MICROECONOMICS: COMMODITY TRADING – THE EDGEWORTH BOX MULTIMARKET THEORY: 2-PERSON, 2-GOODS TRADES Example Two traders, Anne and Ben, deal in two commodities, say Apples and Bananas. Anne has apples and bananas; we say she holds the commodity bundle (, ). Likewise Ben has apples and bananas. They get together to trade by exchange or barter in these commodities. At this stage no money is involved. To solve this ‘game’ we first need to consider the traders preferences for different possible commodity bundles. Indifference curves and Utility An indifference curve for Anne joins (, ) points in the − plane which Anne regards as equally preferable. Each indifference curve partitions the plane into two regions, the one further from the origin being preferred to the one closer. (Anne would like more of (, ), rather than less!) Indifference curves do not cross, and they can only touch at infinity. One common example of a class of indifference curves is ( − )( − ) = 2, for > 0. Increasing increases the distance of the preferred region from the origin. Hence looks like a measure of preference. A good indifference curve should be concave, as here (2 2⁄ > 0 when > 0 and > 0). This is justified by the concept of satiation. A consumer is likely to prefer equal amounts of the two goods, than large amounts of one and little of the other. Utility Assigned to each indifference curve is a utility, a number such that is monotonically increasing with the preference of the indifference curves. Thus we define (, ) as the utility of the unique indifference curve through the point (, ). This is Anne’s utility function for (, ). In the example above Preferred Region ( − )( − ) = 2 Curves (, ) = + . Indifference curves are defined by (, ) = 0 = Trading and the Edgeworth Box Anne and Ben trade only with each other, so the total number of apples and bananas remains fixed. + = 0, + = 0, constant throughout trading We define an 0 × 0 Edgeworth Box in which Anne’s origin is at the bottom left, whilst Ben’s origin is at the top right. Any point on the diagram represents simultaneously the commodity bundle of both traders. Let ′ and ′ denote the indifference curves for Anne and Ben respectively through their initial holdings. Anne considers only points to the ‘north-east’ of ′ as these are the points where her utility is improved from (, ). Likewise Ben considers only points to the ‘south-west’ of ′. Edgeworth suggested that if a trade were to occur at a point say, the tangent of Anne’s indifference curve at P must be parallel to the tangent to Ben’s indifference curve at P. If not, Anne and Ben could find some other point where both have higher utility values than at . Edgeworth suggested that the ‘solution’ to trading is a curve ′ lying between ′ and ′, with characteristics that any point on ′ the tangents of the respective indifference curves through this point are parallel. This curve ′ is called the contract curve. In some sense the contract curve is like the negotiation set of a co-operative (or non zero-sum) game. The curve ′ is player ’s fall back position, whilst player has a fall back position ′, i.e. their maximin status-quo point. Any point in between which does not have tangents parallel to the indifference curves passing through it, can be dominated by one or other of the players. Finding the Contract Curve Given an indifference curve (, ) = 0, we have = + = 0 ⇒ = − ( ⁄ ) ( ⁄ ) as its gradient. The contract curve is the line along which the gradients of the two indifference curves are equal. Hence we require ( ) = ( ) ⇒ ( ⁄ ) ( ⁄ ) = ( ⁄ ) ( ⁄ ) . To find the endpoints of the contract curve, we find the points where it meets the curves (, ) = (, ) and (, ) = (, ). Example Rupert and Conrad meet to trade companies. Rupert has 8 newspaper companies and Conrad has 2, whereas Rupert has one printing press but Conrad has 9. Let = Rupert 's Newspaper holdings, = Rupert 's printing press holdings, = Conrad's Newspaper holdings, = Conrad's printing press holdings. Their respective utility functions are given by = + 4 , = 9 + . i) Write down the constraints on the variables. ii) Draw the Edgeworth box. Sketch the two indifference curves with a value of 2 3⁄ and the contract curve. iii) Suppose a price mechanism is used for the transaction (see later). Find, by fair means or foul, the final values of their respective holdings. i) There are 10 Newspaper companies and 10 printing presses, so + = 10 + = 10 We will use these to plot the Edgeworth box in ( , ) space. ii) The box has vertices at (0, 0), (10, 0), (0, 10) and (10, 10). The initial point is at ( = 8, = 1). Rupert’s indifference curve is + 4 = 2 3 , (1) ⇒ 3 = 2 + 8 ⇒ ( − ⁄ )( − ⁄ ) − ⁄ = . Conrad’s indifference curve is 9 + = (10 − )(10 − ) 100 − 9 − = 2 3 (2) ⇒ 100 − 10 − 10 + = 200 3⁄ − 6 − 2 3⁄ ⇒ − 4 − 28 3⁄ + 100 3⁄ = 0 ⇒ ( − ⁄ )( − ) − ⁄ = . To find the contract curve we need = 4 2 ( + 4)2 , = 2 ( + 4)2 , = 2 (9 + )2 , = 9 2 (9 + )2 . The contract curve is defined by the respective gradients being equal. Hence the contract curve is given by ⁄ ⁄ = ⁄ ⁄ ≡ 2 4 2 = 9 2 2 . Here ≥ 0, ≥ 0, ≥ 0 and ≥ 0, so we can take the positive square root of this equation, to give 2 = 3 , ⇒ (10 − ) = 6(10 − ) ⇒ 5 + 10 − 60 = 0, ⇒ + − = . () To find the endpoints, we wish to find where those points where equations (1) & (2) are equal to equation (3). From equation (3), = 12 2 + . Substituting into (1) gives (12 (2 + )⁄ ) (12 (2 + )⁄ ) + 4 = 2 3 ⇒ 12 12 + 4(2 + ) = 2 3 ⇒ 12 = 8 + 8(2 + ) 3⁄ ⇒ 28 3⁄ = 40 3⁄ ⇒ = ⁄ . Hence = 12 × 10 7 (2 + 10 7⁄ )⁄ = 120 (14 + 10)⁄ = . For you to check; (2) = (3) ⇒ = ⁄ and = ⁄ . The contract curve does not give a unique solution: There are two ways – a Price Mechanism and a Welfare Function – of finding a unique result. The Price Mechanism Suppose that Anne and Ben decide that apples and bananas each have a price and respectively. (20 3⁄ , 5 2⁄ ) (5, 10 3⁄ ) . . = (8,1) (10 3⁄ , 10 3⁄ ) Anne then has a budget constraint. If her initial commodity bundle was apples and bananas, then after trading she would want her final commodity bundle (, ) to satisfy + = + , otherwise she loses (or gains!) money. This constraint is a line through her initial point with gradient − ⁄ . Since the total number of apples and bananas is constant, we have + = 0, + = 0. Now Ben’s corresponding budget constraint is ( − ) + ( − ) = ( − ) + ( − ) ⇒ + = + . So the two players would look for a point on the contract curve where + = + is satisfied. This would give a unique point on the contract curve which would be the solution of the trading game. However, we don’t actually need to know the prices of the items! It turns out that under ideal market conditions, that the gradients of the utility functions are parallel to the straight line budget constraint. We can use this observation to find a solution to the trading game. Let us return to the problem of the press barons, Rupert and Conrad. We know − ⁄ ⁄ = = − 2 4 2 . (A) (see earlier) Originally Rupert had = 8 newspapers and = 1 printing press. So his budget constraint would be + = 8 + . (1) (Here = price of a newspaper title, and = price of a printing press.) Differentiating gives + = 0 ⇒ = − . Now from (1) we have + = 8 + ⇒ − = ( − 8) ( − 1) . (B) Under ideal market conditions one would expect (A) = (B), ⇒ − 2 4 2 = ( − 8) ( − 1) . (C) The contract curve we found earlier was + 2 − 12 = 0. (D) Solving for and using (C) and (D) gives = . = . , which lies on the contract curve, between the two initial utility curves (as expected). (Can you have 0.859 of a newspaper title or 0.908 of a printing press?) The Welfare Function The second means of identifying a unique solution on the contract curve is by means of a welfare function, that we shall denote by . This is a subjective function. It is devised from the two utility functions and , constructed by someone in authority (e.g. the Government). It summarises the desirability of one person or group receiving some good/benefit over another person or group. The aim is to maximise , with respect to and . Example Let the welfare function (, ) take the form = + , where and are constants. We wish to maximise . So we require = + = 0, = + = 0, ⇒ = − ⁄ ⁄ = − ⁄ ⁄ ⇒ ⁄ ⁄ = ⁄ ⁄ . This is just the equation of the contract curve (see earlier for the definition). So the maximum lies on the contract curve. In the press baron example = + 4 , = 9 + , = 4 2 ( + 4)2 , = × = 2 (9 + )2 × −1. This comes about because + = 10 ⇒ ⁄ = −1. Now as = 10 − and = 10 − , we have = − (10 − ) 2 (100 − 9 − )2 . So at the maximum + = 0, [ ( + ) ] = [ ( − ) ( − − ) ]. This constraint, plus the contract curve + − = , gives a unique solution. However, we do need some values for and . Let = 25 and = 64. Taking positive square roots gives 10 ( + 4) = 8(10 − ) (100 − 9 − ) ⇒ 1000 − 90 − 10 2 = 80 2 + 320 − 8 − 32 2 ⇒ 680 − 82 + 22 2 − 80 = 0 ⇒ 680 + 22 2 − (82 + 80) = 0 Eliminating using the contract curve gives 680 + 22 2 − 12 ( + 2) (82 + 80) = 0 ⇒ (340 + 11)( + 2) = 6(82 + 80) ⇒ 11 2 − 130 + 200 = 0 (11 − 20)( − 10) = 0. The solution = 10 lies outside the range ∈ [10 7⁄ , 5 2⁄ ] prescribed by the initial utility curves (see earlier diagram). But the solution = 20 11⁄ is fine. The corresponding point = . … = . … lies on the contract curve between the two initial utility curves.
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