ECC4660 Assignment 1
James Ottaway 27726878
31 March 2016
Question 1
Solow Swan, Cobb-Douglas function
a
Find expressions for \(k^*\),\(y^*\) and \(c^*\)
\(Y_t=K^\alpha_t(A_tL_t)^{1-\alpha}\)
\(A_{t+1} \equiv (1+g)A_t\), technological growth
\(L_{t+1} \equiv (1+n)L_t\), labour force growth
\(K_{t+1} = (1-\delta)K_t + I_t\), \(I_t = S_t\), \(S_t = sY_t\) capital accumulation \(\therefore K_{t+1} = (1-\delta)K_t + sY_t\)
\(C_t = (1-s)Y_t\)
\(y_{t} = \frac{Y_{t}}{AL} , k_{t} = \frac{K_{t}}{AL}\), per effective unit of labour
\(\frac{Y_{t}}{A_tL_t} = \frac{K_{t}^\alpha (A_{t}L_{t})^{1 - \alpha}}{A_tL_t}\)
\(y_{t} = K_{t}^\alpha (A_{t}L_{t})^{- \alpha}\)
\(y_{t} = k_{t}^\alpha\)
Capital Accumulation in market clearing
\(K_{t+1} = (1 - \delta)K_{t} + sY_{t}\) , \(S_t = I_t\)
\(K_{t+1} = (1-\delta)K_t + s(K_t^\alpha(A_tL_t)^{1-\alpha})\)
\(\frac{K_{t+1}}{A_{t}L_{t}}\) = \(\frac{K_{t+1}}{A_{t+1}L_{t+1}}\) \(\times\) \(\frac{A_{t+1}L_{t+1}}{A_{t}L_{t}}\) , recall \(A_{t+1} \equiv (1+g)A_t\) and \(L_{t+1} \equiv (1+n)L_t\)
\(\frac{K_{t+1}}{A_{t}L_{t}}\) = \(\frac{K_{t+1}}{A_{t+1}L_{t+1}}\) \(\times\) \(\frac{(1+g)A_{t}(1+n)L_{t}}{A_{t}L_{t}}\) , cancel out \(A_t\) and \(L_t\)
\(\frac{K_{t+1}}{A_{t}L_{t}}\) = \(\frac{K_{t+1}}{A_{t+1}L_{t+1}}\) \(\times\) \((1+g)(1+n)\)
\(k_{t+1} = (1+g)(1+n)k_{t+1}\)
\(k_{t+1} = \frac{(1-\delta)}{(1+n)(1+g)}k_t + \frac{s}{(1+n)(1+g)}f(k_t) \equiv g(k_t)\)
Steady State
Capital
\(k^* = g(k^*)\)
\(k^* = \frac{(1-\delta)}{(1+n)(1+g)}k^* + \frac{s}{(1+n)(1+g)}f(k^*)\)
\([(1+n)(1+g)-(1-\delta)]k^* = sf(k^*)\)
\((n+g+ng+\delta)k^* = sf(k^*)\) , \(ng \approx 0\)
\(\therefore (n+g+\delta)k^* = sf(k^*)\)
\(sk_t^\alpha = (n+g+\delta)k_t\)
\(k^* = (\frac{s}{n+g+\delta})^\frac{1}{1-\alpha}\)
Output
\(y^* = k^{*\alpha}\), recall \(k^* = (\frac{s}{n+g+\delta})^\frac{1}{1-\alpha}\)
\(\therefore k^{*\alpha} = ((\frac{s}{n+g+\delta})^\frac{1}{1-\alpha})^\alpha\)
\(y^* = ((\frac{s}{n+g+\delta})^\frac{1}{1-\alpha})^\alpha\)
\(y^* = (\frac{s}{n+g+\delta})^\frac{\alpha}{1-\alpha}\)
Consumption
\(C_t = (1-s)Y_t\)
\(c^* = y_t - sy_t\), recall \(y_t = k^\alpha _t\) and \(sf(k^*) = (n+g+\delta)k^*\) in steady state
\(c^* = k^{*\alpha}_t - (n+g+\delta)k^*\)
b
Golden Rule
\(c^* = f(k^*) - sf(k^*)\)
\(c^* = f(k^*) - (n+g+\delta)k^*\)
\(c^* = k^{*\alpha}_t - (n+g+\delta)k^*\)
\(\frac{\partial c^*}{\partial k^*} = \alpha k^{*\alpha -1} - (n+g+\delta) = 0\), set to zero to maximise
\(\alpha k_{GR}^{\alpha -1} = n+g+\delta\)
\(k_{GR}^{\alpha -1} = \frac{n+g+\delta}{\alpha}\)
\(k_{GR} = (\frac{n+g+\delta}{\alpha})^{\frac{1}{\alpha -1}}\)
c
Golden Rule Saving
\(c = k^\alpha - sk(n+g+\delta)\)
\(\frac{\partial c}{\partial k} = \alpha k^{\alpha -1} - s(n+g+\delta)\)
\(\alpha k^{\alpha -1} - s(n+g+\delta) = 0\)
\(\alpha \frac{k^\alpha}{k} = s(n+g+\delta)\)
\(\alpha k^\alpha = sk(n+g+\delta)\) , recall \((n+g+\delta)k^* = sf(k^*)\)
\(\alpha k^\alpha = sk^\alpha\), cancel out \(k^\alpha\)
\(s = \alpha\)
d
d-i
The Rate of technological progress rises
A higher technological progress rate shifts \((n+g+\delta)k\) increasing the steady state of capital and resulting in higher outputs.
