Eco 405 HW 4 - State University Of New York At Buffalo

Eco 405 Hw 4sean Fahlestate University Of New York At Buffalospring

Analyze various aspects of production functions, consumer behavior, and demand functions based on the provided questions. The assignment includes determining returns to scale, cost minimization, analyzing the nature of goods, deriving demand functions, and plotting demand curves within the context of microeconomic theory.

Paper For Above instruction

Understanding the relationships between input factors, cost minimization, and consumer preferences is fundamental in microeconomics. The set of problems provided offers an excellent opportunity to analyze production functions, derive demand functions, and interpret consumer behavior through theoretical models.

Returns to Scale and Marginal Product Analysis

The first set of questions prompts us to evaluate whether given production functions exhibit increasing, constant, or decreasing returns to scale, along with the behavior of marginal products as input factors are varied.

(a) q = 3L + 2K

Here, the production function is linear in both L and K. This suggests constant returns to scale because doubling inputs L and K will double output q. The marginal product of each factor, holding the other constant, is constant: ∂q/∂L = 3 and ∂q/∂K = 2. As inputs increase, the marginal products remain unchanged, indicating no diminishing or increasing returns.

(b) q = (2L + 2K)^{1/2}

The function is homogeneous of degree 1/2, meaning that doubling all inputs multiplies output by 2^{1/2} ≈ 1.414. Since the degree of homogeneity is less than one, this function exhibits decreasing returns to scale. The marginal product with respect to L or K decreases as each respective input increases, because the marginal product is proportional to 1/2 times the function’s form, which diminishes as inputs grow.

(c) q = 3LK^2

This function is homogeneous of degree 3:1, since if you double both inputs, output increases by a factor of 2^3 = 8, indicating increasing returns to scale. The marginal product of L is ∂q/∂L = 3K^2, which increases with higher K, implying increasing marginal returns overall when inputs are scaled up. Similarly, the marginal product of K is ∂q/∂K = 6LK, which increases with larger L, reinforcing increasing returns.

(d) q = L^{1/2} K^{1/2}

This is a Cobb-Douglas function, homogeneous of degree 1, indicating constant returns to scale. Marginal products diminish as inputs grow because the derivatives, ∂q/∂L = (1/2)L^{-1/2}K^{1/2} and ∂q/∂K = (1/2)L^{1/2}K^{-1/2}, decrease with increased input levels, illustrating diminishing marginal returns for each factor.

(e) q = 4L^{1/2} + 4K

This production function is a sum of two terms: one with decreasing returns (L^{1/2}) and one with constant returns (K). Doubling all inputs does not necessarily double output due to the non-homogeneous nature of the sum, implying neither increasing nor decreasing returns to scale definitively. However, the dominant behavior depends on the relative sizes of L and K; if K is large, the function behaves more linearly, thus approximating constant returns for changes in K, whereas the L component exhibits diminishing returns.

Cost Minimization with a Production Function

Given the production function F(K, L) = KL^2, with input prices of $10 for capital and $15 for labor, the goal is to determine the cost-minimizing combination of K and L for any given output level q.

To minimize costs, we set up the Lagrangian:

C = 10K + 15L, subject to the constraint q = KL^2.

From the constraint, express K: K = q / L^2. Substitute into the cost function:

C = 10 * (q / L^2) + 15L.

Take the derivative with respect to L and set it to zero for minimization:

∂C/∂L = -20q / L^3 + 15 = 0.

Solving for L:

-20q / L^3 + 15 = 0 ⇒ 20q / L^3 = 15 ⇒ L^3 = 20q / 15 = (4/3)q.

Thus, the optimal L:

L* = [(4/3) q]^{1/3}.

Corresponding K:

K = q / (L)^2 = q / [( (4/3) q )^{2/3}] = q^{1/3} * ( (4/3) )^{-2/3}.

These formulas give the cost-minimizing input levels for any level of output q, illustrating the principles of cost minimization based on the marginal productivity and input prices.

