If we are making choices for choosing from more than two goods, it is convention to use vectors to vectors to represent our choice variables. As such, the utility function remains same (except we have more X’s), the budget constraint can be compressed as M=PX, where X is vector of [X1,X2,X3…Xn] and P is the corresponding price vector for the n-goods, P= [P1,P2,P3….Pn]. After calculation, the Marshallian Demand X* will be a function of the prices for all goods, and the income/wealth.
When the utility function is U(X,Y,Z)=XaYbZc, we say it is in Cobb-Douglas form. With Cobb-Douglas form, we will see eventually, X*=(a/(a+b+c))*M/Px, Y=(b/(a+b+c))*M/Py, Z*=(a/(a+b+c))*M/Pz. That means, the total amount spend on each of the good, is the proportion of the income determined by the subscript of that good in the utility equation. Typically, the sum of the superscript equals to one.
Non-Differentiable Utility Curve
When the utility is U(X,Y,Z)=Max (X,Y,Z), we will see that the solved equation will be, X=M/Px, Y=Z=0, if and only if Px<Py and Pz. That means, with such utility function, we will devote all the income in one goods that will maximizes our utility. If the utility function is U(X,Y,Z)=Max (2X,Y,Z), we do it in the same manner, calculate, which one will be the greatest, 2M/Px, M/Py, or M/Pz. In these case, we will have X=M/Px, Y and Z =0, if and only if 2Px<Py and Pz.
Leontief: Another Case of Non-Differentiable Utility
When the utility function is U(X,Y,Z)=Min(2X,Y,Z), we call it has the Leontief preference. With Leontief preference, we will have 2X*=Y*=Z*. As such, we can find the Marshallian demand by simply substitute this equilibrium condition into the budget constraint. For example, in this case we want to get X*, so we will have M=X*Px+Y*Py+Z*Pz= X*Px+2X*Py+2X*Pz=X*(Px+2Py+2Pz)=M, and accordingly X*(P,M)= M/(Px+2Py+2Pz). The feature of such Leontief function is, when you have any goods that more than the minimal one, the marginal utility from consuming more will be zero; and the marginal utility will be infinite if you increase the consumption of the good which is less than the other. Demonstrated by the graph:
If we are optimizing inter-temporal choice, that is, optimal decision making across periods, we do the same way as single period utility maximization instead now we have multiple constraints. For example, we do the retirement planning, that is how much you should save in order to maximize lifetime utility. Suppose you have two periods, young and old respectively, and you can only gain income M when you are young and you can save S from the M to be consumed when you get old. Since the saving from young earns a interest rate r, the total budget for consumption in period old is (1+r)*S. Thus, we have budget constraint for two periods :
Young: M-S=X1*PX+Y1*PY Old: S(1+r)=X2*PX+Y2*PY
Here, we assumed price is consistent across time, and the subscript (1 and 2) indicates the consumption during periods of young and old. There is time value of money, so you earn interest on saving; therefore you also discount future utility from consumptions. With a discount factor ƅ (ƅ<1), we have present value of lifetime utility:
U(X,Y)=X^2+Y^2; Life time Utility=U(X1,Y1)+b*U(X2,Y2)
To solve this familiar maximization problem, we again form the Lagrangean, with more than one constraint. Then, we treat two periods with two constraints separately, as we maximize period one’s utility subject to period one’s constraint, then do it for period two. Finally, we solve for the optimal saving:
Now we put these solved Marshallian demand, X1*,X2*,Y1*,Y2*, back into the lifetime utility, then we just maximize life time utility function with respect to S, and we will find out the optimal Saving that maximizes total utility. In this way we solved the inter-temporal optimization with multiple constraints.
The value function is what we call the indirect utility function, which is the utility function using the solved Marshallian Demands. U=(X,Y)=(X*,Y*). The solve X*,Y* is function of (P,M), so we have U*=(X*(P,M),Y*(P,M))=V*(P,M). The value function gives us the maxima utility we can achieve for any given P and M, as it implies that all our demand are already optimized by the Marshallian Demand for any given P and M.
Envelope theorem: , . The envelope theorem states that the FOC of a function will be the same as the function in the constrained form (in this Case, Lagrangean). We are taking FOC with respect to M and P, as V is function of M and P. Moreover, the famous Roy’s Identity states that, the Marshallian Demand is: