0
$\begingroup$

Consider a consumer with a Cobb-Douglas utility function over three goods of the form $u=(x_1+z_1)^\alpha \cdot (x_2+z_2)^\beta \cdot (x_3+z_3)^\gamma$ with $\alpha, \ \beta, \ \gamma, \ z_1, \ z_2, \ z_3>0$, and $\alpha+\beta+\gamma=1$. The consumer has income $m$, and pays prices $p_1, p_2, p_3$ for the three goods.

Find the Marshallian demands for goods 1, 2 and 3 for this consumer, using the Lagrangian method.

$\endgroup$
2
$\begingroup$

Hint: I won't give a full answer since this is a very specific question, but I can give you a long outline here. A Lagrangian is set-up with an objective function (in this case utility) plus or minus (depending on Kuhn Tucker constraints, which we won't worry about) a constraint function (in this case, income) which is set to zero and then multiplied by the Lagrange multiplier, which we will denote $\lambda$.

So we set up the Lagrangian function. (Plug in the utility function yourself.)

$$\mathcal{L} = u(x_1, x_2, x_3) - \lambda (p_1x_1 + p_2x_2 + p_3x_3 - m)$$

Solve for the partial derivatives $\frac{\partial \mathcal{L}}{\partial x_1}$, $\frac{\partial \mathcal{L}}{\partial x_2}$, and $\frac{\partial \mathcal{L}}{\partial x_3}$ and set them equal to zero (You can also solve for the partial of $\lambda$, but it just gives you back the budget constraint, which I skip to). For example, you get for your first derivative:

$$\frac{\partial \mathcal{L}}{\partial x_1} = \alpha (x_1 + z_1)^{\alpha - 1} - \lambda p_1 = 0$$

Once you solve the three partials, the easiest thing for you to do is solve for $\lambda$ with respect to the other variables for all three partials, and then figure out how to substitute them into the budget constraint. You should solve for each $x$ in terms of the prices, maybe some $z$'s, the greek letters $(\alpha, \beta, \gamma)$ and income $m$. (I tried working some of it out; enjoy your algebra.)

| improve this answer | |
$\endgroup$
0
$\begingroup$

The marshallian demand functions are solutions to the utility maximization problem. That is, the solution to $$ \max u(\textbf{x})\\ s.t. \textbf{p}\cdot \textbf{x} = m $$ Where boldface is used to represent vectors.

In your problem, you would use your utility function, and $\textbf{p}\cdot \textbf{x} = p_1x_1 + p_2x_2 + p_3x_3$

The other answer provides a link that I believe shows how to derive the formulas. I just wanted to explicitly state that marshallian demands are solutions to the utility maximization problem. If you know how to solve a maximization problem (using FOC and/or lagrangians), you can derive the marshallian demands.

| improve this answer | |
$\endgroup$
-2
$\begingroup$

The Marshallian demand functions are basically partial derivatives of the Cobb-Douglas utility function. You should consider that you want to maximize spending first, then derive the functions to get the optimal prices, demand and a equilibrium with both - not sure if I used the correct words.

Nevertheless, here's an example with 2 products, here, from SE.

| improve this answer | |
$\endgroup$
  • 1
    $\begingroup$ Marshallian demand functions are not partial derivatives of the utility function. (Not even in the Cobb-Douglas case.) Additional steps are needed. $\endgroup$ – Giskard Jan 6 '16 at 9:07
  • $\begingroup$ Poor misuse of words. But the example I gave there is useful to understand the steps. There's more exemplifications in that question and they're related because the problem is fairly similar. $\endgroup$ – gsilvapt Jan 6 '16 at 17:05

Not the answer you're looking for? Browse other questions tagged or ask your own question.