\(f(x) = f(x_0) + \frac{f'(0)}{1!} (x - 0)\)
\(+ \frac{f''(0)}{2!} (x - 0)^2\)
For any \(f(x) \approx\) \(f(x_0) + \frac{f'(x_0)}{1!} (x - x_0) ...\)
And this is the Taylor Approximation, and this is a TANGENT LINE
The first two terms are the slope/derivatives
Test-like questions with this:
\(f(x) = y = \ln x\)
Using the Taylor approximation:
\(f(x) \approx \ln x_0 + \frac{1}{x_0} \cdot \frac{1}{1!} (x- x_0) - \frac{x_0^{-2}}{2!} (x - x_0) ^2\)
A 2nd order Taylor approximation is just a parabola with well-chosen parameters. It's only the X that is changing, but \(x_0\) is the steady state, original endowment, etc. Will need to be able to crank out simple ones-- good for local analysis but dangerous in aggregate systems.
Basic Firm Profit Maximization & Derived Demand
One input, Cobb-Douglas: \(y = zK^ \alpha\)
Cost, capital rental: \(rK\)
Profit, output minus costs: \(\pi = zK ^\alpha - rK\)
We can't solve for two variables with only one equation. Why do economists love taking first derivatives and setting them equal to zero? Math trick to get a second equation.
\(\frac{d \pi}{dK} = \alpha z K^{\alpha - 1} - r = 0\)
\(\frac{r}{\alpha z} = K^{\alpha - 1}\)
\(K* = (\frac{r}{\alpha z})^{\frac{1}{\alpha - 1}}\)
\(K* = (\frac{\alpha z}{r})^{\frac{1}{1- \alpha}}\)
Remember, when everything is multiplied, exponents are elasticities:
Taking logs and derivatives,
\(\ln K* = (\frac{1}{1 - \alpha}) \ln \alpha \frac{1}{1- \alpha} \ln z\)
\(- \frac{1}{1 - \alpha} \ln r\)
\(\frac{d \ln K*}{x \ln r} = \frac{1}{1 - \alpha}\), which is the elasticity of capital demand.
What is alpha? The weakness of diminishing returns, or a tendency toward more constant returns if \(0 < \alpha < 1\). So if alpha is close to one, then the firm is very close to constant returns to scale, and doubling capital would mean doubling output. If you double the size of your firm them, output might almost double (costs will certainly double). Tells us that the size of our firm is incredibly responsive to the interest rate when alpha is close to one (how much capital is invested and bought).
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