Why does the LRAC (long-run average cost) curve intersect with the SRAC (short-run average cost) curve at exactly one point? I understand why there's at least one intersection (it's because, given an output quantity, we can find the optimal value of capital $k$ that minimizes the short-run total cost, and this is by definition the optimal LRTC for the given $y$. Since $y$ is common to both, the point of intersection of the LRTC and SRTC is the point of intersection of the LRAC and the SRAC). I also know that $SRAC \text{ at } y \geq LRAC \text{ at } y$. But I don't know why the tangency is at only one point.
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$\begingroup$ Hi! 1. Are you sure you mean to write "intersect" and not something like "share a point of tangency"? 2. Can you please support your claim with a reference? It does not seem to be generally true; but a reference would help by providing context (and show that you did research before posting). $\endgroup$– GiskardCommented Oct 18, 2022 at 16:12
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$\begingroup$ @Giskard 1. Yes, it's a point of tangency as I mentioned in the last line (or by, "one point of intersect" and "SRAC $\geq$ LRAC", which together point to a tangency point). 2. Varian, p. 406 (last paragraph). Another link: Cost curve, Wikipedia. If you can please show me the counter-example, it would be even more enlightening (in the sense, I would know which assumptions I am making incorrectly). $\endgroup$– Rick_MortyCommented Oct 18, 2022 at 19:00
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$\begingroup$ Your claim that there is always only one point of tangency is supported by this sentence from Varian: "f the short-run cost is always greater than the long-run cost and they are equal at one level of output"? Notice that it starts with the word "if". Not sure where on the Wikipedia page you want me to look, I am not going to read the whole thing. $\endgroup$– GiskardCommented Oct 19, 2022 at 5:06
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$\begingroup$ @Giskard It's a highlighted link. If you're using Chrome, it should direct you to the exact sentence. Otherwise, please see the second point here. $\endgroup$– Rick_MortyCommented Oct 22, 2022 at 8:06
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A counterexample:
Consider the production function $f(x_1,x_2) = x_1 + x_2$, and assume input prices of 1. The long-run cost function is $C(y) = y$. Assume that in the short run $x_2 = 5$. Then the short-run cost function is $C_s(y) = \max(5,y)$.
LRAC is 1 for all output levels, SRAC is $5/y$ for $y<5$ and 1 for all other output levels. LRAC and SRAC coincide at all points $y\geq 5$.
