Question

An economy is described by the standard Solow model without technological progress and without population growth. You are given the information that the savings rate dropped to a lower level in this economy, but you don’t know by how much it did so. Suppose that prior to the drop in s the economy was in a steady-state with a capital stock per worker higher than the Golden Rule level.

a. In a graph which should include the production function, the investment function and the depreciation function (all in per-worker terms) show how the economy is affected by this drop in the savings rate. Make sure to label each axis, all curves and steady states clearly!

b. Sketch the relationship between the savings rate and the steady-state consumption per worker. Show in this graph the effect of the drop in the savings rate. Remember that the information given at the beginning of the question still holds!

c. Provide an analysis about how consumption per worker, capital per worker and output per worker behave over time. That is, draw time paths that show the behavior of each of these variables immediately before and then after the drop in the savings rate.

Answer 1

SOLOW GROWTH MODEL

SOLOW GROWTH MODEL

Start with a Constant Returns to Scale (CRTS) generation work: Y = f (K,L). CRTS infers that by increasing each contribution by some factor "z", yield changes by a various of that equivalent factor: zY = f ( zK, zL)

For this situation, let z = 1/L. That implies:

Y * 1/L = f (K * 1/L, L * 1/L)

or then again

Y/L = f (K/L, 1)

characterize y = Y/L and k = K/L, with the goal that the generation capacity would now be able to be composed as

y = f (k),

where y is yield per specialist and k is capital per laborer.

A graphical delineation of the generation connection is:

The production function shows the production of goods. We now look at the demand for goods. The demand for goods, in this simple model, consists of consumption plus investment:

y = c + i

where y = Y/L; c = C/L; and i = I/L.

Investment, as always, creates additions to the capital stock.

The consumption function in this simple model is: C = (1 – s) Y,

which can be rewritten as c = (1 – s) y, where “s” is the savings rate and 0 < s < 1.

Going back to the demand for goods, y = c + i, we can rewrite this as

y = (1 – s) y + i

y = y – sy + i

so, y – y – sy = i

which means that sy = i: savings equals investment.

We can now put our knowledge to use by looking at a simple model of growth.

Investment adds to the capital stock (investment is created through savings):

i = sy = s f(k)

The higher the level of output, the greater the amount of investment:

Assume that a certain amount of capital stock is consumed each period: depreciation takes away from the capital stock. Let “d“be the depreciation rate. That means that each period d*k is the amount of capital that is “consumed” (i.e., used up):

We can now look at the effect of both investment and depreciation on the capital stock:

Dk = i – dk, which is stating that the stock of capital increases due to additions (created by investment) and decreases due to subtractions (caused by depreciation). This can be rewritten as Dk =s* f(k) – dk.

The steady state level of capital stock is the stock of capital at which investment and depreciation just offset each other: Dk = 0:

if k < k^* then i > dk , so k increases towards k^*

if k > k^* then i <dk , so k decreases towards k^*

Once the economy gets to k^*, the capital stock does not change.

The Golden Rule level of capital accumulation is the steady state with the highest level of consumption. The idea behind the Golden Rule is that if the government could move the economy to a new steady state, where would they move? The answer is that they would choose the steady state at which consumption is maximized. To alter the steady state, the government must change the savings rate.

Since y = c + i,

then c = y – i

which can be rewritten as c = f(k) – s f(k)

which, in the steady state, means c = f(k) – dk. This indicates that to maximize consumption, we want to have the greatest difference between y and depreciation.

Since we want to maximize c = f(k) – dk, we take the first derivative and set it equal to zero:

Since we are looking at incremental changes in k, dk = 1, which leaves us with

the result that at the Golden Rule, the marginal product of capital must equal the rate of depreciation: MP_K =d.

The generation capacity demonstrates the creation of products. We presently take a gander at the interest for products. The interest for products, in this straightforward model, comprises of utilization in addition to venture:

y = c + I

where y = Y/L; c = C/L; and I = I/L.

Venture, as usual, makes augmentations to the capital stock.

The utilization work in this straightforward model is: C = (1 – s) Y,

which can be changed as c = (1 – s) y, where "s" is the investment funds rate and 0 < s < 1.

Returning to the interest for products, y = c + I, we can revise this as

y = (1 – s) y + I

y = y – sy + I

in this way, y – y – sy = I

which implies that sy = I: reserve funds rises to venture.

We would now be able to put our insight to use by taking a gander at a straightforward model of development.

