UNIVERSITY OF DELHI
M.A. Economics: Summer Semester 2014 Course 004: Macroeconomic Theory
Problem Set 2: Solow Model (with & without Technical Progress)
1. Consider a standard Solovian economy where technology (…rm-speci…c as well as aggregate) at timet is represented by the following Cobb Douglas production function:
Yt= (Kt) :(Nt)1 ; 0< <1:
There is no population growth in this economy; total population is constant at N for all t: Let us also assume that capital stock fully depreciates upon production (100% depreciation). People save and invest a constant proportion^sof their labour income and a constant proportion~sof their capital earnings where s <^ s:~ All savings are which is automatically invested which augments next period’s capital stock.
(a) (i) Derive the market wage rate (wt) and the market rental rate for capital (rt) in this economy at any point of time t. Derive the corresponding aggregate savings function and the concimmitant dynamic equation for the capital labour ratio for this economy.
(ii) Find out the corresponding steady state level of per capita income.
(b) Now suppose the government imposes a poportional tax on the market interest rate at a constant rate such that the e¤ective return to capital now becomes(1 )rt:Thts total tax revenue thus collected, Tt; is then redistributed to all the agents as a wage subsidy such that the e¤ective wage rate (per worker) now becomes(wt+ t)where tN =Tt:
(i) Derive the precise value of the per capita wage subsidy tas the function of economy’s capital output ratio.
(ii) Derive the corresponding aggregate savings function and the concimmitant dynamic equation for the capital labour ratio for this economy.
(iii) Find out the corresponding steady state level of per capita income.
(iv) How does the steady state level of per capita income changes if the governmentincreasesthe tax rate from to 0?Explain the economic logic behind your answer.
2.(a) De…ne the golden rule and the concept of dynamic e¢ ciency in the context of Solow model.
(b) Consider the standard Solow model with a given savings ration (s), a given rate of growth of population (n), no depreciation, and a Cobb-Douglas production technology:
Yt= (Kt) :(Nt)1 ; 0< <1:
(i) Derive the corresponding values of the steady state per capita capital stock and the golden rule value of the per capita capital stock in terms of the parameters of the system.
(ii) Lets= 1=2; = 1=2;n= 1=3. Is the corresponding steady state dynamically e¢ cient?
(iii) Lets= 1=8; = 1=4;n= 1=2. Is the corresponding steady state dynamically e¢ cient?
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3. Consider the following modi…ed version of the Solow model where output is produced by identical …rms using a CRS technology:
Yt= (Kt) :(ANt)1 ; 0< <1:
Each …rm takes the labour productivity factor (A) as given and maximizes pro…t so as to equate the marginal products of labour and capital to the market wage rate and market interest rate respectively.
The entire output is thus distributed to the households in the form of wage income (wN) and interest income (rK). Suppose the wage-earners consume their entire income and save nothing; while the interest-earners save their entire income and consume nothing. The aggregate savings so generated is automatically invested in physical capital formation, which augments the stock of physical capital over time (no depreciation of capital):
Kt+1=St+Kt
Labour force grows at a constant exogenous raten.
(a) (i) De…nekt Kt Nt
as the average capital-labour ratio in the economy and derive the dynamic equation forkt.
(ii) Plot the corresponding phase diagram with on the horizontal axis. Show that there exists a unique nontrivial steady state which is stable.
(iii) Verify that this non-trivial steady state is always dynamically e¢ cient.
(b) Assume now that the labour productivity term depends on the average capital-labour ratio in the following way:
At= Kt
Nt
;
where >0and + (1 )>1.
(i) Detive the dynamic equation for kt and plot the corresponding phase diagram with on the horizontal axis.
(ii) Show that in this case, there still exists a unique nontrivial steady state, but now the steady state is unstable.
(iii) Argue that due to this instability, the initial position of the economy now matters: if the economy starts with relatively low initial capital-labour ratio then it experiences negative growth over time; if it starts with relatively high initial capital-labour ratio then it experiences perpetual positive growth.
4. Consider a Solovian economy where technology (…rm-speci…c as well as aggregate) at timetis represented by the following Cobb Douglas production function:
Yt= (Kt) :(Nt)1 +AKt; 0< <1:
whereAis constant. Population grows at a constant raten. There is no depreciation of capital stock.
(i) Derive the dynamic equation for the capital labour ratio for this economy for this economy.
(ii) Derive the parametric con…guarations under which the economy will exhibit perpetual grwoth of per capita income?
(iii) What would the corresponding rate of growth of per capita income and aggregate income?
5. Consider an economy populated by H identical households indexed by h - each constisting of a single member. The representative member of household h lives exactly for two periods. He has a total endowment of 1 unit of time in the …rst period of his life, of which he works for (1 L^h) units and spends rest of the time enjoying leisure. Out of the wage income he consumes some and saves the rest to earn an interest income in the second period of his life. Apart from the …rst period wage income
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and the second period interest income, he has no other sources of income in either periods of his life.
The agent cares for his current consumption (C1h), his future consumption (C2h) and the amount of leisure enjoyed in the …rst period( ^Lh):
Let the utility function of householdhbe:
M ax:
fC1h;C1h;L^hg(C1h) (C2h)1 + log( ^Lh); 0< <1:
(a) Given the current wage rate (W), current price level (P), expected future interest rate (~r) and future price level (P~), write down the …rst and second period budget constraint of the household and from these two derive the life time budget constraint of the household.
(b) Derive the optimal consumption and leisure choices of the household in terms of all the current and future market prices (W; P;~r;P) which the household treats as exogenous.~
(c) Derive the corresponding aggregate consumption function for the current period. Does it look like a Keynesian consumption function? Comment.
(d) Derive the corresponding aggregate labour supply function. Does it look like a Classical labour supply function? Comment.
6. Consider a modi…ed version of the Solow model where technology (…rm-speci…c as well as aggregate) at timet is represented by the following AK production function:
Yt=AKt;A >0:
Capital stock fully depreciates upon production (100% depreciation). People save and invest a constant proportion s of their income. All savings are automatically invested which augments next period’s capital stock.
Population growth in this economy is endogenous and depends on the per capita income in the following way: At zero per capita income there is no population growth. As per capita income increases, rate of growth of population initially increases upto some given level of per capita income (because of lower mortality rate and higher fertility). Beyond this level, as per capita income increases further, population growth rate starts decreasing (since mortality rate has now fallen to zero while fertility rate starts to decline); …nally population growth becomes zero altogether beyond another (higher) level of per capita income. This kind of population dynamics is captured by the following function:
Nt+1 = [1 +n(yt)]Nt;
wheren(yt) = Byt(1 yt) for05y <1;
0 fory=1:
Let us assume that the following parametric con…guaration holds:
1< sA <1 +B 4:
(a) Derive the dynamic equation for per capita income (expressing yt+1 as a function of yt):
(b) Derive the expression for 4y yt+1 yt function. Given the above parametric restriction, show that yt+1Ryt according assAR1 +n(yt):
(c) Plot then(yt)function. Using the sign of4y as derived above, show that there exits two steady state levels of per capita income - one locally stable, one locally unstable - such that if the economy starts below some threshold income level then it converges to the stable steady state and per capita income does not grow in the long run. On the other hand, if the economy starts above the threshold income level then per capita income increases perpetually.
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