# Production function and elasticity

Let $$y=x_1^\alpha x_2^\beta$$ where $$\beta=1-\alpha$$ be a Cobb-Douglas production function.

Find the elasticity of the optimal demand functions (for minimizing production cost) for both goods wrt. $$w_2/w_1$$ ($$w_1,w_2$$ are the respective prices of the inputs). What does this say for our spending on $$x_1$$ compared to the total cost?

My try

I found the elasticity of $$x_2^*/x_1^*$$ wrt $$w_2/w_1$$ to be 1. Now, I have to find a value for the ammount spend on good $$x_1$$. How can I do that with my elasticity? I think it has to be constant, but still no clue on how to find such a value even though I have found the elasticity to be 1.

edit: I know it has to be constant.

• What have you done already? You have to derive conditional demand as it is called - not optimal demand - could you please show your derivations? Commented Dec 22, 2020 at 12:30

The first order conditions equate marginal revenue per factor to the price of that factor:

\begin{align} p\cdot\alpha\frac{y}{x_1} &= w_1\\ p\cdot\beta\frac{y}{x_2} &= w_2, \end{align}

Where I used the property of power function $$(x^n)'_n = n \frac{x^n}{x}$$.

Divide the second FOC by the first to get the relation between the relative prices and the relative factor demands:

$$\frac{\beta}{\alpha}\frac{x_1}{x_2} = \frac{w_2}{w_1}. \tag{A}$$

From this relation we can draw two conclusions:

1. Rewrite (A) in log form:

$$- \ln \frac{x_2}{x_1} + \ln \frac{\beta}{\alpha} =\ln \frac{w_2}{w_1},$$

And using the log definition of elasticity $$\epsilon_y^x = \frac{\mathrm{d}\ln y}{\mathrm{d}\ln x}$$ we come to the conclusion, that relative factor demand is decreasing in relative factor prices with unit elasticity:

$$\frac{\mathrm{d}\ln x_2/ x_1}{\mathrm{d}\ln w_2/w_1} = -1.$$

1. Multiply both sides of (A) by $$\frac{x_2}{x_1}$$ :

$$\frac{\beta}{\alpha} = \frac{w_2x_2}{w_1x_1}. \tag{B}$$

Rearrangement (B) says that the expenditures on different factors are proportional to their respective input elasticities, i.e. if our total spending on factor 1 is $$\\\\alpha$$ then we must spend $$\\\\beta$$ on factor 2.

The total cost $$C$$ is allocated in the same proportion, i.e. for a general Cobb-Douglas production function, spending on factor 1 is

$$w_1 x_1 = \frac{\alpha}{\alpha+\beta}C,$$

or simply $$w_1 x_1 =\alpha C$$ if $$\alpha+\beta=1$$, i.e. if the production function is homogeneous of degree 1.

• I found the $MP$ and multiplied it by $x_1$ to aquire the total cost which gave me $\alpha \cdot y$ and thus the total factor used is just $\alpha$. Is this not correct and much faster?
– user31331
Commented Dec 23, 2020 at 17:10
• $MP$ of $x_1$ that is
– user31331
Commented Dec 23, 2020 at 17:12
• MP, if you mean marginal product, is measured in real terms, units of output per units of factor. If you multiply this amount by $x_1$ you get something measured in units of output. Cost and spending on the other hand is always measured in monetary units, so you lack a price (that of the final product). It might be that in a certain problem the price is equal to 1, so you make no mistake in math, but it is better to pay due to the accounting logic. Commented Dec 23, 2020 at 17:36
• Found this for better explanation of what I mean; Kevin Clinton, September 2004, "A useful production function: Cobb-Douglas". His second paragraph explains what I mean and thus that $MP_1(x_1)$ would give the usage of that input relative to all the usage. This is what I think is intended here. (the pdf is just a copy paste on google and should pop right up).
– user31331
Commented Dec 23, 2020 at 17:39
• well, if no one is asking for a proof, you totally should use the properties of Cobb-Douglas function. If you have a Cobb-Douglas function $x_1^\alpha x_2^\beta x_3^\gamma ...$ with $\alpha+\beta+\gamma + .. = 1$ then the total production cost is allocated between factors in proportion $\alpha : \beta : \gamma : ...$. Elasticity from the first part does not play any role here. Commented Dec 23, 2020 at 17:55