Marshallian demand help [closed]

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.

closed as off-topic by Giskard, FooBar, optimal control, cc7768, EnergyNumbersJan 7 '16 at 11:21

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• "This question does not meet the standards for homework questions as spelled out in the relevant meta posts. For more information, see our policy on homework question and the general FAQ." – Giskard, FooBar, optimal control, cc7768, EnergyNumbers

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.)

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.

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.

• Marshallian demand functions are not partial derivatives of the utility function. (Not even in the Cobb-Douglas case.) Additional steps are needed. – Giskard Jan 6 '16 at 9:07
• 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. – gsilvapt Jan 6 '16 at 17:05