Consumer Behavior and Good Types

Next, the theory behind goods—inferior, normal, and elastic—delves into how consumption reacts to price and income changes.

(a) An inferior good means that as income increases, demand decreases. Therefore, the statement that a decrease in the price of y must cause y's consumption to increase if x is inferior is false, because the consumption of y depends on its own price and income effects rather than necessarily on the inferior nature of x.

(b) If an increase in y's price causes the consumption of x to decrease, then x might not necessarily be inferior; this could be a substitution effect or related to other factors. The statement that x must be inferior is invalid, as demand responses depend on broader substitution and income effects rather than just the price change.

(c) If x is a normal good, y's price elasticity depends on multiple factors, but there is no direct implication that y must be price elastic, making this statement false. Elasticity is determined by the consumer’s preferences, substitution effects, and relative prices, not solely on whether x is normal.

Utility Maximization and Demand Function Derivations

Using the utility function u(x, y) = x^{0.5} + y^{0.5}, the goal is to derive the Marshallian demand functions and analyze demand behavior.

(a) Deriving the Marshallian demand involves solving the utility maximization problem under the budget constraint px x + py y = I. The Lagrangian is:

ℒ = x^{0.5} + y^{0.5} + λ (I - px x - py y).

First-order conditions give:

∂ℒ/∂x: 0.5 x^{-0.5} = λ * px,

∂ℒ/∂y: 0.5 y^{-0.5} = λ * py,

and the budget constraint: px x + py y = I.

Dividing the first two equations gives:

(0.5 x^{-0.5}) / (0.5 y^{-0.5}) = (λ px) / (λ py) ⇒ x^{-0.5} / y^{-0.5} = px / py.

Simplify:

(y / x)^{0.5} = px / py ⇒ y / x = (px / py)^2.

Substitute y = x * (px / py)^2 into the budget constraint:

px x + py [x * (px / py)^2] = I,

px x + y = I, so

px x + py x (px / py)^2 = I,

px x + x * (px)^2 / py = I,

x (px + (px)^2 / py) = I,

x = I / [px + (px)^2 / py] = I * py / (px py + px^2).

Similarly, y = x * (px / py)^2.

(b) For py = 10 and I = 100:

Demand for x: x = (100 10) / (px 10 + px^2) = 1000 / (10 px + px^2).

The demand for y can then be obtained via y = x * (px / 10)^2.

(c) The Engel curves describe how demand varies with income at fixed prices.

For fixed px = 5 and py = 10: x = (I 10) / (5 10 + 25) = (10 I) / (50 + 25) = (10 I) / 75 = (2/15) I.

Similarly, y can be derived with the same substitution.

Demand Curves for a Simple Utility Function

Finally, for the utility function u(x, y) = 2x + y, which is linear, the consumer prefers to allocate income to maximize utility subject to prices and income constraints. Given the prices: py = 10, income I = 100:

Budget constraint: 5x + 10y = 100.

Utility maximization involves choosing x and y within budget to maximize 2x + y.

To maximize utility, note that utility increases by 2 units per x and 1 unit per y. The consumer will prioritize buying as many x units as possible until the budget runs out or until the marginal utility per dollar equalizes. The marginal utility per dollar for x is 2/5, and for y is 1/10; since 2/5 > 1/10, the consumer prefers x until reaching the budget limit:

Maximum x: x = 100 / 5 = 20 units, with y = 0.

If the consumer buys 20 units of x, remaining income is zero, and total utility is:

U = 2(20) + 0 = 40.

If the consumer spends less on x, more can be allocated to y, but since utility per dollar is higher for x, optimal choice involves spending all income on x, giving the Marshallian demand for x as x = 20, y=0 for this particular setup, illustrating a corner solution in linear utility case.

Summary and Conclusions

This analysis has demonstrated vital microeconomic concepts such as the nature of returns to scale, cost minimization strategies, demand derivations, and the economic interpretation of goods based on consumer responses. These foundational models aid in understanding production efficiencies and consumer behavior, which are critical in both theoretical and applied economics.

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