Intertemporal Expenditure Minimization
Utility Maximization over Consumption in Two Periods
- Utility:
- Budget Today and Tomorrow Together:
We found the indirect utility given optimal choices:
% previous solution, indirect uitlity
The Expenditure Minimization Problem
We can represent the budget constraint and utility function (objective function) graphically. When we plug the optimal choices back into the utility function, we have the indirect utility.
- 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: 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. To solve the problem, we write down the Lagrangian, and solve a problem with three choices.
We have three partial derivatives of the lagrangian, and at the optimal choices, these are true:
Optimal Relative Allocations of Consumptions in the First and Second Periods
Bringing the firs two conditions together, we have:
Optimal Expenditure Minimization Consumption Choices
Using the third first order condition, and the optimal consumption ratio, we have:
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: % The Lagrangian given U_at_c_opti found earlier
lagrangian = (c2 + (1+r)*c1 - lambda*( log(c1) + beta*log(c2) - U_at_c_opti));
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
f_budget = z1*(1+r) + z2 - (1+r)*c1;
f_indiff = exp((U_at_c_opti-log(c1))/(beta));
fplot(f_budget, [0, (z1 + z2/(1+r))*1.25]);
fplot(f_indiff, [0, (z1 + z2/(1+r))*1.25]);
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');
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');
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'})