Utility Maximization and Intertemporal Consumption

Model Components and Maximization Problem

Suppose we have a household who will income tomorrow, and has dollar income income today. He needs to determine how much to save/borrow. There is no uncertainty in this problem, we solve the problem with uncertainty again in: Protofolio Choice: Investments in Risky (stocks) and Safe (bank) Assets, and Financing Risky Investments with Bank Loans.

We can write down the model where we maximize utility over choices :

• Utility:
• Budget Today:
• Budget Tomorrow:

Note: Does the interest rate have any effects when there are no inheritances in the second period ()? Change the second period inheritances in the code below to analyze the effect of interest rate.

Open set for Choice set

Even though the budget constraint seems to allow for 0 consumption today and tomorrow, but log utility is not defined at 0, hence the maximization problem is undefined at and . Hence, the actual choice set for is an open interval:

• , (which means 0 and b are not in the domain)

If you choose save or more, consumption today will be undefined (death today). If you borrow more than endowment tomorrow divided by interest rate, you will not be able to pay back your debts (we assume no default). Given this choice set, we could view this as a constrained maximization problem, but the constraints never bind.

Finding Optimal Choices--Brute Force Grid

% Model Parameters

beta = 0.95;

r = 0.05;

wealth_1 = 10;

wealth_2 = 1;

% Generate a Vector of Points

choice_grid_count = 1000;

% This creates 100 equi-distance points, not at 0 and b, but between 0 and b

save_grid = linspace(-wealth_2/(1+r)+0.0001, wealth_1-0.0001, choice_grid_count);

% Evaluate utility

utility_at_savegrid = log(wealth_1 - save_grid) + beta*log(wealth_2 + save_grid*(1+r));

% Find Max

[max_utility, max_utility_index] = max(utility_at_savegrid);

% max_utility is the highest utility onthe choice grid

max_utility

max_utility = 3.3628

% out of the choice grid points, which nth choice grid gives highest utility

max_utility_index

max_utility_index = 488

% we can find the savings level at the index

optimal_savings_choice = save_grid(max_utility_index)

optimal_savings_choice = 4.3868

% Plot

figure();

hold on;

plot(save_grid, utility_at_savegrid);

scatter(optimal_savings_choice, max_utility, 'filled');

xlabel('Borrow Save Choices along Grid');

ylabel('Utility at different savings choices');

title({'Optimal Borrow Save choice (red dot) on feasible choice grid';...

['optimal Borrow Save = ', num2str(optimal_savings_choice)]});

xlim([-wealth_2/(1+r), wealth_1])

plot(ones(size(save_grid))*0, utility_at_savegrid, 'k--');

grid on;

Analytical Solution

syms z beta r x

funcU = log(z - x) + beta*log((z/2) + x*(1+r))

funcU = 

dUdx = diff(funcU, x)

dUdx = 

xOpti = solve(dUdx==0, x)

xOpti = 

Supply Curve For Capital

z=10;

beta=0.92;

grid_points = 21;

% Rate Vector

r = linspace(1.0,1.2,grid_points);

% Supply Curve

% use the . for division because it is a vector divided by another vector

s=(z*beta*(1+r)-(z/2))./((1+r)*(1+beta));

% Plot

figure();

plot(s,r);

xlabel('Savings (Saved at Bank, to be lent out to firms)');

ylabel('Interest Rate');

title({'Inverse Supply For Capital'});

grid on;