Intertemporal Expenditure Minimization

Utility Maximization over Consumption in Two Periods

• Utility:
• Budget Today and Tomorrow Together:

The Expenditure Minimization Problem

• Indirect Utility function:

Note that the indirect utility is a function of the price r that households face, and the resources the have available--their income--. We can also write:

Similar to the firm's profit maximization and cost minimization problems, which gave us similar optimality conditions, we can solve the household's expenditure minimization problem given , and the optimal choices will be the same choices that gave us initially. Specifically:

• such that:

First Order Conditions of the Constrained Consumption Problem

Note again we already know the solution of this problem from: unconstrained problem. What we are doing here is to resolve the problem, but now directly for and , rather than b. But the results are the same because once you know b you know the consumption choices from the budget, and vice-versa. The solution method here is more complicated because we went from an one-choice problem in unconstrained problem to a three choice problem below. But the solution here is more general, allowing us to have addition constraints that can not be easily plugged directly into the utility function.

• , then,
• , then,
• , then,

Optimal Relative Allocations of Consumptions in the First and Second Periods

Optimal Expenditure Minimization Consumption Choices

Subsequently, we can obtain optimal expenditure minimization and b.

Compuational Solution

Solving the problem with the same parameters as before given , we will get the same solutions that we got above:

syms c1 c2 lambda

% The Lagrangian given U_at_c_opti found earlier

lagrangian = (c2 + (1+r)*c1 - lambda*( log(c1) + beta*log(c2) - U_at_c_opti));

% Derivatives

d_lagrangian_c1 = diff(lagrangian, c1);

d_lagrangian_c2 = diff(lagrangian, c2);

d_lagrangian_mu = diff(lagrangian, lambda);

GRADIENTmin = [d_lagrangian_c1; d_lagrangian_c2; d_lagrangian_mu]

GRADIENTmin = 

solu_min = solve(GRADIENTmin(1)==0, GRADIENTmin(2)==0, GRADIENTmin(3)==0, c1, c2, lambda, 'Real', true);

soluMinC1 = double(solu_min.c1);

soluMinC2 = double(solu_min.c2);

soluMinLambda = double(solu_min.lambda);

disp(table(soluMinC1, soluMinC2, soluMinLambda));

soluMinC1 soluMinC2 soluMinLambda _________ _________ _____________ 1.5288 1.4447 1.6053

Graphical Representation

At a particular , we have a specific numerical value for . We have different bundles of that can all achieve this particular utility level . We can draw these and see visually where the optimal choices are. So we have two equations that we can draw:

• Budget:
• Indifference at : , which is:

Think of as the x-axis variable, and as the y-axis variable, we can plot them together

% Numbers defined before, and U_at_c_opti found before

syms c1

% The Budget Line

f_budget = z1*(1+r) + z2 - (1+r)*c1;

% Indifference at V*

f_indiff = exp((U_at_c_opti-log(c1))/(beta));

% Graph

figure();

hold on;

% Main Lines

fplot(f_budget, [0, (z1 + z2/(1+r))*1.25]);

fplot(f_indiff, [0, (z1 + z2/(1+r))*1.25]);

% Endowment Point

scatter(c1_opti, c2_opti, 100, 'k', 'filled', 'd');

plot(linspace(0,c1_opti,10),ones(10,1) * c2_opti, 'k-.', 'HandleVisibility','off');

plot(ones(10,1) * c1_opti, linspace(0,c2_opti,10), 'k-.', 'HandleVisibility','off');

% Optimal Choices Point

scatter(z1, z2, 100, 'k', 's');

plot(linspace(0,z1,10),ones(10,1) * z2, 'k-.', 'HandleVisibility','off');

plot(ones(10,1) * z1, linspace(0,z2,10), 'k-.', 'HandleVisibility','off');

% Labeling

ylim([0, (z1 + z2/(1+r))*1.25])

title(['beta = ' num2str(beta) ', z1 = ' num2str(z1) ', z2 = ' num2str(z2) ', r = ' num2str(r) ' '])

xlabel('consumption today');

ylabel('consumption tomorrow');

legend({'Budget', 'Utility at Optimal Choices', 'Optimal Choice', 'Endowment Point'})

grid on;