Consider government spending in Solow growth model. The production function take

Question

Consider government spending in Solow growth model. The production function takes the form as

Yt = F(Kt, Gt, Lt) = Kt Gt ,Lt1--

where > 0, > 0. Kt is aggregate capital stock. Lt is aggregate labor used in the production. Government spending G can be interpreted as infrastructure or other productive services. The resource constraint is

Ct + It + Gt = Yt = F(Kt, Gt, Lt)

Assume that government spending is financed by proportional income tax at rate r , such that Gt = rYt, and that private consumption and investment are fraction (1 - s) and s of the disposable household income (Yt - Gt), where s is the exogenous saving rate. The law of motion is given by Kt+1 = (1-) Kt + It

Where is the non negative depreciation rate. Population growth is given by Lt+1 / Lt = n

1. Write the production function, the law of motion, and the resource constraint in per capita term?

2. Write the production function in per capita term as a function of capital per capita and government spending per capita. What can you conclude from it?

3. Compute the growth rate of capital per capita and its steady state value. (as a function of all the parameters in the model)

4. Given the steady state of capital per capital K* What is the tax rate r that maximizes the steady state value of capital?

Explanation / Answer

Per capita production function=y = (k)a(g)b

k= per capita capital

g=per capita government expenditure

per capita law of motion = k(t+1) = (1-d)k(t) + i(t)

i= per capita investment

Resource constraint = c(t) + g(t) + i(t) : all in per capita terms

2. capital per capita will be used in 'a' proportion and 'b' proportion of government expenditure per capita

3. growth rate of capital per capita = i(t) - d k(t)

skagb = dk

((sgb)/d)1-a=k*

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