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Optimal choice with a budget constraint

January 20, 2012

One of the basic premises of ‘normal’ preferences in microeconomics is that we assume that more is better. In other words, if we are considering two goods, X and Y, and we have a choice between a bundle of 3X and 4Y, or 10X and 25Y, we are going to prefer the second because we get more of both.

However most of the time there is a cost involved with both of the goods, and we don’t have unlimited resources. The more we spend on good X the less we spend on good Y. So we will have a budget constraint depending on the resources available to us. We could either spend all our budget on good X, or all on good Y, or on some combination of the two.

We usually illustrate the budget constraint on a diagram along with indifference curves indicating our preferences.

Here we have two goods, X and Y, and a budget constraint indicated by the black line. All combinations of bundles between the origin and the budget line are affordable. Anything on the budget line is just affordable, it means we are using up all our budget on the combination of goods X and Y chosen. Anything inside the budget line (ie closer to the origin) is affordable and we will have some of the budget to spare. Anything outside the budget line (ie further away from the origin than the line) is not affordable.

There are four bundles shown, A, B, C and D, each on a different indifference curve. The most preferable bundle is D, because this is on the highest indifference curve, furthest away from the origin, but this bundle lies outside our budget constraint so it is unaffordable.

Bundle A is inside our budget constraint so this is not using our full resources and we can easily move to a more preferable bundle (like B or C) simply by spending more. B and C are both bundles that use up the full budget, but C is preferable to B as it is on a higher indifference curve.

C is the optimal choice when faced with the budget constraint given here, as it is the indifference curve that is tangential to the budget constraint. This means that at point C, the slope of the indifference curve, or the marginal rate of substitution, is equal to the slope of the budget line.

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