d-ii
The saving rate rises
A higher saving rate shifts \(sf(k)\) upwards increasing the steady state of capital and resulting in higher outputs.
Question 2
Cobb-Douglas Diamond Model \(g=0\)
\(Y_t = K^\alpha _t(A_tL_t^D)^{1-\alpha}\)
\(L_{t+1} = L_t(1+n)\), labour force growth
\(U(C_{1t},C_{2t+1}) =log(C_{1t})+\beta log(C_{2t+1})\)
Budget Constraints (pre-tax):
\(C_{1t} + S_t = W_tA_t\)
\(C_{2t+1} = (1+r_{t+1})S_t\)
a
Government taxes the young \(\tau\) and pays to old, so they receive \((1+n)\tau\)
a-i
\(U(C_{1t},C_{2t+1}) =log(C_{1t})+\beta log(C_{2t+1})\)
\(\frac{\partial U}{\partial S_t} = U_{1(C_{1t},C_{2t+1})}\frac{\partial C_{1t}}{\partial S_t} + U_{2(C_{1t},C_{2t+1})}\frac{\partial C_{2t+1}}{\partial S_t} = 0\)
\(\Rightarrow \frac{1}{(C_{1t})}(-1) + \frac{\beta 1}{C_{2t+1})}(1 + r_{t+1}) = 0\)
\(\Rightarrow \frac{C_{2t+1}}{C_{1t}} = \beta (1 + r_{t+1})\)
\(\Rightarrow \frac{(1 + r_{t+1})s_t + (1 + n)\tau }{w_t A_t – \tau – s_t} = \beta (1 + r_{t+1})\)
\(\Rightarrow (1 + r_{t+1})s_t + (1 + n)\tau = \beta (1 + r_{t+1})( w_t A_t – \tau – s_t)\)
\(\Rightarrow (1 + r_{t+1})s_t + (1 + n)\tau = \beta (1 + r_{t+1}) w_t A_t – \beta (1 + r_{t+1})\tau – \beta (1 + r_{t+1})s_t\)
\(\Rightarrow [(1 + r_{t+1})+ \beta (1 + r_{t+1})]s_t = \beta (1 + r_{t+1}) w_t A_t – \beta (1 + r_{t+1})\tau – (1 + n)\tau\)
\(\Rightarrow s_t = \frac{\beta (1 + r_{t+1} ) w_t A_t – \beta (1 + r_{t+1} )\tau -(1 + n)\tau }{(1 + r_{t+1})+ \beta (1 + r_{t+1})}\)
\(\Rightarrow s_t = \frac{\beta (1 + r_{t+1}) w_t A_t – \tau (\beta r_{t+1}+ \beta -n-1)}{(1+\beta )(1 + r_{t+1})}\)
\(\frac{\partial S_t}{\partial \tau} = -\frac{\beta r_{t+1}+\beta -n-1}{(1+\beta)(1+r_{t+1})}\)
\(= -\frac{\beta}{1+ \beta}-\frac{1+n}{(1+\beta)(1+r_{t+1})}\), substitute for \(s_t\)
\(c_{1t} = w_tA_t – \tau – (\frac{\beta (1 + r_{t+1}) w_t A_t – \beta (1 + r_{t+1} )\tau -(1 + n)\tau }{(1+\beta )(1 + r_{t+1})})\)
\(c_{2t+1} = (1 + r_{t+1})(\frac{\beta (1 + r_{t+1}) w_t A_t – \beta (1 + r_{t+1})\tau -(1 + n)\tau }{(1+\beta )(1 + r_{t+1})})+ (1 + n)\tau\)
\(\frac{\partial c_{1t}}{\partial \tau} = – 1 – \frac{\partial s_t}{\partial \tau}\)
\(= – 1 – \frac{\beta }{(1+\beta )} – \frac{(1+n)}{(1+\beta )(1 + r_{t+1})}\)
\(\frac{\partial c_{2t+1}}{\partial \tau} = (1+n) + \frac{\partial s_t}{\partial \tau}\)
\(= (1+n) – \frac{\beta }{(1+\beta )} – \frac{(1+n)}{(1+\beta )(1 + r_{t+1})}\)
Tax \(\tau\) has a negative effect on private saving. If this is offset by social security is unknown.
