In addition to being the lowest cost combination of L and K that produces 1000 units of output, pt. A involves a tangency point between the isoquant and isocost. That is, the slopes of these curves at pt. A are equal. The slope of the isocost is w/r, or –1/5. The slope of the isoquant is the ratio of the marginal products, MPL/MPK, which is given as the marginal rate of technical substitution (MRTS). Using calculus, it is possible to derive the MRTS as –K/L. Point A satisfies the condition that K/L = 1/5.

We can solve for K* and L* at pt. A, using (1), (2a) and the fact that, at pt. A, K/L = 1/5. First, substitute (2a) into (1) and the equation K/L = 1/5. We're left with:

10L + 50(1,000,000/L) = TC
and

(1,000,000/L)/L = 1/5



Solve the second equation for L, substitute that result into the first equation to get the lowest value for TC (TC*).

TC* = $44,721.36



Once you have TC*, you can substitute this value into the isocost equation above (10L + 50,000,000/L = TC) and then solve for L* (rounded to the nearest whole number).

L* = 2,236



Going back to (1), we can substitute in L* and TC*, to get K*.

K* = 447

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