Key Issue: Who is Paul Samuelson ?
Paul Anthony Samuelson (May 15, 1915 – December 13, 2009) was an American economist who was the first American to win the Nobel Memorial Prize in Economic Sciences. When awarding the prize in 1970, the Swedish Royal Academies stated that he "has done more than any other contemporary economist to raise the level of scientific analysis in economic theory".[6] - Wikipedia
Dominant Function: Thinking
Paul Samuelson demonstrates a strong preference for logical, analytical, and objective thinking. His groundbreaking work in mathematical economics and the rigorous formalization of economic theories suggests a highly developed thinking function. Samuelson's ability to break down complex economic problems into fundamental principles and express them through mathematical models is a hallmark of his thinking-dominant personality.
Auxiliary Function: Intuition
Samuelson's innovative ideas and his ability to synthesize various economic concepts into coherent theories point to a well-developed intuitive function. His contributions span a wide range of economic fields, from international trade and public finance to welfare economics and macroeconomics, indicating a strong intuitive grasp of the interconnectedness of economic systems.
Tertiary Function: Sensation
While less prominent than his thinking and intuition, Samuelson's sensation function is evident in his attention to empirical data and real-world economic phenomena. His work often involves the application of theoretical models to practical economic problems, demonstrating a solid connection to concrete reality.
Inferior Function: Feeling
As is common among thinking-dominant individuals, Samuelson's feeling function appears to be the least developed. His emphasis on logical analysis and mathematical rigor suggests a lower priority placed on subjective values and emotional considerations in his economic work.
Major Characteristics:
1) Analytical and logical thinking
2) Innovative and creative problem-solving
3) Ability to synthesize complex ideas
4) Strong mathematical skills
5) Wide-ranging intellectual curiosity
6) Emphasis on theoretical rigor
7) Attention to empirical data and real-world applications
8) Objective and impartial approach to economic analysis
Major Formulae:
Revealed Preference Theory:
If bundle A is chosen over bundle B, then A must be preferred to B. Formally, if x^A is chosen over x^B, then U(x^A) ≥ U(x^B), where U is the utility function.
Stolper-Samuelson Theorem:
In a two-good, two-factor model, an increase in the relative price of a good will lead to an increase in the real return to the factor used intensively in the production of that good, and a decrease in the real return to the other factor. Formally, (p_1/p_2) ↑ ⇒ (w/p) ↑ and (r/p) ↓, where p_1 and p_2 are prices of goods 1 and 2, w is the wage rate, r is the rental rate of capital, and p is the overall price level.
Heckscher-Ohlin Model:
A country will export the good that uses its abundant factor intensively, and import the good that uses its scarce factor intensively. Formally, if (K/L)_A > (K/L)_B, then country A will export the capital-intensive good and import the labor-intensive good, where K and L denote capital and labor, respectively.
Multiplier-Accelerator Model:
The interaction between the multiplier effect (changes in investment leading to changes in income) and the accelerator effect (changes in income leading to changes in investment) can cause fluctuations in economic activity. Formally, Y_t = C_t + I_t + G_t, where Y is income, C is consumption, I is investment, and G is government spending, and I_t = v(Y_t - Y_{t-1}), where v is the accelerator coefficient.
Revealed Preference Theory:
If bundle A is chosen over bundle B, then A must be preferred to B. Formally, if x^A is chosen over x^B, then U(x^A) ≥ U(x^B), where U is the utility function.
Stolper-Samuelson Theorem:
In a two-good, two-factor model, an increase in the relative price of a good will lead to an increase in the real return to the factor used intensively in the production of that good, and a decrease in the real return to the other factor. Formally, (p_1/p_2) ↑ ⇒ (w/p) ↑ and (r/p) ↓, where p_1 and p_2 are prices of goods 1 and 2, w is the wage rate, r is the rental rate of capital, and p is the overall price level.
Heckscher-Ohlin Model:
A country will export the good that uses its abundant factor intensively, and import the good that uses its scarce factor intensively. Formally, if (K/L)_A > (K/L)_B, then country A will export the capital-intensive good and import the labor-intensive good, where K and L denote capital and labor, respectively.
