econagent provides utility functions and composite goods in economics.
You can install the development version of econagent from GitHub with:
# install.packages("devtools")
devtools::install_github("UchidaMizuki/econagent")library(econagent)
library(tidyverse)- Utility functions:
util_cobb_douglas(): Cobb-Douglas utility functionutil_leontief(): Leontief utility functionutil_ces(): Constant elasticity of substitution (CES) utility functionutil_cet(): Constant elasticity of transformation (CET) utility function
util_calibrate(): Fit parameters of utility functions to the given prices and quantities.
# Sample data
prices <- c(4, 2)
income <- 12
quantity_x <- 2
quantity_y <- util_2goods_budget(prices, income)(quantity_x)
quantities <- c(quantity_x, quantity_y)cobb_douglas <- util_cobb_douglas() |>
util_calibrate(prices, quantities)
cobb_douglas
#> <Cobb-Douglas>
#> function(quantities, efficiency, weights) {
#> efficiency * prod(quantities^weights, na.rm = TRUE)
#> }
#> <bytecode: 0x0000016d65cae040>
#> <environment: 0x0000016d65c63970>
#> (
#> efficiency = 6
#> weights = [1] 0.6666667 0.3333333
#> ...
#> )
leontief <- util_leontief() |>
util_calibrate(prices, quantities)
leontief
#> <Leontief>
#> function(quantities, efficiency, weights) {
#> efficiency * min(quantities / weights, na.rm = TRUE)
#> }
#> <bytecode: 0x0000016d65413b98>
#> <environment: 0x0000016d654285f0>
#> (
#> efficiency = 3
#> weights = [1] 0.5 0.5
#> ...
#> )
ces_minus_1_5 <- util_ces(substitution = -1.5) |>
util_calibrate(prices, quantities)
ces_minus_1_5
#> <CES(-1.5)>
#> function(quantities, substitution, homogeneity, efficiency, weights) {
#> efficiency *
#> sum(weights * quantities^substitution, na.rm = TRUE)^(homogeneity /
#> substitution)
#> }
#> <bytecode: 0x0000016d6507c658>
#> <environment: 0x0000016d65075900>
#> (
#> substitution = -1.5
#> homogeneity = 1
#> efficiency = 6
#> weights = [1] 0.6666667 0.3333333
#> ...
#> )util_2goods_indifference()returns the function of indifference curve.util_2goods_budget()returns the function of budget line.
util_2goods_indifference(cobb_douglas, cobb_douglas(quantities))(1:6)
#> [1] 8.0000000 2.0000000 0.8888889 0.5000000 0.3200000 0.2222222
util_2goods_budget(prices, income)(1:6)
#> [1] 4 2 0 -2 -4 -6#> Warning: package 'geomtextpath' was built under R version 4.6.1
util_2goods_utility()returns the function of total utility or marginal utility with a given quantity of good Y.gradient = FALSE: Total utility (default)gradient = TRUE: Marginal utility for good X
# Total utility
util_2goods_utility(cobb_douglas, quantities[[2]])(1:6)
#> [1] 7.559526 12.000000 15.724448 19.048813 22.104189 24.961006
# Marginal utility
util_2goods_utility(cobb_douglas, quantities[[2]], gradient = TRUE)(1:6)
#> [1] 5.039684 4.000000 3.494322 3.174802 2.947225 2.773445
prices_new <- c(2, 2)
# Price effect
quantities_new <- util_demand(cobb_douglas, prices_new, income = income)
quantities_new
#> [1] 4 2
# Substitution effect
quantities_substitution <- util_demand(
cobb_douglas,
prices_new,
utility = cobb_douglas(quantities)
)
quantities_substitution
#> [1] 2.519842 1.259921
# Income effect
quantities_new - quantities_substitution
#> [1] 1.480158 0.740079
A composite good aggregates several elementary (or already-composite) goods into one new good through a utility/production function, giving it its own derived price (a price index) and quantity. Composite goods can themselves feed into a further composite, forming a tree — e.g. an input-output-table style nested production structure.
goods_by(): Build a tree of goods from a data frame, grouped by the given variables.goods_compose(): Aggregate a group of goods into one composite good using a utility function (e.g.util_cobb_douglas(),util_leontief(),util_ces()).goods_produce(): Given a target quantity at some node, derive the implied quantities of its input goods down the tree.goods_consume(): Given income at some node, derive the quantities consumed at each level of the tree.goods_reprice(): Propagate a price change at the leaves up the tree, recomputing composite price indices.goods_trade_iceberg()/goods_trade_update(): Apply (or update) an iceberg trade cost as another kind of composite-good transformation.
As an example, consider two industries — manufacturing and services — each producing gross output from value added (labor and capital, combined with a Cobb-Douglas function) and intermediate input bought from the other industry (combined with a Leontief function, i.e. fixed proportions):
labor_capital <- tibble::tibble(
output_sector = rep(c("manufacturing", "services"), each = 2),
input_sector = rep(c("labor", "capital"), times = 2),
quantity = c(60, 40, 30, 20)
)
value_added <- labor_capital |>
goods_by(output_sector, input_sector) |>
goods_compose(util_cobb_douglas(), node = factor("value_added"))
intermediate_input <- tibble::tibble(
output_sector = c("manufacturing", "services"),
input_sector = c("services", "manufacturing"),
quantity = c(15, 10)
) |>
goods_by(output_sector, input_sector)
industry <- rbind(intermediate_input, value_added) |>
goods_compose(util_leontief())
industry
#> # Goods: 10 nodes and 3 features
#> # Trees:
#> # output_sector [2]
#> # ├─input_sector [2]
#> # └─value_added [2]
#> # └─input_sector [4]
#> . price quantity utility
#> <node> <dbl> <dbl> <list>
#> 1 <output_sector> manufacturing 1 115 <Leontief>
#> 2 <output_sector> services 1 60 <Leontief>timbr::climb() flattens the tree so every level can be inspected at
once:
industry |>
timbr::climb(output_sector, input_sector) |>
tibble::as_tibble()
#> # A tibble: 6 × 5
#> # Groups: output_sector [2]
#> output_sector input_sector price quantity utility
#> <chr> <chr> <dbl> <dbl> <list>
#> 1 manufacturing services 1 15 <NULL>
#> 2 services manufacturing 1 10 <NULL>
#> 3 manufacturing labor 1 60 <NULL>
#> 4 manufacturing capital 1 40 <NULL>
#> 5 services labor 1 30 <NULL>
#> 6 services capital 1 20 <NULL>goods_produce() derives the input quantities implied by a target
output quantity, working down the tree:
quantities_new <- tibble::tibble(
output_sector = c("manufacturing", "services"),
quantity = c(150, 90)
)
industry |>
goods_produce(quantities_new) |>
timbr::climb(output_sector, input_sector) |>
tibble::as_tibble()
#> # A tibble: 6 × 5
#> # Groups: output_sector [2]
#> output_sector input_sector price quantity utility
#> <chr> <chr> <dbl> <dbl> <list>
#> 1 manufacturing services 1 19.6 <NULL>
#> 2 services manufacturing 1 15 <NULL>
#> 3 manufacturing labor 1 78.3 <NULL>
#> 4 manufacturing capital 1 52.2 <NULL>
#> 5 services labor 1 45 <NULL>
#> 6 services capital 1 30 <NULL>goods_consume() derives the quantities consumed at each level implied
by a given income:
incomes <- tibble::tibble(
output_sector = c("manufacturing", "services"),
income = c(200, 100)
)
industry |>
goods_consume(incomes) |>
timbr::climb(output_sector, input_sector) |>
tibble::as_tibble()
#> # A tibble: 6 × 5
#> # Groups: output_sector [2]
#> output_sector input_sector price quantity utility
#> <chr> <chr> <dbl> <dbl> <list>
#> 1 manufacturing services 1 26.1 <NULL>
#> 2 services manufacturing 1 16.7 <NULL>
#> 3 manufacturing labor 1 104. <NULL>
#> 4 manufacturing capital 1 69.6 <NULL>
#> 5 services labor 1 50 <NULL>
#> 6 services capital 1 33.3 <NULL>goods_reprice() propagates a leaf price change up the tree: raising
the price of labor in manufacturing raises the composite price of
manufacturing’s gross output, while leaving services (which doesn’t use
manufacturing’s labor) unaffected:
prices_new <- tibble::tibble(
output_sector = "manufacturing",
input_sector = "labor",
price = 1.5
)
industry |>
goods_reprice(prices_new)
#> # Goods: 10 nodes and 3 features
#> # Trees:
#> # output_sector [2]
#> # ├─input_sector [2]
#> # └─value_added [2]
#> # └─input_sector [4]
#> . price quantity utility
#> <node> <dbl> <dbl> <list>
#> 1 <output_sector> manufacturing 1.24 115 <Leontief>
#> 2 <output_sector> services 1 60 <Leontief>