Dynamic Fiscal Policy – Summer Term 2025
Worksheet 1
Simulation of a Pay-as-you-go Social Security Scheme
Consider the following two-overlapping-generations model:
A. Demographics
The working population Lt grows at a constant exogenous rate n. Accordingly for the cohort born in t the following relation holds:
B. Households
Households are described by a representative individual with full anticipation that plans over a complete life cycle consisting of two periods. Young households, in their first life cycle period, form the working population and spend their wage wt either on consumption ct(1) or on savings st(1) . In their second life cycle period households retire and live off their savings from the first life cycle period. The utility function depends on first period consumption ct(1) and second period consumption ct(2)+1 :
C. Firms
The firms underly full competition and produce output according to the following Cobb-Douglas production function:
Yt , Kt and Lt represents production, capital and labor on an aggretated level.
Furthermore, there is:
Whereby Yt , Kt and Lt represent aggregate output, aggregate capital input and aggregate work input in period t.
D. Capital Market
In presence of depriciation δ but in absence of external trade the capital market equilibrium condition is:
Exercise 1
a) Derive the household’s lifetime budget constraint,
b) the Marshallian demand functions for first and second period consumption
c) as well as optimal savings.
Exercise 2
a) Derive the capital market equilibrium condition given by Equation (7).
Hint: In a closed economy the difference between capital stock in period t and period t+1 is equal to the economy’s savings St . The latter consists of the savings of the young generation St(1) and the savings of the old generation St(2) . Notice, that for matters of utility maximization, the old genereation completely dissaves its capital.
b) Derive the model’s dynamic equation of motion kt+1(kt , kt+1) under the as- sumption of full competition in the absence of depriciation (δ = 0). Simplify this equation as far as possible.
c) Derive the steady-state capital intensity k* .
d) How does population growth influence capital intensity in the steady state? Is this influence also evident outside the steady state? Justify.
Exercise 3
Assume n = 0.4, α = 0.25, β = 0.6 and δ = 0.
a) Calculate the economy’s steady-state capital intensityas well as the capital coefficient
b) Is the capital coeffcient within a realistic range?
c) Is the economy on a dynamically efficient growth-path? Justify your answer.
Exercise 4:
Introduce a government sector into the model. Assume the government to provide a mandatory pay-as-you-go social security system with a constant per capita contribution τt = τ ∀ t levied on the working population and a per head transfer trt payed to the older generation. Further assume the following relation between contributions and transfers:
a) Derive the new lifetime budget constraint.
b) Show that the introduction of the per capita contribution does not distort household consumption decisions. What can be concluded from this finding? Justify your answer using the income and substitution effect.
c) Derive the new Marshallian demand functions for first and second period consumptions.
d) Derive the new dynamic equation of motion kt+1(kt , kt+1) under the assump- tion of full competition in the absence of depriciation (δ = 0). Simplify this equation as far as possible.
e) Explain on which factors this dynamic equation of motion depends. To what extent does the introduction of a social security system for old age influence the development of capital intensity?
Important working instructions:
1) Allow a sufficient amount of time for the assignment and work thoroughly and properly.
2) Make sure that your handwriting is legible. Please pay special attention to the distinction of upper and lower case letters.
3) Make sure to provide a complete documentation of your thoughts and arithme- tic steps. Therefor consecutively number all equations you refer to.
4) Make sure to assign the results to the corresponding exercise. You will not get any points for results in the wrong place!
5) Elaborate the intuition of your formal statements, taking the underlying as- sumptions of the model as well as the implications of your results into account.
6) Group-work is expressly permitted. Make use of this opportunity! However, be aware that each student has to hand in a separate handwritten solution. Obvious copying may lead to the deduction of points or failing the course.