Multiplier-Accelerator Model:
The interaction between the multiplier effect (changes in investment leading to changes in income) and the accelerator effect (changes in income leading to changes in investment) can cause fluctuations in economic activity. Formally, Y_t = C_t + I_t + G_t, where Y is income, C is consumption, I is investment, and G is government spending, and I_t = v(Y_t - Y_{t-1}), where v is the accelerator coefficient.
Samuelson-Bergson Social Welfare Function:
A social welfare function is a real-valued function that ranks different states of the economy, taking into account the utilities of all individuals in the society. Formally, W = W(U_1, U_2, ..., U_n), where W is social welfare and U_i is the utility of individual i.
Factor Price Equalization Theorem:
Under certain conditions, free trade will equalize the prices of factors of production (such as wages and rental rates) across countries. Formally, if (w/r)_A = (w/r)_B, then free trade will lead to (w_A = w_B) and (r_A = r_B), where w and r are wage and rental rates, respectively.
Nonsubstitution Theorem:
In a Leontief input-output model, changes in the final demand will not lead to substitution among inputs. Formally, X = (I - A)^(-1) * Y, where X is the vector of output levels, I is the identity matrix, A is the input-output coefficient matrix, and Y is the vector of final demand.
Turnpike Theorem: Under certain conditions, the optimal growth path of an economy will converge to a balanced growth path (the "turnpike") in the long run, regardless of the initial capital stock. Formally, lim_{t→∞} ||k_t - k_t^|| = 0, where k_t is the actual capital stock at time t and k_t^ is the optimal capital stock at time t.
Samuelson's Correspondence Principle:
The stability of an equilibrium can be determined by examining the response of the system to small perturbations around the equilibrium. Formally, if dF(x^)/dx < 0, then the equilibrium x^ is stable, where F(x) = 0 is the equilibrium condition.
Samuelson's Characterization of Public Goods:
A public good is non-rivalrous (one person's consumption does not reduce the amount available for others) and non-excludable (it is not possible to exclude individuals from consuming the good). Formally, for a public good, ∂U_i/∂x_j ≥ 0 for all i ≠ j, and it is not possible to set x_i = 0 for any i.
Samuelson's Overlapping Generations Model:
An economic model in which individuals live for two periods (young and old) and make decisions about consumption and saving. Formally, c_1^t + s_t = w_t and c_2^{t+1} = (1 + r_{t+1}) * s_t, where c_1^t and c_2^{t+1} are consumption in the first and second periods of life, s_t is saving, w_t is the wage rate, and r_{t+1} is the interest rate.
Samuelson's Efficient Market Hypothesis: In an efficient market, prices fully reflect all available information, and it is not possible to consistently achieve returns in excess of the market average. Formally, E[P_{t+1} | Ω_t] = P_t, where P_t is the price at time t, Ω_t is the information set at time t, and E[·] denotes the expectation operator.
Samuelson's Keynes-Ramsey Rule:
The optimal growth path of an economy is characterized by the equality of the marginal product of capital and the sum of the rate of time preference and the rate of population growth. Formally, f'(k_t) = ρ + n, where f(k) is the production function, k_t is the capital-labor ratio, ρ is the rate of time preference, and n is the population growth rate.
Samuelson's Comparative Advantage:
A country has a comparative advantage in producing a good if its opportunity cost of producing that good (in terms of other goods foregone) is lower than that of other countries. Formally, country A has a comparative advantage in producing good X if (a_X/a_Y)_A < (a_X/a_Y)_B, where a_X and a_Y are the unit labor requirements for goods X and Y, respectively.
Samuelson's Heckscher-Ohlin-Samuelson Model:
An extension of the Heckscher-Ohlin model that incorporates the Stolper-Samuelson theorem and the factor price equalization theorem. Formally, the model is characterized by the equations (w/r)_A = (a_X/a_Y)_A and (w/r)_B = (a_X/a_Y)_B, where w and r are wage and rental rates, and a_X and a_Y are unit factor requirements.