a-ii
\(k_{t+1}\) as a function of \(k_t\) is it bigger/smaller when social security is absent i.e if \(\tau = 0\)?
\(K_{t+1} = L_tS_t\)
\(k_{t+1} = \frac{K_{t+1}}{A_{t+1}L_{t+1}} = \frac{L_tS_t}{A_{t+1}L_{t+1}}\) ,recall \(A_{t+1} \equiv (1+g)A_t, L_{t+1} \equiv (1+n)L_t\) and \(g=0\)
\(= \frac{S_t}{(1+n)A_t}\)
\(S_t= \frac{\beta}{1+\beta}(W_tA_t-\tau) - \frac{(1+n)\tau}{(1+r_{t+1})(1+\beta)}\)
\(\Rightarrow k_{t+1} = \frac{\beta}{(1+\beta)(1+n)A_t}W_tA_t - \frac{\beta}{(1+\beta)(1+n)A_t}\tau - \frac{1+n}{(1+r_{t+1})(1+\beta)(1+n)A_t}\tau\)
\(= \frac{\beta}{(1+\beta)(1+n)}f(k_t) - f'(k_t)k_t - \frac{\beta \tau}{(1+\beta)(1+n)A+t} - \frac{\tau}{(1+r_{t+1})(1+\beta)A_t}\)
As \(r_{t+1} = f(k_{t+1})\)
\(\Rightarrow k_{t+1} = \frac{\beta}{(1+\beta)(1+n)}f(k_t)-f'(k_t)k_t - \frac{\beta \tau}{(1+\beta)(1+n)A_t} - \frac{\tau}{(1+r_{t+1})(1+\beta)A_t}\)
\(\Rightarrow k^* \downarrow\)
a-iii
If \(k^* \leq k_{GR}\) then a marginal increase in \(\tau\) can be used to tax young individuals, discouraging saving and bringing capital down to the golden rule, and subsidise the old, like the pension or “pay-as-you-go-social-security”. This would make all individuals better off.
If \(k^* > k_{GR}\) then saving and capital is below the golden rule. Government can subsidise young individuals to encourage saving with a tax on the old. This would only make the younger generation better off.
b
Government taxes the young \(\tau\) and uses \(\tau\) to purchase capital. Old possess capital, so they receive \((1+r_{t+1})\tau\)
b-i
Effect of tax on saving
\(\frac{∂U}{∂s_t} = U1(c_{1t}, c_{2t+1}) \frac{∂c_{1t}}{∂s_t} + U2(c_{1t}, c_{2t+1}) \frac{∂c_{2t+1}}{∂s_t} = 0\)
\(\Rightarrow \frac{1}{C_{1t}}(-1) + \beta \frac{1}{C_{2t+1}}(1 + r_{t+1}) = 0\)
\(\Rightarrow \frac{C_{2t+1}}{C_{1t}} = \beta (1 + r_{t+1})\)
\(\Rightarrow \frac{(1 + r_{t+1})s_t + (1 + r_{t+1})\tau }{(w_t A_t – \tau – s_t)} = \beta (1 + r_{t+1})\)
\(\Rightarrow (1 + r_{t+1})s_t + (1 + n)\tau = \beta (1 + r_{t+1})( w_t A_t – \tau – s_t)\)
\(\Rightarrow (1 + r_{t+1})s_t + (1 + r_{t+1})\tau = \beta (1 + r_{t+1}) w_t A_t – \beta (1 + r_{t+1})\tau – \beta (1 + r_{t+1})s_t\)
\(\Rightarrow [(1 + r_{t+1})+ \beta (1 + r_{t+1})]s_t = \beta (1 + r_{t+1}) w_t A_t – \beta (1 + r_{t+1})\tau – (1 + r_{t+1})\tau\)
\(\Rightarrow s_t = \frac{\beta (1 + r_{t+1}) w_t A_t –(\beta -1)(1+ r_{t+1})\tau }{(1 + r_{t+1})+ \beta (1 + r_{t+1})}\)
\(\Rightarrow s_t = \frac{\beta (1 + r_{t+1} ) w_t A_t –(\beta -1)(1+ r_{t+1})\tau }{(1+\beta )(1 + r_{t+1})}\)
\(\frac{∂s_t}{∂\tau } = – \frac{(\beta -1)(1+r_{t+1})}{(1+\beta )(1 + r_{t+1})}\)
\(= – \frac{\beta }{1+\beta }\)
Tax \(\tau\) has a negative impact on private saving. The scale of the effect is determined by \(\beta\), the discount factor.
