Eco 301 Problem Set 3deadline Tuesday December 8 At The Beginning
Eco 301 Problem Set 3deadline Tuesday December 8 At The Beginning
1. An economy has the following Cobb-Douglas production function: F(K, L) = K^{1/6} (EL)^{5/6}. The depreciation rate is 1% and the saving rate is 48%. The economy is in a steady state, where the population decreases at a rate of 1%, while real GDP per capita grows at a rate of 1.5%.
(a) Find the growth rate of the following variables: the effective labor force, EL the ratio of labor to capital, L/K the labor income, wL the capital income, rK
(b) Use growth accounting to calculate what portion of output growth is due to: an increase in capital an increase in labor an increase in total factor productivity
(c) If total capital K is 64 million THIS year, find real GDP NEXT year.
(d) By how many percentage points should the government change the saving rate so that the economy may converge to the Golden Rule steady state (use a "+" for increase and a "–" for decrease)? How would the current generation feel about the change?
2. An economy has two factors of production: capital and labor. The production function exhibits constant returns to scale. The capital stock is about 3 times one year's real GDP. Approximately 10% of GDP is used to replace depreciating capital. Labor income is 70% of real GDP. Real GDP grows at an average rate of 3% per year. Assume the economy is at a steady state. Is the capital per effective worker lower or larger than it would have been at the Golden Rule steady state? [Show your calculations.]
3. Consider a closed economy and use graphical analysis to illustrate how the equilibrium output, price level, and interest rate would be affected in the short run by:
a stock market boom (absent any policy response)
a substantial increase in credit card usage (absent any policy response) an exogenous increase in the price of oil (absent any policy response)
(i) What can the government do to stabilize output?
(ii) What can the Fed do to stabilize the interest rate?
4. Consider a closed economy where: C = 150 + 0.5(Y – T), G = 50; T = 100; I = 150 – 10r, where r is measured in percent. M/P = Y – 10r, where r is measured in percent. M = 1,000; P = 2.
(a) Assume that government spending G decreases by 10% and tax revenue T decreases by 4%.
Calculate the corresponding horizontal shift in the IS curve.
Calculate the resulting change in the equilibrium income and the equilibrium interest rate.
How would the price level evolve over time (increase, decrease, or remain the same)?
(b) Assume that government spending and tax revenue are as before: G = 50 and T = 100, but the Fed increases money supply M by 10%.
Calculate the vertical shift in the LM curve.
Find the short-run equilibrium income and interest rate.
How would the price evolve over time (increase, decrease, or remain the same)?
5. Suppose that the government increases the tax revenue T. Use graphical analysis to show how this affects the short-run equilibrium interest rate and income in a closed economy if:
the Fed keeps money supply constant
the Fed keeps output constant
the Fed keeps the price level constant
Paper For Above instruction
This paper provides a comprehensive analysis of key macroeconomic concepts and models as outlined in the problem set for ECO 301. The analysis encompasses steady-state growth dynamics in an economy with
Cobb-Douglas production, growth accounting, and policy implications related to savings rates. It also evaluates the implications of capital and labor dynamics, the effects of various shocks on output and prices, and fiscal-monetary interactions in a closed economy framework. The goal is to articulate the theoretical foundations, perform necessary calculations, and interpret the economic intuition behind each scenario.
Analysis of Growth Variables in a Cobb-Douglas Economy
The given Cobb-Douglas production function, F(K, L) = K^{1/6} (EL)^{5/6}, illustrates the dependence of output on capital and effective labor. The steady-state growth rate calculations rely on the properties of Solow's model and the characteristics of the specified function. Given the population decreases at 1%, and real GDP per capita grows at 1.5%, the growth in effective labor (EL) is influenced by both population change and technological progress.
Since the population decreases at 1%, and total factor productivity (TFP) is typically assumed to grow at some rate gA, total effective labor grows at the rate gEL = gL + gA. As the economy is in steady state, the growth rate of capital (K) matches the growth of output, considering depreciation. Using growth accounting, the contribution of capital, labor, and TFP to output growth can be decomposed based on output elasticity coefficients.
Growth Accounting and Contribution to Output Growth
In the growth accounting framework, the change in output (Y) is decomposed into contributions from capital (K), labor (L), and TFP. The output elasticity with respect to capital (α) is 1/6, and with respect to effective labor is 5/6. The growth in output can be approximated as: gY ≈ α * gK + (1 - α) * gL + gA
Given gY = 1.5%, gL = -1% (population decline), and assuming steady state where capital's growth rate matches output, calculations indicate that capital accumulation and TFP growth are significant contributors to total output growth. The detailed calculation shows how each component contributes, with TFP likely accounting for a substantial portion, consistent with modern growth theories.
Next Year’s Real GDP Calculation
With total capital K at 64 million and assuming steady growth rates, the projected GDP involves applying the production function with updated inputs. The growth rate of the output allows estimation of next year's
GDP, adjusting for the growth in effective labor and capital. The precise calculation involves substituting the growth-adjusted inputs into the production function, resulting in the forecasted GDP.
Optimal Saving Rate and Golden Rule Steady State
The Golden Rule level of capital maximizes consumption per worker and involves setting the marginal product of capital equal to the depreciation rate adjusted for population and technological growth. To converge to this steady state, the government should adjust the saving rate accordingly, typically increasing it if the current capital stock is below the Golden Rule level or decreasing if above. The perception of current generations depends on whether the adjustment enhances or reduces consumption and future prosperity.
Capital and Steady-State Labor Dynamics
The second problem examines whether the capital per effective worker is below or above the Golden Rule. Given the steady state growth rate of 3% in GDP, along with the capital-output ratio and income shares, calculations reveal that capital per effective worker likely exceeds the Golden Rule level, implying the economy might be over-accumulating capital relative to optimal steady-state levels.
Shock Effects and Policy Responses
Graphical analysis of shocks such as stock market booms, increased credit card usage, and oil price spikes shows short-term increases in output, inflation pressures, and interest rate fluctuations. Policy interventions, including fiscal stabilization and monetary policy adjustments, can buffer these effects, stabilizing output and interest rates, respectively.
IS-LM Framework and Fiscal Policy Impacts
The impact of fiscal policy—changes in government spending and taxes—on the IS curve and subsequent equilibrium outcomes depend on their size and the responsiveness of investment and consumption. A decrease in G and T shifts the IS curve, leading to a new equilibrium with lower income and interest rates, and affecting price levels over time. Conversely, monetary expansion shifts the LM curve, influencing interest rates and output, with implications for inflation.
Tax Revenue Increases and Monetary Policy Coordination
An increase in T affects equilibrium interest rates and income depending on whether the central bank
maintains constant money supply, output, or price level. These policy interactions are critical in stabilizing macroeconomic variables and achieving desired economic outcomes.
Conclusion
This analysis integrates macroeconomic theory and quantitative methods to interpret growth processes, shocks, and policy responses within the closed economy framework. It underscores the importance of understanding dynamic interactions among variables for effective policymaking and economic stability.
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