Speculation adds to the capital stock (venture is made through investment funds):

I = sy = s f(k)

The higher the degree of yield, the more noteworthy the measure of venture:

Expect that a specific measure of capital stock is expended every period: devaluation detracts from the capital stock. Let "d"be the devaluation rate. That implies that every period d*k is the measure of capital that is "expended" (i.e., spent):

We would now be able to take a gander at the impact of both venture and devaluation on the capital stock:

Dk = I – dk, which is expressing that the supply of capital increments because of augmentations (made by speculation) and diminishes because of subtractions (brought about by deterioration). This can be reworked as Dk =s* f(k) – dk.

The consistent state level of capital stock is the supply of capital at which speculation and deterioration simply balance one another: Dk = 0:

in the event that k < k*, at that point I > dk , so k increments towards k*

in the event that k > k*, at that point I <dk , so k diminishes towards k*

When the economy gets to k*, the capital stock doesn't change.

The Golden Rule level of capital amassing is the enduring state with the most significant level of utilization. The thought behind the Golden Rule is that if the administration could move the economy to another unfaltering state, where might they move? The appropriate response is that they would pick the enduring state at which utilization is amplified. To modify the enduring state, the administration must change the reserve funds rate.

Since y = c + I,

at that point c = y – I

which can be revised as c = f(k) – s f(k)

which, in the unfaltering state, implies c = f(k) – dk. This demonstrates to boost utilization, we need to have the best contrast among y and devaluation.

Since we need to boost c = f(k) – dk, we take the principal subsidiary and set it equivalent to zero:

Since we are taking a gander at gradual changes in k, dk = 1, which leaves us with

the outcome that at the Golden Rule, the minor result of capital must rise to the pace of deterioration: MPK =d.

Presenting Population Growth

Give "n" a chance to represent development in the work power. As this development happens, k = K/L decreases (because of the expansion in L) and y = Y/L likewise decines (additionally because of the expansion in L).

Along these lines, as L grows­, the adjustment in k is presently:

Dk = s*f(k) – d*k – n*k,

where n*k speaks to the reduction in the capital stock per unit of work from having more work. The relentless state condition is currently that s*f(k) = (d+n) * k:

In the unfaltering state, there's no adjustment in k so there's no adjustment in y. That implies that yield per laborer and capital per specialist are both consistent. Since, be that as it may, the work power is developing at the rate n (i.e., L increments at the rate "n"), Y (not y) is likewise expanding at the rate "n". Additionally, K (not k) is expanding at the rate n.

Presenting Technological Progress

We will accept that mechanical advancement happens as a result of expanded effectiveness of work. That thought can be consolidated into the creation work by just expecting that every period, work can deliver more yield than the past period:

Y = f (K, L*E)

where E speaks to the effectiveness of work. We will accept that E develops at the rate "g". As yet expecting steady comes back to scale, the creation capacity would now be able to be composed as:

y = Y/L*E = f ( K/L*E , L/L*E ) = f (k), where k = K/L*E

We are currently taking a gander at yield for every effectiveness unit of work and capital per productivity unit of work.

Since k = K/L *E, we can perceive how k changes after some time:

where, the indication of the principal term on the right, kdis negative since capital is being devoured by deterioration (dK/K <0).

The unfaltering state condition is adjusted to mirror the mechanical advancement:

Dk = s*f(k) – (d+g+n)*k,

when Dk = 0 (i.e., at the unfaltering state), s*f(k) = (d+g+n)*k.

At the consistent state, y and k are steady. Since y = Y/L*E, and L develops at the rate n while E develops at the rate g, at that point Y must develop at the rate n+g. Essentially since k = K/L*E, K must develop at the pace of n+g.

The Golden Rule level of capital gathering with this increasingly convoluted model is found by boosting utilization at an unfaltering state, which yields the accompanying connection:

which basically shows that the minor result of capital net of deterioration must rise to the aggregate of populace and innovative advancement.

Model:

Let Y = K1/3(LE)2/3

with s = .25, n = .01, d=.1, and g = .015

The creation work, since it displays CRTS, can be changed as

To locate the enduring state, review that Dδk = 0, so s*f(k) = (d+n+g) k

which can be revamped as:

s/(d+n+g) = k/f(k)

Since f(k) = k1/3, this can be revamped as:

With this incentive for k*, we can discover y* = (k*)1/3 = 1.41, and c* = y* - s y* = 1.06.

To locate the Golden Rule level of capital amassing, review that at the GR,

MPK =(d+n+g).

Since Y = K1/3(LE)2/3 at that point

Since, at the Golden Rule, the above determined MPK must rise to (d+n+g),

Since k** = 4.35,

y** = k1/3 = 1.63

c** = y** - .125k** = 1.088

s** = 1 – (c**/y**) = .333

The graph depicting this would be: