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PPC will not expand outward if the country preferred more consumer goods than capital goods.

The total amount of resources in an economy at any given point of time is fixed. If a country preferred more consumer goods, more resource has to be allocated to produce the consumer goods. It will lead to a sacrifice of the production of capital goods. In a long run, the production of consumer goods is difficult to expand since the capital goods’ restriction.

This is my answer to the question. But I'm not sure if it's good enough. Any suggestion would be appreciated.

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    $\begingroup$ In whatever class setting you are looking at, is housing considered a capital good or a consumer good? Because if housing is a capital good (and in a sense it is, because it produces housing services), then a taste shift where households preferred more consumer goods could involve shifting out the PPC if it meant less housing. $\endgroup$ – BKay Jan 28 at 17:13
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    $\begingroup$ If however, capital goods are only used to produce and not consume, what does it mean for consumers to "prefer them less"? If they don't enter into the utility function how would you change preferences to make them prefer them less? $\endgroup$ – BKay Jan 28 at 17:16
  • $\begingroup$ I'm not sure. The question does not specify. I would consider it as a consumer good. $\endgroup$ – user8314628 Jan 28 at 17:17
  • $\begingroup$ If we don't consider depreciation, could we assume the capital goods we invested is fixed? Namely, if we don't spend more resource on it, the amount of capital goods would not change anymore. Instead, the amount of consumer goods will increase since we spend all the resource on it. $\endgroup$ – user8314628 Jan 28 at 17:21
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To keep things simple, let's assume that inputs to production other than capital goods are constant over time, and that production technology is regarded as one of those inputs (ie there is no technical progress). Let's also assume that the production function is such that output (across the whole range of the production possibility frontier) is an increasing function of capital when other inputs are constant: in symbols $\partial{Y}/\partial{K}>0$.

The question then is whether capital and therefore output can increase given the country's "preference for more consumer goods than capital goods". If this simply means that its chosen output combination includes a higher proportion of consumer goods than capital goods, then that is certainly possible. Suppose for example that output consists of $80\%$ consumer goods and $20\%$ capital goods, that the annual capital-output ratio is $2$, and that $5\%$ of capital is lost each year due to wear and tear. Then the annual loss of capital due to wear and tear is $2 \times 5\% = 10\%$ of output, and capital goods increase each year by $20\% - 10\% = 10\%$. Thus the production-possibility frontier would expand outwards.

If however the country's preference implies that output contains less than $10\%$ of capital goods, then (on the above assumptions) the production possibility frontier would contract inwards over time.

Note that the above takes no account of the fact that, in practice, capital goods are not infinitely malleable, that is, the production of different goods requires different sorts of capital goods. One might pursue that point by considering how a country's preference for consumer goods might influence its choice of what sort of capital goods to produce (if indeed it produces any).

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