Question

Consider the Solow grow model. Suppose for each unit of savings, the government consumes a fraction τ , so only the fraction 1 − τ would accumulate the capital stock. In other words, the law of motion for capital becomes:

K¹= (1 − δ)K + (1 − τ )sY

where δ is the depreciation rate, s is the saving rate, and Y is aggregate output. Suppose production function is Y = zF(K, N). Follow the same steps we did in class to derive the equation that determines the steady state capital under this new law of motion. Then, mathematically show whether output per capita falls or rises with τ , i.e., determine the sign of dy/dτ . Explain your result briefly.

Answer 1

The low of motion of capital is given as :

Assume , d = depreciation rate , t = government consumption fraction , s = saving rate

K_t+1 = (1-d)*K_t + (1-t)*s*zF(K_t,N_t) ---- (1)

Divide both side by N_t , we get :

(K_t+1 /N_t)= (1-d)*(K_t /N_t)+ (1-t)*s*z(F(K_t,N_t)/N_t)

= > (1+n) k_t+1 = (1-d)*k_t+ (1-t)*s*zf(k_t) { as F(K/N ,1) = 1/N F(K,N) as F exhibits constant returns to scale , also i assume k = K/L , also N_t+1 = (1+n)* N_{t ,} where n is the rate of population rowth}

At steady state : k_t = k_t+1 = k^*

=> k^* (n+d) = (1-t)*s*z*f(k^*)

Now if t increase , then (1-t) decrease , therefore :

=> k^* (n+d) > (1-t)*s*z*f(k^*) --- (2)

The inequality can be converted into the equality if k^* decrease, it is because when it will happen then f(k^*) increases {as capital exhibits decreasing returns in the solow model} and hence RHS of (2) will increase. And if k^* decrease, then LHS will also decreas. Hence these 2 process will continue until the equality would be maintained.