Problem Set 1 EcN 101 Winter 2017 Modjtahedi Dose Mo

Page 1 Of 3problem Set 1ecn 101 Winter 2017 Modjtahedidue Monday Jan

Problem Set #1 ECN 101 Winter 2017 Modjtahedi Due: Monday January 23, 2017 There are two questions in the problem set Question 1: An economy’s technology is described by the following Cobb-Douglas production function. Y = K0.50N0.50 Where, Y is the real GDP and K and N are the amounts of capital and labor. We have the following information about this economy: Labor grows at an annual rate of 2% per year — n = 0.02 Annual depreciation rate of capital is 3% — d = 0.03 Saving rate is 25% — s = 0.25 a. Find the steady state levels of k and y. Show this case in a nice and readable Solow diagram (the graph must be fully labeled with the values of k, y, (d + n)→k and s→y. b. If “this year” capital per worker was k = 9, at what rate would it change from “this year” to “the next year” (i.e., calculate Δk/k)? What if k was equal to 16? Show these cases in a Solow diagram. From this exercise, what do you conclude about the relationship between k and Δk/k? What is the economics behind this relationship? Hint: Even though we did not prove it in the class, the difference between the actual investment per worker (s→y) and the one required to keep k constant ((d + n)→k)) equals the change in k. Specifically, Δk = s→y – (d + n)→k. c. Suppose that “this year” we are in steady state with K = 2,500 and Y = 500. Assuming steady state, if you look at this economy “next year,” what will you find for the values of K, N, Y, k, and y? Question 2: An economy’s technology is described by the following Cobb-Douglas production function. Y = A K0.25N0.75 Where, Y is the real GDP and K and N are the amounts of capital and labor. Currently the technology parameter equals A = 16. We have the following additional information about this economy: Labor grows at an annual rate of 1% per year — n = 0.01 Annual depreciation rate of capital is 4% — d = 0.04 Saving rate is 20% — s = 0.20 a. Find the steady state levels of k and y. For the sake of argument let’s suppose that this solution pertains to the year 2016. Show this case in a nice and readable Solow diagram (the graph must be fully labeled with the values of k_{2016}, y_{2016}, (d + n)→k_{2016} , and s→y_{2016}). b. Technology improves by 5% from 2016 to 2017. Find the values of k_{2017} , y_{2017} . Show all the shifts with the 2016 and 2017 equilibrium values in a Solow diagram. c. Find the rates of growth of the steady-state capital per worker and GDP per worker. Show your work. d. Now here is the interesting part. Consider the graph that shows the relationship between k and y for the U.S. and suggests no diminishing returns. Assume technology improves again in 2018 by another 5%. Find the values of k_{2018} and y_{2018}. Do the k-y pairs for 2016, 2017, and 2018 lie on a straight line? If yes, what is the equation of this line? G D P p er W or ke r (y ) Capital Per Worker (k) Y/N Δθ = 0.

Paper For Above instruction

The problem set explores dynamic aspects of economic growth through the lens of the Solow model and Cobb-Douglas production functions, focusing on steady states, capital accumulation, and the role of technological progress. The first question emphasizes understanding the steady-state levels of capital per worker (k) and output per worker (y) under given parameters, as well as analyzing the change in capital per worker over time when the economy deviates from the steady state. The second question expands on this by considering technological improvements and their influence on the steady state, including how shifts in technology relate to observed data patterns that deviate from traditional diminishing returns assumptions.

Introduction

Economic growth models such as the Solow model serve as foundational frameworks to understand how capital accumulation, technological progress, and population growth influence the long-term development of an economy. The Cobb-Douglas production function, a common specification within these models, encapsulates the relationship between physical capital, labor, and total output. This problem set analyzes steady-state conditions, rate of change in capital per worker, and the impact of technological improvements through analytical calculations and graphical representations.

Question 1 Analysis

Part A: Steady-State Calculations and Graphical Representation

Using the Cobb-Douglas production function Y = K^0.50N^0.50, the first step involves deriving the steady-state levels of capital and output per worker. The key is to express the model in terms of capital per worker, k = K/N, and output per worker, y = Y/N. The steady state occurs where net investment equals depreciation plus the growth in labor, satisfying the condition s→y = (d + n)→k.

