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Under what conditions do the following production functions exhibit decreaseing, constant, or increasing return to scale. START X=2 K K Test a q = L +K 2 2 Constant Returns 4 2 Constant Returns b CO 2 Constant Returns 10 2 Constant Returns Part a is the simplest to answer. I will show two ways to get to an answer. 12 10 2 Constant Returns One way is to look at a production function q = f(L, K) and see if f(xL, xK)> 1 then xL + xK = x(L+ K) = xq. So, this function shows constant returns to scale always. 4 Constant Returns Now lets do the analysis in a table. CO O A N 4 Constant Returns 4 Constant Returns 10 4 Constant Returns CO O 12 4 Constant Returns 2 6 Constant Returns 10 6 Constant Returns CO On A 6 Constant Returns 14 6 Constant Returns 10 6 Constant Returns 12 6 Constant Returns Q U .A W N H O UA W N H O UA W N H O UA W N K O UA W NKOUAWNH 2 8 Constant Returns 8 Constant Returns 14 co On A 8 Constant Returns 8 Constant Returns 10 8 Constant Returns 12 8 Constant Returns 12 10 Constant Returns 14 10 Constant Returns CO O A N 16 10 Constant Returns 10 Constant Returns 10 10 Constant Returns 12 10 Constant Returns 14 2 12 Constant Returns 4 12 Constant Returns 18 6 12 Constant Returns 20 8 12 Constant Returns 22 10 12 Constant Returns 12 24 12 12 Constant ReturnsMight just play with this functions for different values of a and b. Try a + b 1. See what comes of it. q = f(L, K) = L+L'K+K a : 1 X = 2 F( XL,XK) = XL + (XL)*(xK) +xK =XL +xL x-K+xK b= 2 atb>1 =xl+ xatb Lokb + xK L K L K Test 12 2 Increasing returns Excel's (and many other programs) may not able to detect equality of real numbers. It's a rounding issue. If a + b =1 then 22 2 Increasing returns A W N H x(L + LOK' + K = xf ( L, K) = xq 32 2 Increasing returns This implies constant returns to scale. 42 2 Increasing returns If atb > 1 then x > x so there would be increasing returns to scale. 11 52 10 2 Increasing returns 7 38 4 Increasing returns W N H U NNNP XL+ xatb Lokb + xK > xf ( L, K ) = xq 12 4 Increasing returns 17 106 4 Increasing returns Finally, if a+b