A consumer consume only two goods $X$ and $Y$ and is in equilibrium. Show that when the price of good $X$ rises, the consumer buys less of good $X$ . Use utility analysis.

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Solution

In case of two commodities, the consumer's equilibrium is attained at the point where the utility derived from each additional unit rupee spent on each of the goods is equal. That is, Marginal Utility of a Rupee spent on the good $x (i . e, \frac{M U_{x}}{P_{x}})$ is equal to the Marginal Utility of a Rupee spent on good $Y (i . e, \frac{M U_{y}}{P_{y}})$ , which, in turn, is equal to the Marginal Utility of Money(MUm).
That is, $\frac{M U_{x}}{P_{x}} = M U_{m}$
Howevwe, when the price of commodity x rises, the ratio of marginal utility to price of $(\frac{M U_{z}}{P_{z}})$ becomes lower than that of Y, that is $\frac{M U_{x}}{P_{x}}$ < $\frac{M U_{y}}{P_{y}}$
In such a case, the consumer rearranges his consumption combination such that the equality is again restored. He would decrease his consumption of commodity X. With this decreases, marginal utility of x rises, As a result, the ratio of marginal utility to the price of X rises. The consumer would continue decreasing the consumption of commodity X till the equality between the ratio of marginal utility to price in case of X and Y is again reached.

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A consumer consume only two goods X and Y and is in equilibrium. Show that when the price of good X rises, the consumer buys less of good X. Use utility analysis.

A consumer consume only two goods $X$ and $Y$ and is in equilibrium. Show that when the price of good $X$ rises, the consumer buys less of good $X$ . Use utility analysis.