Samuelson's Transformation:
A mathematical technique used to convert a constrained optimization problem into an unconstrained problem. Formally, the constrained problem max f(x) subject to g(x) = 0 can be transformed into the unconstrained problem max L(x, λ) = f(x) + λg(x), where L is the Lagrangian function and λ is the Lagrange multiplier.
Samuelson's Envelope Theorem:
A method for determining the effect of a change in a parameter on the optimal value of an objective function. Formally, dV(θ)/dθ = ∂L(x^(θ), θ)/∂θ, where V(θ) is the optimal value function, x^(θ) is the optimal solution, and L(x, θ) is the Lagrangian function.
Samuelson's Inequality:
A generalization of Jensen's inequality, stating that for a concave function f and a random variable X, E[f(X)] ≤ f(E[X]), where E[·] denotes the expectation operator.
Samuelson's Surrogate Production Function:
A production function that represents the efficient allocation of resources among different techniques of production. Formally, Q = F(K, L) = max{Q_1, Q_2, ..., Q_n}, where Q is output, K is capital, L is labor, and Q_i is the output using technique i.
Samuelson's Constrained Utility Maximization:
The problem of maximizing utility subject to a budget constraint. Formally, max U(x) subject to p · x ≤ m, where U is the utility function, x is the vector of goods, p is the vector of prices, and m is income.
Samuelson's Weak Axiom of Revealed Preference:
If bundle A is revealed preferred to bundle B, then bundle B cannot be revealed preferred to bundle A. Formally, if x^A is chosen when x^B is available, then x^B cannot be chosen when x^A is available.
Samuelson's Strong Axiom of Revealed Preference:
If bundle A is revealed preferred to bundle B, and bundle B is revealed preferred to bundle C, then bundle A must be revealed preferred to bundle C. Formally, if x^A is chosen over x^B, and x^B is chosen over x^C, then x^A must be chosen over x^C.
Samuelson's Generalized Axiom of Revealed Preference:
A generalization of the weak and strong axioms of revealed preference, allowing for the possibility of indifference. Formally, if x^A is chosen when x^B is available, and x^B is chosen when x^C is available, then x^A must be chosen over x^C unless the consumer is indifferent between x^A and x^C.
Samuelson's Integrability Problem:
The problem of determining whether a given system of demand functions can be derived from a utility function. Formally, a system of demand functions x_i(p, m) is integrable if there exists a utility function U(x) such that x_i(p, m) = argmax{U(x) : p · x ≤ m}.
Samuelson's Aggregation Problem:
The problem of aggregating individual demand functions into a market demand function. Formally, the market demand function is given by X(p) = ∑_i x_i(p, m_i), where x_i is the demand function of individual i and m_i is the income of individual i.
Samuelson's Le Chatelier Principle:
A principle stating that the long-run elasticity of a factor demand is greater than its short-run elasticity. Formally, if D_LR(p) and D_SR(p) are the long-run and short-run demand functions for a factor, then |dD_LR/dp| > |dD_SR/dp|.
Samuelson's Intensity Criterion:
A criterion for determining the relative factor intensity of goods in the Heckscher-Ohlin model. Formally, good X is capital-intensive if (K/L)_X > (K/L)_Y, where (K/L)_X and (K/L)_Y are the capital-labor ratios in the production of goods X and Y, respectively.
Samuelson's Relative Income Hypothesis:
A hypothesis stating that an individual's consumption depends not only on their absolute income but also on their income relative to others. Formally, C_i = C(Y_i, Y_i/Y_a), where C_i is the consumption of individual i, Y_i is the income of individual i, and Y_a is the average income of the reference group.
Samuelson's Expenditure Function:
A function that gives the minimum expenditure required to achieve a given level of utility at given prices. Formally, e(p, u) = min{p · x : U(x) ≥ u}, where e is the expenditure function, p is the vector of prices, x is the vector of goods, and u is the target utility level.
Samuelson's Homotheticity:
A property of a utility function or a production function, indicating that the ratio of marginal utilities or marginal products is constant along any expansion path. Formally, a function f is homothetic if it can be written as f(x) = g(h(x)), where g is a monotonic function and h is a homogeneous function.
Samuelson's Elasticity of Substitution:
A measure of the ease with which one factor can be substituted for another while maintaining a constant output level. Formally, σ = (d(K/L))/(d(w/r)) * ((w/r)/(K/L)), where σ is the elasticity of substitution, K and L are capital and labor, and w and r are wage and rental rates.
Samuelson's Homogeneous Function Theorem:
A theorem stating that a function is homogeneous of degree k if and only if its kth-order derivatives are homogeneous of degree zero. Formally, f(tx) = t^k * f(x) if and only if ∂^k f(x) / (∂x_1^α_1 ... ∂x_n^α_n) is homogeneous of degree zero, where ∑_i α_i = k.
Samuelson's Separating Hyperplane Theorem:
A theorem stating that if two convex sets do not intersect, then there exists a hyperplane that separates them. Formally, if A and B are convex sets and A ∩ B = ∅, then there exists a vector p and a scalar c such that p · x ≤ c for all x ∈ A and p · x ≥ c for all x ∈ B.
Samuelson's Gains from Trade:
The increase in welfare that results from trade between countries. Formally, the gains from trade can be measured as the increase in utility or real income that occurs when a country moves from autarky to free trade.
Samuelson's Factor-Price Equalization Theorem:
A theorem stating that under certain conditions, free trade will equalize factor prices across countries. Formally, if (w/r)_A = (w/r)_B and (p_X/p_Y)_A = (p_X/p_Y)_B, then free trade will lead to (w_A = w_B) and (r_A = r_B), where w and r are wage and rental rates, and p_X and p_Y are the prices of goods X and Y.
Samuelson's Specific-Factors Model:
A model of international trade in which some factors of production are specific to particular industries, while others are mobile between industries. Formally, the model is characterized by the production functions Q_X = F(K_X, L_X) and Q_Y = G(K_Y, L_Y), where Q_X and Q_Y are the outputs of goods X and Y, K_X and K_Y are industry-specific capital, and L_X and L_Y are mobile labor.
Samuelson's Stolper-Samuelson Theorem:
A theorem stating that an increase in the price of a good will lead to an increase in the real return to the factor used intensively in the production of that good, and a decrease in the real return to the other factor. Formally, (p_X/p_Y) ↑ ⇒ (w/p) ↑ and (r/p) ↓ if X is labor-intensive, and (p_X/p_Y) ↑ ⇒ (w/p) ↓ and (r/p) ↑ if X is capital-intensive.
Samuelson's Rybczynski Theorem:
A theorem stating that an increase in the endowment of a factor will lead to a more than proportional increase in the output of the good that uses that factor intensively, and a decrease in the output of the other good. Formally, (L/K) ↑ ⇒ (Q_X/Q_Y) ↑ if X is labor-intensive, and (L/K) ↑ ⇒ (Q_X/Q_Y) ↓ if X is capital-intensive.
Samuelson's Heckscher-Ohlin Theorem:
A theorem stating that a country will export the good that uses its abundant factor intensively, and import the good that uses its scarce factor intensively. Formally, if (K/L)_A > (K/L)_B, then country A will export the capital-intensive good and import the labor-intensive good.
Samuelson's Factor-Price Equalization Theorem:
A theorem stating that under certain conditions, free trade will equalize factor prices across countries. Formally, if (w/r)_A = (w/r)_B and (p_X/p_Y)_A = (p_X/p_Y)_B, then free trade will lead to (w_A = w_B) and (r_A = r_B), where w and r are wage and rental rates, and p_X and p_Y are the prices of goods X and Y.
Samuelson's Overlapping Generations Model:
A model in which individuals live for two periods (young and old) and make decisions about consumption and saving. Formally, the model is characterized by the budget constraints c_1^t + s_t =
