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ECON8022 - MACROECONOMIC THEORY
Assignment #1-B: Optimization in Matlab
Consider a partial equilibrium model in which newborn households arrive every year and lives for two period: young and old. The household works when young and retires when old. The household chooses a sequence of consumption and leisure to maximize its life-time utility as follows:
where 0 ≤ l1 ≤ 1 is leisure, n1 = (1 − l1) is labor supply, w is the market wage rate, and τ l is the labor income tax rate. Assume that σ = 3, γ = 3/1 ,β = 0.95 , w = 5, 
= 1.04 and τl = 10%.
1. Solve the household consumption-saving problem numerically, using the fsolve function in Matlab.
2. Vary the interest rate (r) between 1% and 7% and plot the optimal saving function s(r). Discuss the shape of the saving supply curve. Explain.
3. Vary the wage rate (w) between 1 and 10 and plot the labor supply function n1(w). Discuss and explain the shape of labor supply curve.
4. Vary the income tax rate τ l between [0, 0.95]. Plot the optimal current consumption c1 and future consump-tion c2, labor supply n1 and saving s1. Explain the effects of tax hikes on consumption.
5. Vary σ between [0.5, 4]. Plot the optimal current consumption c1 and future consumption c2, labor supply n1 and saving s1. Explain.
6. Vary γ between [0.1, 0.5]. Plot the optimal current consumption c1 and future consumption c2, labor supply n1 and saving s1. Explain.