HTSEFP: Multivariable differential calculus

1Cobb-Douglas extremisation¶

There are two main possibilities:

• You want to maximise the production function, which looks like this: $f(x,y) = ax^{\alpha}by^{1-\alpha}$. The constraint is the amount of money available for labour and capital, and you are given the cost per unit capital and labour. Let's say you have \$50,000 available, and it costs \$100 per unit labour and \$150 per unit capital. So the constraint is $50,000 = 100x + 150y$ where x is the units of labour and y is the units of capital. In other words, $g(x,y) = 100x+150y-50,000 = 0$
• You want to minimise the cost function for a given production function. So $f(x,y) = 100x+150y$ is the cost function and $g(x,y,) = ax^{\alpha}by^{1-\alpha} - \beta = 0$ is the production function.

It may also then be necessary to justify that your answer is a minimum/maximum using the sufficiency test.

Then, estimate how a change in one function will affect the extrema of the other function.

1.1General solution¶

For the first possibility, we use Lagrange multipliers:

$$L = f + \lambda g = ax^{\alpha}by^{1-\alpha} + \lambda [100x+150y-50000] = 0$$

Then find the partial derivatives of L with respect to x and y to solve for lambda. Once you get two formulae involving lambda, equate them (you don't really care about the value of lambda just yet). You then get a relationship between x and y, which you can plug into the constraint function to solve for x and y. So you get an answer in terms of the units of capital and labour that would give you max production.

For the second possibility, we again use Lagrange multipliers, but slightly differently:

$$L = f + \lambda g = 100x+150y + \lambda [ax^{\alpha}by^{1-\alpha} - \beta] = 0$$

Again, we find the partial derivatives of L with respect to x and y to solve for lambda, then get a relationship between x and y, and it's the same process as before (although finding the actual answer might necessitate the use of a calculator, making this type of question less likely than the previous).

The sufficiency test:

$$H = L_{xx}g^2_y - 2L_{xy}g_xg_y + L_{yy}g^2_x$$

Calculate the necessary partial derivatives and plug in the values of x and y at this point. If H is negative, you have a maximum; if H is positive, you have a minimum; if it is 0, the test fails.

To estimate how a change in one function might affect the other, we'll have to calculate lambda. Start with $\Delta g$, which is given; we want to find $\Delta f$. Assume $\Delta g \approx dg, \, \Delta f \approx df$. So:

$$dg = \frac{\partial g}{\partial x}\,dx + \frac{\partial g}{\partial y}dy \quad df = \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y}dy$$

Since $dL = \frac{\partial L}{\partial x}dx + \frac{\partial L}{\partial y}dy = 0$, and $\frac{\partial L}{\partial x} = \frac{\partial f}{\partial x} + \lambda \frac{\partial g}{\partial x}$ (same for y)

Then we have $df + \lambda dg = 0\, \therefore df = -\lambda dg$

So solve for lambda using the values of x and y you find earlier, then multiply it by negative $\delta g$ to get your estimate.

1.2Examples¶

• Winter 2011 midterm, question 1 (first type)
• Assignment 1, question 19 (the same question as was on the midterm)
• Assignment 2, question 20 (second type)

2Extremising a function over a region¶

Given a function $f(x,y)$ over a region (most likely a disk), find the extrema.

2.1General solution¶

First, we look for local extrema - when the partial derivatives are equal to 0. Evaluate f at those points, and record the values.

Now, parametrise the edge of the region. Assuming that it is a disk of radius a, as in $x^2+y^2 \le a^2$, then we can parametrise it as

$$x = a\cos \theta,\,y = a\sin\theta$$

Now substitute those functions of x and y into f(x,y) to get a function in terms of theta. Now find the derivative of that function, and solve for $f(\theta) = 0$. Plug the values of x and y that result into f(x.y). From the values, determine which is a global maximum, which is a global minimum, and which are only local extrema.

2.2Examples¶

• Winter 2011 midterm, question 2

3Implicit differentiation for a system¶

In the general case, you're given something like $F(x,y,z,u,v) = 0$ and have to express the answer in terms of Jacobians. In the specific case, you're given the actual function.

3.1General solution¶

Later

3.2Examples¶

• Later