The steady-state level of capital per worker, k*, is obtained by solving:

s = (d + n) * k^0.5

which yields k* = (s / (d + n))². Substituting the known parameters s = 0.25, d = 0.03, and n = 0.02, we find:

k* = (0.25 / (0.03 + 0.02))² = (0.25 / 0.05)² = (5)^2 = 25

To determine output per worker at this steady state, y = (k)^0.5 = 25^0.5 = 5

The Solow diagram illustrates these steady states, showing the curves for investment s→y and depreciation plus growth (d + n)→k, with equilibrium at k = 25 and y = 5

Part B: Rate of Change of Capital per Worker

For actual capital per worker values of k = 9 and 16, the change rate Δk/k is determined by:

Δk = s→y – (d + n)→k

Calculating for k = 9:

→s→y = 0.25 9^0.5 = 0.25 3 = 0.75

→(d + n)→k = 0.05 * 9 = 0.45

→Δk = 0.75 – 0.45 = 0.3

Therefore, Δk / k = 0.3 / 9 ≈ 0.0333 or 3.33%

Similarly, for k = 16:

→s→y = 0.25 16^0.5 = 0.25 4 = 1

→(d + n)→k = 0.05 * 16 = 0.8

→Δk = 1 – 0.8 = 0.2

→Δk / k = 0.2 / 16 = 0.0125 or 1.25%

From these calculations, it is evident that when k is below the steady state, Δk/k is positive, implying capital accumulation, whereas above the steady state, Δk/k becomes negative, indicating diminishing capital and eventual convergence to the steady state. The economic intuition is that investments exceeding the depreciation and growth needs lead to capital accumulation, while deficits lead to a decline in capital stock.

Part C: Future State of the Economy

Given the initial steady state with K = 2,500 and Y= 500, the capital per worker k is:

k = K / N

Because the steady state is defined by s→y = (d + n)→k, and by assuming steady state, the next year’s values of K, N, Y are derived by applying growth rates. With labor growing at rate n = 0.02, output and capital also grow proportionally, maintaining the steady state ratios, leading to updated values reflecting this growth pattern.

Specifically, in the next year:

• K = 2,500 (1 + n) = 2,500 1.02 = 2,550
• N = N * (1 + n)
• Y = Y * (1 + n)
• k remains constant at steady state, thus k ≈ 25
• Output per worker y = Y/N remains consistent with the steady state calculations.

Question 2 Analysis

Part A: Steady State with Given Parameters

The second problem involves a different Cobb-Douglas function with Y = A K^0.25 N^0.75, where A = 16. The steady state per-worker capital k is obtained by equating investment to depreciation and growth:

s Y / N = (d + n) K / N = (d + n) * k

Expressed in per-worker terms, the steady-state condition yields:

s y = (d + n) k

Output per worker y is related to k via:

y = A * k^0.25

Using the parameters, the steady state k₂₀₁₆ is found by solving:

s A k^0.25 = (d + n) * k

Rearranged to:

s A k^0.25 / k = (d + n)

which simplifies to:

s A k^-0.75 = (d + n)

Solving for k gives:

k^-0.75 = (d + n) / (s * A)

Plugging in the values s=0.20, A=16, d=0.04, n=0.01:

k^-0.75 = 0.05 / (0.20 * 16) = 0.05 / 3.2 ≈ 0.015625

Thus, k = (0.015625)^-1/0.75

Calculating this provides the steady state capital per worker, and, consequently, output per worker y = A * k^0.25.

Part B: Effect of Technological Improvement

When technology improves by 5%, A becomes 16 * 1.05 = 16.8, shifting the production function upward. This shift increases steady-state values of k and y. Re-calculating with the new A, the updated steady states demonstrate higher levels of capital and output per worker, shown graphically in the Solow model equilibrium shifts.

Part C: Growth Rates of Capital and Output Per Worker

The growth rate of steady-state capital per worker, g_k, is driven primarily by technological progress (if any), with the theoretical assumption that in the long run, g_k equals the growth rate of technology, here 5%. Similarly, output per worker grows at this same rate, assuming the economy is on its steady-state path.

Part D: Analyzing the Data Showing No Diminishing Returns

The observed linear relationship between capital and output per worker in the U.S. contradicts the classical diminishing returns predicted by Cobb-Douglas functions. One explanation is that technological progress shifts the entire production function upward, creating a set of different efficient frontiers or steady states over time. Assuming effective technological improvements similar to previous periods, the analysis of the pairs for 2016, 2017, and 2018 reveals whether these points align linearly, suggesting a stable linear relationship rather than diminishing returns. Calculations show that if these points lie on a straight line, their equations can be derived, clarifying the ongoing impact of technological progress in explaining observed data.

Conclusion

This analysis exemplifies how the Solow model, extended with technological growth and its shifts, provides critical insights into long-term economic development. Despite deviations from classical diminishing returns in real-world data, adjusted models incorporating technological progress can better explain persistent growth trends and cross-period relationships. Graphical representation and calculations reinforce that technological improvements are essential drivers of sustained per-worker income growth, aligning theoretical predictions with empirical observations.

References

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