# Marshallian demand with Leontif preferences

Consider a utility function on the form $$u(q_{1},q_{2},q_{3}) = min\{\alpha ln(q_{1}) + (1 - \alpha) ln(q_{2}), ln(q_{3})\}$$

I know that optimal behaviour requires $$\alpha ln (q_{1}) + (1 - \alpha) ln(q_{2}) = ln(q_{3})$$

I tried substituting this into the budget constraint (after raising both sides to the power of $$e$$)

$$p_{1}q_{1} + p_{2}q_{2} = y$$

But I am unsure how to proceed in finding the Marshallian demand, since I have two variables $$(q_{1}, q_{2})$$ and only one equation. I have tried different transformations but my main problem is that it feels like I'm missing an equation.

Any help is greatly appreciated.

EDIT: I just re-read the question and I misspecified the budget constraint! The consumer only spends on $$q_{1}$$ and $$q_{2}$$.

• So since $\ln(0)$ is not defined but $\lim_{q_3 \to 0} \ln(q_3) = -\infty$ the task is to maximize w.r.t. $q_1,q_2$ the formula $$\min\{\alpha \ln(q_{1}) + (1 - \alpha) \ln(q_{2}), -\infty\} = -\infty?$$ – Giskard Oct 28 '18 at 11:36
• I believe we just have to assume some exogenous non-zero $q_{3}$. In the question I am given that $q_{1}$ and $q_{2}$ represent tea and coffee and $q_{3}$ represents sugar. I tried a transformation such that the UMP became min$\{\phi_{1} + \phi_{2}, \phi_{3}\}$ and thus at optimum $\phi_{1} + \phi_{2} = \phi_{3}$. Wouldn't this mean that the agent chooses the cheaper one of $\phi_{1}$ and $\phi_{2}$? – Gensys Oct 28 '18 at 11:45

So, I have not really worked out the maths behind solving this. I prefer to take a shortcut and just compute it numerically. Here is what I have.

These are the contour surfaces and the budget constraint. Notice how the solution will be at the "kinked" curve in the middle of each surface. The levels 1 and 2 are only for example. The third level, u = 3.92762 is related to the next part of my answer. Also, BC is the budget curve with income = 300. I have chosen $$\alpha = 0.5$$ and $$P_1 = 2, P_2=3, P_3=1$$.

Now, using the parameters mentioned in the last paragraph, the maximum is found to be at $$q_1 = 65.8896, q_2 = 39.1448, q_3 = 50.7863$$ and the maximized utility is $$u = 3.92762$$. I assumed that all goods are strictly consumed at positive amounts.

I am attaching my Mathematica code below, so that you can replicate and modify these results as you want

u = Min[a*Log[q1] + ((1 - a)*Log[q2]), Log[q3]]

BC = p1*q1 + p2*q2 + p3*q3
a = 0.5; p1 = 2; p2 = 3; p3 = 1;

ContourPlot3D[{u == 1, u == 2, u == 3.92762, BC == 300}, {q1, 0,
100}, {q2, 0, 100} , {q3, 0, 100}, AxesLabel -> Automatic,
ContourStyle -> {Opacity[0.5], Opacity[0.5], Opacity[0.5], Opacity }, Mesh -> None, PlotLegends -> "Expressions"]

NMaximize[{u, BC <= 300, q1 > 0, q2 > 0, q3 > 0}, {q1, q2, q3}, Reals]