Another comparative static exercise is to keep income (and all other prices) constant and study what happens to quantity demanded of good 1 as its own price changes. For example, if the price decreases the budget set will expand, or the budget line will swing outward around the intercept along the Q2-axis. If we connect all optimal bundles for various price ratios, as p1 changes, we can trace out a so called price−offer curve (or price-consumption line). This is illustrated in figure shown below, where p11 > p21 > p31. In the case illustrated the quantity demanded of good 1 increase as its price decreases, this means of course that the demand function is negatively sloped. This is obviously the case with the demand function derived from Cobb-Douglas preferences:
Q1 (Y, p1) = (a. Y)/p1
We previously defined the concept of an “own” price elasticity, as the percentage change in the quantity demanded of a good divided by the percentage change in its own price:
As before we assume that Y0 = 10, p01 = 1, and U(Q) = Q1.Q2, the initial quantity demanded is therefore Q01 = 5. Assume now that the own price increases to p11 = 1.10. The new quantity is therefore, Q11 = 10/(2.1.1) = 4.545. The change in quantity demanded is ΔQ1 = (4.545 − 5) = −.455, and the change in price is equal to Δp1 = +0.1. Putting these numbers into the formula for the price elasticity, we get,
We should note that this is actually a measure of the average elasticity between the initial and final points along the demand curve. If we made the price increase much smaller, e.g., Δp1 = +0.001, the change in quantity demanded would be, ΔQ1 = −.004995 and the own price elasticity would be:
ε1p1 = - (- .004995/0.001) (1/5) = 0.999. Actually the price elasticity at the initial point is simply equal to 1 (the same value as the income elasticity so it is easy to remember), which we’ll find if we consider very small changes in the price.
In the general case we can also calculate the cross − price elasticity of demand; this is the change in the quantity demanded of good 1 as a response to a given change in the price of good 2:
If the cross-price elasticity between goods 1 and 2 is positive, the two goods are gross−substitutes, i.e., an increase in the price of good 2 will increase the demand for good 1 (and vice versa). If the cross-price elasticity is negative, the two goods are gross−complements. Note, that in the Cobb-Douglas case the two goods are independent (i.e., neither substitutes nor complements), since the price of good 2 does not enter the demand function for good 1.
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