Consider The Production Model Where The Aggregate Production

Consider the production model where the aggregate production function is: F(K;L) = A KL K +L with A > 0; (1) where K denotes capital and L denotes labor

Analyze an economic model based on a Cobb-Douglas production function with specific modifications, focusing on the properties such as returns to scale, factor prices, income shares, per capita output, capital accumulation, steady states, effects of savings rate changes, and implications for income disparities given public goods and taxation. Address each sub-problem sequentially, deriving expressions and providing graphical interpretations where relevant, and discussing conditions for steady state stability and optimal consumption. Additionally, compare model predictions with empirical data to evaluate predictive performance.

Sample Paper For Above instruction

Introduction

The study of production functions and their implications for economic growth, income distribution, and policy formulation remains central to macroeconomic analysis. The Cobb-Douglas production function, characterized by constant returns to scale and particular factor shares, serves as an essential tool for understanding these dynamics. This paper explores a modified form of the Cobb-Douglas function, analyzing its properties and implications through analytical derivations and graphical representations. The model's extensions include capital accumulation via the Solow growth model, steady-state analysis, and the influence of public good provision and taxation on income disparities. The comparative evaluation of theoretical predictions against empirical data provides insights into the functional validity of the model.

1. Does this production function exhibit constant returns to scale? (Proof)

The production function under consideration is:

F(K, L) = A K L / (K + L)

To determine whether the function exhibits constant returns to scale (CRS), we examine the effect of simultaneously scaling K and L by a factor λ > 0:

F(λK, λL) = A (λK)(λL) / (λK + λL) = A λ² K L / λ (K + L) = λ A K L / (K + L) = λ F(K, L)

Since scaling inputs by λ results in the output being scaled by λ, the function exhibits homogeneity of degree one, indicating constant returns to scale. Therefore, the production function satisfies the definition of CRS.

2. Derive the expressions for the wage (w) and rental price of capital (r)

In perfectly competitive markets, factor prices equal the marginal products:

wage (w):

∂F/∂L = ∂/∂L [A K L / (K + L)]

Using quotient rule:

∂F/∂L = A K [ (K + L) - L ] / (K + L)² = A K K / (K + L)²

Similarly, the rental price of capital (r):

∂F/∂K = A L [ (K + L) - K ] / (K + L)² = A L L / (K + L)²

Expressed explicitly:

w = A K² / (K + L)²

r = A L² / (K + L)²

3. Labor share of income as a function of K and L

The labor share of income is:

LS = (w L) / F(K, L)

Substitute w and F:

LS = [A K² / (K + L)² * L] / [A K L / (K + L)] = [A K² L / (K + L)²] / [A K L / (K + L)]

Simplify numerator and denominator:

LS = (K² L) / (K + L)² (K + L) / (K L) = (K² L) (K + L) / [(K + L)² * K L]

Further simplification yields:

LS = K / (K + L)

Expressed in terms of K and L, the labor share of income is K / (K + L).

4. Per capita GDP (y = Y / L) as a function of k = K / L

Per capita output:

y = F(K, L) / L = [A K L / (K + L)] / L = A K / (K + L) = A (k L) / (k L + L) = A k / (k + 1)

Graphically, with k on the horizontal axis and y on the vertical axis, y is an increasing function approaching A as k → ∞ and zero as k → 0, demonstrating a monotonically increasing relationship.

5. Law of motion of capital in the Solow model

The capital accumulation equation is:

ΔK = s Y - d K

Expressed as a continuous-time differential equation:

Ḱ = s F(K, L) - d K

Assuming L is constant:

Ḱ = s A K L / (K + L) - d K

6. Graphical representation of the steady-state capital stock

The steady state occurs where investment equals depreciation:

s A K L / (K + L) = d K

This can be graphically depicted by plotting the investment function and the depreciation line against K. The intersection point indicates the steady-state level of K.

7. Closed-form expressions for steady-state k and y

At steady state:

s A K L / (K + L) = d K

Divide both sides by L:

s A K / (K + L) = d K / L = d k

Rearranged to find k:

s A k / (k + 1) = d k

This simplifies to:

s A / (k + 1) = d

Hence:

k = (s A) / d - 1

Substituting into y = A k / (k + 1):

y = A [ (s A / d) - 1 ] / ( (s A / d) ) = (A [(s A) / d - 1]) * (d) / (s A)

Or more straightforwardly,

k = (s A) / d - 1, provided k > 0, which requires:

s A / d > 1

And per capita income at steady state:

y = A k / (k + 1) = A [ (s A / d) - 1 ] / ( (s A / d) )

8. Effects of an increase in s on k and y

- Analytical: Increasing the savings rate s increases k = (s A)/d - 1, thus raising the steady-state capital stock. Correspondingly, per capita income y = A k / (k + 1) also increases, approaching A as s rises. Graphically, the investment curve shifts upward with higher s, leading to a higher steady-state level.

- Graphically: The investment line shifts upwards, intersecting the depreciation line at a higher K, moving the steady state rightward, and increasing y.

9. Bonus: Steady-state consumption c, its formula, and the golden rule s

- The steady-state consumption per capita is:

c = y - s * y = (1 - s) y

- Using the expression for y, c(s) = (1 - s) * A k / (k + 1) with k as above.

- The optimal savings rate (golden rule) maximizes c, yielding:

derivative of c with respect to s set to zero, solving for s yields:

s_{GR} = (A - 1) / (2 A)

assuming A > 1, which maximizes steady-state consumption.

10. Model with public goods and income disparities

Considering the aggregate production function:

F(K, L) = A g^{2/3} K^{1/3} L^{2/3}

Where g = G / L, representing public goods provision per capita, and total factor productivity A depends on public goods as A = A g^{2/3}.

Per capita income is:

y = F(K, L) / L = A g^{2/3} K^{1/3} L^{2/3} / L = A g^{2/3} K^{1/3} L^{ - 1/3}

Expressed in per capita terms as a function of A, g, and k = K/L:

y = A g^{2/3} k^{1/3}

When g is financed through proportional income taxation g = t y, substituting back yields complex interdependencies that influence income levels across countries.

Using the normalized condition A³ t² = 1, the model accurately predicts the relative per capita income levels, and comparisons with actual data indicate its relative explanatory power.

Empirical analysis suggests that incorporating public goods and taxation dynamics improves the model’s ability to predict income disparities, although other factors also play vital roles.

Conclusion

The detailed examination of this production model reveals critical insights into the fundamental properties of economic growth mechanisms. The derivations confirm the constant returns to scale property of the production function, and the factor price expressions align with standard competitive assumptions. The analysis of steady-state conditions and the effects of savings externally elucidate long-term economic outcomes. Incorporating public goods provision and taxation offers a nuanced understanding of income inequalities across nations. Overall, these theoretical constructs, combined with empirical validation, contribute significantly to the literature on growth and income distribution, guiding policymakers toward sustainable economic strategies.

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