Suppose we have an economy with a fixed labor force and a production function th
ID: 2495899 • Letter: S
Question
Suppose we have an economy with a fixed labor force and a production function that exhibits constant returns to scale, so that the level of capital per worker A- determines the output per worker y. (Here we adopt the convention of Solow in using lowercase letters to indicate per capita quantities.) Let the production function be given by y_1 = m) = f(k_t) = squareroot k_t Assume that there are no imports, exports, or government purchases so that total expenditure is divided between consumption and investment. y_t = c_t + i_t Suppose that each person in this economy saves a fraction.s = 0.4 of income. If we begin with an initial capital stock of k_1 = 9, what are the levels of output y_1, consumption c_1, and investment i_1? We now consider how this economy accumulates capital stock over time. In each year, new capital goods are purchased that become available next year, but at the same time, existing capital depreciates at an annual rate of delta = 0.1 as vehicles break down and machines rust. The yearly change in capital stock is thus given by Delta_k_t+1 = k_t+1 - k_t = i_t - delta k_t Assuming that the savings rate s = 0.4 remains constant, fill in the table below. You may wish to use an Excel spreadsheet to perform these calculations. Looking at the change in capital stock from capable of growing indefinitely? explain this? year to year, is the economy How do diminishing returns to capital help to Find the steady state in this economy. That is, find the level of capital k* such that if the capital stock ever reaches this level, it will remain there forever. Find the level of output y* and consumption c* at the steady state. What happens over time if the capital stock starts below the steady state level k_t k*? Does the savings rate s = 0.4 maximize steady state consumption c*? If not, find the optimal savings rate s* which does.Explanation / Answer
a) Initial Capital stock K1 = 9,
Hence in year 1, level of output, y1 = Square root of K1 = 3
Consumption c1 = 0.6 * y1 = 0.6*3 = 1.8
Investment i1 = 0.4 * y1 = 1.2
I am attaching the table for your reference:
b) Since Delta factor is = 0.1
Hence Delta K1 = 0.1 * 9 = 0.9
Delta K2 = i1 - Delta K1 = 1.2 - 0.9 = 0.3
Hence K2 = K1 + Delta K1 = 9 +0.3 = 9.3
Rest of the calculation can be done as explained in section a.
The table below shows the result:
Formula used :
Output = Sqrt (Capital)
Consumtion = 0.6 * Output
Investment = 0.4* Output
Delta = 0.1 * Capital
Output (t+1) = Output (t) + Investment (t) - Delta (t)
c ) Economy is capable of growing however it will reach a steadt state such that capital stock reaches at its optimum level and maintains there
d) By extending the table in spreadsheet, the steady state is reached at year 132.
Values are as below :
K* = 16
y* = 4
c* = 2.4
Year t Capital Kt Output Yt Consumption Ct Investment It Depreciation Delta Kt 1 9 3 1.8 1.2 0.9 2 9.3 3.04959 1.829754 1.219836 0.93 3 9.589836 3.096746 1.858048 1.238698 0.958984 4 9.869551 3.141584 1.88495 1.256634 0.986955 5 10.13923 3.184216 1.910529 1.273686 1.013923