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The Marginal Rate of Substitution and gains from trade

January 19, 2012

The slope of the indifference curve at a particular point shows us the rate at which the consumer is willing to substitute one good for another in order to retain the same level of utility. This is the marginal rate of substitution, and it is the centrepiece behind ideas of trade and exchange.

To understand the idea of being ‘indifferent’ here, consider the possibility that there are two consumers, me and you, and we both have goods X and Y. I decide that I want some more of good Y, and I am willing to offer you some of my X in order to get it. How much more X you want in order to give up one unit of Y, will depend on a couple of factors – how much you like X compared to Y, and how much X and Y you already have. If you already have loads of X and not much Y, you are likely to be less keen on my offer of trading X for Y than you would be if you had loads of Y and not much X. People generally prefer exchanging the ‘good they have more of’, for the ‘good they have less of’ (apologies for the poor English). This is also known as diminishing marginal rate of substitution and is a property held by most normal convex shaped indifference curves.

Whether you will accept my trade or not depends also on the terms of my exchange. If I offer you 1000 units of X for 1 unit of Y then that might sound like a good trade even if you already have a lot of X. After the trade, you will have a different combination of X and Y, a different ‘bundle’. If this bundle took you to a higher indifference curve then your overall utility has improved, and you would definitely accept the trade. If the bundle took you to a lower indifference curve then your overall utility has got less so you would reject the trade. The marginal rate of substitution is about what rate of exchange you would accept that would leave your overall level of utility unchanged – ie if I want a unit of Y from you, how much X do I need to give you, to keep your level of utility constant. This rate of exchange is given by the slope of the indifference curve, the marginal rate of substitution:

The marginal rate of substitution is the amount of Y you would be willing to give up for a unit of X, in other words the change in Y over the change in X. As you will see, this changes as you move along the indifference curve, in other words as you have different combinations of goods.

At point A, you have a lot of Y and not much X, so here the MRS is very steep, you are willing to give up Y for not much X in return. But down at point B, you have a lot of X and not much Y, and here if you are to give up one of your precious units of Y, you are asking for a lot more X in compensation.

Now you can see what would happen if you were offered trade on fixed terms (ie a certain amount of Y in exchange for a certain amount of X). This is a ‘rate of exchange’. In the diagram below, the black dotted line illustrates a fixed rate of exchange which the consumer is being offered. We will consider that he starts off at bundle A.

Here the terms of trade are very good for the consumer. He can exchange some of good Y for some good X and move to bundle C, and make a simple gain, moving to a higher indifference curve representing a gain of utility. But that would not be the full extent of the possibilities of his gain from trade. He could carry on trading at that rate of exchange all the way up to point D, where he has reached the highest indifference curve possible at this rate of exchange, and the highest level of utility possible at this rate of exchange.

What is special about point D? It is the point at which the MRS for the consumer is the same as the MRS of the rate of exchange offered, the indifference curve is tangential to the rate of exchange.

This shows an important result. When a consumer holds a bundle of goods and is offered trade at any rate of exchange different to his MRS, then he can gain utility by trading at that exchange rate. Only when the rate of exchange offered is exactly equal to his MRS, will he not be able to make any gains by trading at that rate of exchange.

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