b-ii
\(k_{t+1}\) as a function of \(k_t\) is it bigger/smaller when social security is absent i.e if \(\tau = 0\)?
\(K_{t+1} = L_tS_t\),subsituting for \(S_t\)
\(K_{t+1} = L_t\frac{\beta (1 + r_{t+1}) w_t A_t –(\beta -1)(1+ r_{t+1})\tau }{(1+\beta )(1 + r_{t+1} )}\), recall \(k_{t+1} = \frac{K_{t+1}}{A_tL_{t+1}}\)
Recall \(A_{t+1} \equiv (1+g)A_t, L_{t+1} \equiv (1+n)L_t\) and \(g=0\)
\(k_{t+1} = \frac{\beta (1 + r_(t+1) ) w_t –(\beta -1)(1+ r_{t+1})\tau }{(1+\beta )(1 + r_{t+1} )(1+n)}\)
\(= \frac{\beta w_t}{(1+\beta )(1+n)} - \frac{(\beta -1)(1+ r_{t+1})\tau }{(1+\beta )(1 + r_{t+1})(1+n)}\)
\(= \frac{\beta w_t}{(1+\beta )(1+n)} - \frac{(\beta -1)\tau }{(1+\beta )(1+n)}\), substitute for \(w_t\)
\(k_{t+1} = \frac{\beta (1-\alpha)k^\alpha}{(1+\beta )(1+n)} - \frac{(\beta -1)\tau }{(1+\beta )(1+n)}\)
Uncertain. Depends on the size of \(\tau\) and \(\beta\)
Question 3
\(y_t = f(k_t) = \theta _tAk_t^\alpha\) where \(0 < \alpha < 1\) and \(\theta _t\) denotes a stochastic process with a known probability distribution.
\(g(k_{t+1},\theta _t) = k_{t+1} = \frac{1-\delta}{(1+n)(1+g)}k_t + \frac{s}{(1+n)(1+g)}\theta _tAk^\alpha _t\)
\(k_{ss} = (\frac{sA\theta _{ss}}{n+g+\delta})^{\frac{1}{1-\alpha}}\)
First Order Taylor Series Expansion
\(k_{t+1} = k_{ss} + g_{k_t}\)\((k_{ss},\theta _{ss})\)\((k_t - k_{ss})\) \(+ g_{\theta _t}(k_{ss},\theta _{ss})(\theta _t - \theta _{ss})\)
\(g_{kt} = \frac{\partial g}{\partial k_t} = \frac{(1+\delta)}{(1+n)(1+g)} + \frac{s\theta _tA_\alpha}{(1+n)(1+g)}k_t^{\alpha - 1}\)
\(g\theta _t = \frac{\partial g}{\partial \theta _t} = \frac{sA_{kt}^\alpha }{(1+n)(1+g)}\) , \(\theta _{ss} = E(\theta _t)\) , \(E(\theta _t) = 1\)
\(g(k_{ss},\theta _{ss}) = \frac{(1-\delta)}{(1+n)(1+g)} + \frac{s\theta _tA\alpha}{(1+n)(1+g)}k_{ss}^{\alpha -1}\)
#Assign the values
delta <- 0.0075
n <- 0.01
alpha <- 0.36
g <- 0
A <- rep(NA,200)
A[1] <- 3
L <- rep(NA, 200)
L[1] <- 100
for (t in 2:200){
A[t] <- A[t-1] + (1+g)*A[t]}
s <- 0.2
theta <- 1
taylor <- (1-delta)/((1+n)(1+g)) + (s*theta*A[t]*alpha)/((1+n)(1+g))