In the Solow growth model, output Y is produced using capital K and labour L. Assume the production function is Y = √ K √ L, which has total factor productivity constant over time. The change over time in the capital stock is ∆K = I−δK, where I is investment and δ is the depreciation rate. The labour force L is constant over time. Investment I is equal to saving S, which is a fraction s of income. Let k = K/L and y = Y/L denote capital per worker and output per worker. In this question, assume that δ = 1. Capital can be used for production and generates an income for its owner before it depreciates. (a) Show that y = √ k and ∆k = s √ k − k, and solve for the steady-state values of k and y where ∆k = 0. (b) Using a diagram, explain intuitively why there is a steady state where growth in output per worker is zero, and why the economy converges to this steady state in the long run. Total consumption divided by the number of workers is given by c = (1 − s)y. (c) Using your answer to part (a), write down an expression for the steady-state value of c in terms of the saving rate s only. Find the saving rate that maximizes c in steady state. (d) Confirm that ∂Y/∂K = δ = 1 at the saving rate found in part (c), which therefore represents the Golden Rule of capital accumulation. (e) Starting from a steady state where s is greater than the Golden Rule in (c), suppose s is reduced to the Golden Rule level. Plot graphs over time of k, y, and c, and explain why c rises at every point in time. Would the same be true if s was initially below the Golden Rule and then increased? Explain. Suppose that the population is made up of 50% of young workers, and 50% of older people who no longer work, but who own all the capital stock. After taxes and transfers, assume each young worker receives wage income equal to 2y/3, and each old person receives income from ownership of capital equal to y/3. The young save a fraction a of their wage income and consume cy = 2(1−a)y/3 each. The old consume all their income, and thus have consumption co = y/3 each. The national saving rate is s = 2a/3 in this economy. (f) Suppose s is initially greater than the Golden Rule saving rate. If s is reduced by lowering a, explain why not everyone’s consumption will rise at every point in time. Which age group will end up with lower consumption? Assume the government can adjust taxes and transfers to ensure that young workers each receive income wy, with old owners of capital receiving (1 − w)y. The government can choose any income share w of workers between 0 and 1
i need part C,D,E,F,G
In the Solow growth model, output Y is produced using capital K and labour L.
Assume the production function is Y =
√
K
√
L, which has total factor productivity
constant over time. The change over time in the capital stock is ∆K = I−δK, where
I is investment and δ is the
time. Investment I is equal to saving S, which is a fraction s of income.
Let k = K/L and y = Y/L denote capital per worker and output per worker. In this
question, assume that δ = 1. Capital can be used for production and generates an
income for its owner before it depreciates.
(a) Show that y =
√
k and ∆k = s
√
k − k, and solve for the steady-state
values of k and y where ∆k = 0.
(b) Using a diagram, explain intuitively why there is a steady state
where growth in output per worker is zero, and why the economy converges
to this steady state in the long run.
Total consumption divided by the number of workers is given by c = (1 − s)y.
(c) Using your answer to part (a), write down an expression for the
steady-state value of c in terms of the saving rate s only. Find the saving rate
that maximizes c in steady state.
(d) Confirm that ∂Y/∂K = δ = 1 at the saving rate found in part (c), which
therefore represents the Golden Rule of capital accumulation.
(e) Starting from a steady state where s is greater than the Golden Rule
in (c), suppose s is reduced to the Golden Rule level. Plot graphs over time
of k, y, and c, and explain why c rises at every point in time. Would the same
be true if s was initially below the Golden Rule and then increased? Explain.
Suppose that the population is made up of 50% of young workers, and 50% of
older people who no longer work, but who own all the capital stock. After taxes
and transfers, assume each young worker receives wage income equal to 2y/3,
and each old person receives income from ownership of capital equal to y/3. The
young save a fraction a of their wage income and consume cy = 2(1−a)y/3 each.
The old consume all their income, and thus have consumption co = y/3 each.
The national saving rate is s = 2a/3 in this economy.
(f) Suppose s is initially greater than the Golden Rule saving rate. If s
is reduced by lowering a, explain why not everyone’s consumption will rise at
every point in time. Which age group will end up with lower consumption?
Assume the government can adjust taxes and transfers to ensure that young workers each receive income wy, with old owners of capital receiving (1 − w)y. The
government can choose any income share w of workers between 0 and 1.
(g) Suppose the economy is in a steady state with s above the Golden
Rule, but people are not willing to reduce a for the reason seen in part (f).
Explain why the government is able to increase everyone’s consumption at
all points in time by adjusting w. In which direction should w be changed?
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