Question

1. (12.5 points, 15 minutes) A manufacturing firm’s production function is Q = 2KL + 2K. For this production function, MPL = 2K and MPK = 2L + 2. Suppose that the price r of capital services is equal to 1, and let w denote price of the labour services. (a) If the firm is required to produce 10 units of output, for what values of w would a cost-minimizing firm use only capital? (b) Now assume that the rental rate r has changed to $5. Show in the graph the solution from part a) and new cost minimizing combination of labour and capital if the firm wants to stay at the same level of output - 10 units. Comment on your findings. (c) Is the total cost from part b) more or less than total cost from part a)? Explain. (Calculation is not required.)

can you please comment on the findings in part b, and don't forget to explain in part c. Thank you!

Answer 1

Ł= TK + WL+ X(10 – 2KL – 2K) aL =r – 2LX - 2X=0 aK ƏL = w - 2KX=0 al aL 10 – 2KL – 2K = 0 ax T a 2L + 2 2K

$r=1\\ 2K=2Lw+2w\\ substituting\ in\ third\ derivative\\ 2KL+2K=10\\ L(2Lw+2w)+2Lw+2w=10\\ 2L^{2}w+4Lw+2w-10=0\\ L=\frac{-4w\pm \sqrt{16w^{2}-4*2w*(2w-10)}}{2*2w}\\ L=\frac{-4w \pm \sqrt{16w^{2}-16w^{2}+80w}}{4w}\\ L=\frac{-4w \pm \sqrt{80w}}{4w}\\ L=\frac{-4w \pm 4\sqrt{5w}}{4w}\\ L=\frac{-w \pm \sqrt{w}}{w}\\$

since quantity of labor cannot be negative $L\neq \frac{-w - \sqrt{w}}{w}\\$

-W + W demand for labor, L w

$suppose\ r=w=1\\ L=\frac{-w+\sqrt{w}}{w}=\frac{-1+1}{1}=0$

therefore if price of labor equals price of capital firm would only use labor

$if\ r=w=1\\ L=0\\ 2KL+2K=10\\ 2K=10\\ K=\frac{10}{2}=5\\$

now price of capital rises to $ 5 from $ 1

$\frac{r}{2L+2}=\frac{w}{2K}\\ 2rK=2LW+2w\\ 10K=2Lw+2w\\ substituting\ in\ the equation\\ 2KL+2K=10\\ L(\frac{2Lw+2w}{5})+\frac{2Lw+2w}{5}=10\\ 2L^{2}w+2Lw+2Lw+2w=50\\ 2L^{2}w+4Lw+2w-50=0\\ L=\frac{-4w \pm \sqrt{16w^{2}-4*2w*(2w-50)}}{2*2w}\\ L=\frac{-4w \pm \sqrt{400w}}{4w}\\ L=\frac{-4w \pm 20\sqrt{w}}{4w}\\ L=\frac{-w \pm 5\sqrt {w}}{w}$

$w=1\\ L=\frac{-1 + 5}{1}=4\\$

$2KL+2K=10\\ 2K*4+2K=10\\ 8K+2K=10\\ K=\frac{10}{10}=1\\$

$if\ w=5\ \&\ r=1\\ L=\frac{-w + 5\sqrt{w}}{w}=\frac{-5 + 5\sqrt{5}}{5}=\sqrt{5}-1\\ 2KL+2K=10\\ 2K(L+1)=10\\ 2\sqrt{5}K=10\\ K=\frac{10}{2\sqrt{5}}=\sqrt{5}\\$

	L	K
$w=1,r=1$	0	5
$w=1,r=5$	4	1
$w=5,r=5$	1.236	2.236

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