of factor j, at a particular point x, is captured by the marginal rate of technical substitution between these two inputs at x. a. Formally we write MRTS ij (x) = dx j dx i f(x )=f(x), where dx j dx i f(x )=f(x) is the slope of the curve through x given by the conditions x k = x k, k i,j, and f(x ) = f(x). b. We project this curve onto the (x i,x j )-plane, regard it as the graph of a relation expressing x j as a function of x i, and apply Taylor s theorem, obtaining MRTS ij (x) = f i(x) f j (x). c. For the Cobb-Douglas case we have MRTS ij (x) = x j x i. 4. We now suppose that x maximizes profits, and derive calculus-based necessary conditions. 5
a. Consider the curve through x given by the conditions x k = x k, k i,j, and w 1 x 1 +...+w n x n = w 1 x 1 +...+w nx n. b. If we project this curve onto the (x i,x j )-plane, regard it as the graph of a relation expressing x j as a function of x i, and apply Taylor s theorem to the condition that utility cannot be increasing ineither directionalongthiscurve, weobtainmrts ij (x) = f i(x) f j (x) = w i w j, or f i(x) w i = f j(x) w j. c. IntheCobb-Douglascasethisgives x j x i = w i w j orw 1 x 1 =... = w n x n. Insofar as the technology exhibits constant returns to scale, there are three possibilities for profit maximization: i. Average cost is above price, so that profits are maximized by producing nothing. ii. Average cost is below price, so that profits can be made arbitrarily large by increasing the scale. iii. Average cost equals price, in which case any cost-minimizing input bundle achieves zero profits. 5. Inthetwoinputcasewecanwritetheisoquantf(x 1,x 2 ) = f asthegraph of a function x 2 = k(x 1 ). Differentiating the identity f(x 1,k(x 1 )) = ȳ twice and isolating the second derivative of k, we find that k is convex at x 1 if, at x = (x 1,k(x 1 )), we have f 2 2 f 11 2f 1 f 2 f 12 +f 2 1 f 22 < 0. IV. A Pure Exchange Economy A. Many of the central concepts of economic theory are present in the model of two-consumer, two-good pure exchange (no production) known as the Edgeworth box. 1. Let the two consumers be A and B. 6
2. Let the total amounts of the two goods be x and ȳ. 3. An allocation is a pair ((x A,y A ),(x B,y B )) (IR 2 + )2. B. An allocation ((x A,y A ),(x B,y B )) (IR 2 + )2 is feasible if x A + x B = x and y A +y B = ȳ. 1. The set of feasible allocations can be represented as a box in the (x,y)- plane, withthelowerlefthandcornerbeingthepoint((0,0),( x,ȳ)), while the upper right hand corner is the point (( x,ȳ),(0,0)). C. Let the utility functions of the two consumers be u A,u B : IR 2 + IR. 1. An allocation ((x A,y A ),(x B,y B )) is Pareto efficient if it is feasible and there is no other feasible allocation ((x A,y A ),(x B,y B )) that makes one of the consumers better off (u A (x A,y A ) > u A(x A,y A ) or u B (x B,y B ) > u B (x B,y B )) without making either worse off (u A (x A,y A ) u A(x A,y A ) and u B (x B,y B ) u B(x B,y B )). 2. Typically the set of efficient allocations is a curve in the Edgeworth box running from the lower left hand corner to the upper right hand corner. D. We now introduce the concept of private ownership, assuming that the total allocation is originally the sum of the individual endowment bundles (ξ A,ω A ) and (ξ B,ω B ). 1. A Walrasian equilibrium is a price vector p = (p x,p y ), together with an allocation ((x A,y A ),(x B,y B )), satisfying the following conditions: a. (Optimization) (x A,y A ) solves the problem max u A (x,y) subject to p x x + p y y p x ξ A + p y ω A, and (x B,y B ) solves the problem max u B (x,y) subject to p x x+p y y p x ξ B +p y ω B. b. (Market Clearing) ((x A,y A ),(x B,y B )) is feasible. 2. One of the most important (and deepest) theorems of economic theory asserts that, under reasonably general assumptions, for any vector of endowments there is a Walrasian equilibrium. 7
3. Disregarding the endowments in order to ask what allocations could result from Walrasian equilibrium relative to some endowment, we have the notion of a Walrasian allocation, which we define to be a feasible allocation ((x A,y A ),(x B,y B )) such that, for some price vector (p x,p y ), there is no bundle (x,y) with u A (x,y) > u A (x A,y A ) and p x x+p y y p x x A + p y y A, and there is also no bundle (x,y) with u B (x,y) > u B (x B,y B ) and p x x+p y y p x x B +p y y B. First Fundamental Theorem of Welfare Economics: If ((x A,y A ),(x B,y B )) is a Walrasianallocation, say with respect to the price vector (p x,p y ), then ((x A,y A ),(x B,y B )) is Pareto efficient. Proof: Supposing, in order to achieve a contradiction, that the claim is false, let ((x A,y A ),(x B,y B )) be a feasible allocation that makes one of the consumers better off, say u A (x A,y A ) > u A (x A,y A ), without making the other worse off, so that u B (x B,y B ) u B(x B,y B ). Since ((x A,y A ),(x B,y B )) is Walrasian, we must have p x x A +p y y A > p x x A +p y y A and p x x B +p y y B p x x B +p y y B. Since both ((x A,y A ),(x B,y B )) and ((x A,y A ),(x B,y B )) are feasible, we have the following contradictory computation: p x x+p y ȳ = p x (x A +x B)+p y (y A +y B) = (p x x A +p y y A)+(p x x B +p y y B) > (p x x A +p y y A )+(p x x B +p y y B ) = p x (x A +x B )+p y (y A +y B ) = p x x+p y ȳ. 8
E. Primitive notions of equity can be expressed in this framework. 1. An allocation ((x A,y A ),(x B,y B )) is fair (sometimes the term superfair is used) if neither agent prefers the other agent s bundle: u A (x A,y A ) u A (x B,y B ) and u B (x B,y B ) u B (x A,y A ). 2. There exist allocations that are both fair and efficient. One way to obtain such an allocation is to divide the endowment vector equally between the two agents, then let them trade to a Walrasian equilibrium. (Here we are using the theorem asserting that such an equilibrium must exist.) As we have just demonstrated, the resulting allocation is necessarily efficient, and since both agents are maximizing subject to the same constraint, neither can envy the choice of the other, since he or she could also have chosen it. 9