household_c1_c2_constrained_expmin.m

%% Intertemporal Expenditure Minimization
% *back to* <https://fanwangecon.github.io *Fan*>*'s* <https://fanwangecon.github.io/Math4Econ/ 
% *Intro Math for Econ*>*,*  <https://fanwangecon.github.io/M4Econ/ *Matlab Examples*>*, 
% or* <https://fanwangecon.github.io/MEconTools/ *MEconTools*> *Repositories*
%% 
% We previously solved for the unconstrained household's savings and borrowing 
% problem:  <https://fanwangecon.github.io/Math4Econ/derivative_application/K_save_households.html 
% unconstrained problem>. And also the <https://fanwangecon.github.io/Math4Econ/opti_hh_constrained_brsv/household_borrow_constrained.html 
% constrained optimization> problem with asset choice.
%% Utility Maximization over Consumption in Two Periods
% We solved the <https://fanwangecon.github.io/Math4Econ/opti_hh_constrained_brsv/household_c1_c2_constrained.html 
% constrained utility maximization probem> already. 
%% 
% * *Utility*: $U(c_{1}, c_{2}) = \log(c_{1}) + \beta \cdot \log(c_{2})$
% * *Budget Today and Tomorrow Together*: $c_{1}\cdot(1 + r) + c_{2} = Z_1 \cdot 
% (1+r) + Z_2$
%% 
% We found the indirect utility given optimal choices:

clear all
% previous solution, indirect uitlity
U_at_c_opti = 0.75563;
c1_opti = 1.5288;
c2_opti = 1.4447;
% parameters
beta = 0.90;
z1 = 1;
z2 = 2;
r = 0.05;
%% 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*: $V(c^*_1(r,Z_1,Z_2), c^*_2(r,Z_1,Z_2)) = V(r,Z_1,Z_2)$
%% 
% Note that the indirect utility is a function of the price $r$ that households 
% face, and the resources the have available--their income--$Z_1,Z_2$. We can 
% also write:
%% 
% * $V(r,Z_1,Z_2) = \max_{c_1, c_2} U(c_1, c_2; r, Z_1, Z_2)$
%% 
% Similar to the firm's profit maximization and <https://fanwangecon.github.io/Math4Econ/opti_firm_constrained/profit_maximize.html 
% cost minimization problems>, which gave us similar optimality conditions, we 
% can solve the household's expenditure minimization problem given $V^*$, and 
% the optimal choices will be the same choices that gave us $V^*$ initially. Specifically:
% 
% $$\min_{c1, c2} \left( c_2 + (1+r) c_1\right)$$
%% 
% * such that: 
%% 
% $$\log(c_1) + \beta \log(c_2) = V^{\ast}(r, Z_1, Z_2)$$
%% First Order Conditions of the Constrained Consumption Problem
% Note again we already know the solution of this problem from: <https://fanwangecon.github.io/Math4Econ/derivative_application/K_save_households.html 
% unconstrained problem>. What we are doing here is to resolve the problem, but 
% now directly for $c_1$ and $c_2$, 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 <https://fanwangecon.github.io/Math4Econ/derivative_application/K_save_households.html 
% 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. 
%% 
% * $\mathcal{L} = c_2 + (1+r) c_1 - \mu \left( \log(c_1) + \beta \log(c_2) 
% - V^*( r) \right)$
%% 
% We have three partial derivatives of the lagrangian, and at the optimal choices, 
% these are true: 
%% 
% * $\frac{\partial \mathcal{L}}{\partial c_1} = 0$, then, $\frac{\mu}{c^{\ast}_{1}} 
% =   (1 + r)$
% * $\frac{\partial \mathcal{L}}{\partial c_2} = 0$, then, $\frac{\beta\cdot 
% \mu}{c^{\ast}_{2}} =   1$
% * $\frac{\partial \mathcal{L}}{\partial \mu} = 0$, then, $\log(c^{\ast}_1) 
% + \beta \log(c^{\ast}_2) = V^{\ast}(r, Z_1, Z_2)$
%% Optimal Relative Allocations of Consumptions in the First and Second Periods
% Bringing the firs two conditions together, we have:
%% 
% * $\frac{\beta}{c^{\ast}_{2}} =  \frac{1}{c^{\ast}_{1}\cdot (1+r)}$
% * $\frac{c^{\ast}_1}{c^{\ast}_{2}} =  \frac{1}{\beta\cdot (1+r)}$
% * $c^{\ast}_1 =  \frac{1}{\beta\cdot (1+r)} \cdot c^{\ast}_{2}$
%% 
% This is the same as for the constrained utility maximization problem: <https://fanwangecon.github.io/Math4Econ/opti_hh_constrained_brsv/household_c1_c2_constrained.html 
% constrained utility maximization probem>.
%% Optimal Expenditure Minimization Consumption Choices
% Using the third first order condition, and the optimal consumption ratio, 
% we have:
%% 
% * $\log(\frac{1}{\beta\cdot (1+r)} \cdot c^{\ast}_{2}) + \beta \log(c^{\ast}_2) 
% = V^{\ast}(r, Z_1, Z_2)$
% * $c^{\ast}_{2} = \exp\left( \left(V^{\ast}(r, Z_1, Z_2) + \log(\beta\cdot(1+r)) 
% \right) \cdot \frac{1}{1+\beta}\right)$
%% 
% Subsequently, we can obtain optimal expenditure minimization $c_1$ and $b$.
% 
% The solutions here are *Hicksian*, these are the dual version of the Marshallian 
% problem from:  <https://fanwangecon.github.io/Math4Econ/opti_hh_constrained_brsv/household_c1_c2_constrained.html 
% constrained utility maximization probem>
%% Compuational Solution
% Solving the problem with the same parameters as before given $V^*$, 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]
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));
%% Graphical Representation
% At a particular $r,Z_1,Z_2$, we have a specific numerical value for $V^*$. 
% We have different bundles of $c_1, c_2$ that can all achieve this particular 
% utility level $V^*$. We can draw these and see visually where the optimal choices 
% are. So we have two equations that we can draw:
%% 
% * Budget: $(1) c_2  + (1+r) c_1 \le Z_1(1+r) + Z_2$
% * Indifference at $V^*$: $V^* = \log(c_1) + \beta \log(c_2)$, which is: $c_2 
% = \exp \left( \frac{V^* - \log(c_1)}{\beta}\right)$
%% 
% Think of $c_1$ as the x-axis variable, and $c_2$ 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;
%% 
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