Webpack SyntaxError: Unexpected token =>

I just moved my project from one machine to another, and when I run webpack --watch I get the following error:

C:\Users\joe_coolish\AppData\Roaming\npm\node_modules\webpack\bin\webpack.js:186
                outputOptions.children = options.map(o => o.stats);
                                                       ^^
SyntaxError: Unexpected token =>
    at exports.runInThisContext (vm.js:73:16)
    at Module._compile (module.js:443:25)
    at Object.Module._extensions..js (module.js:478:10)
    at Module.load (module.js:355:32)
    at Function.Module._load (module.js:310:12)
    at Function.Module.runMain (module.js:501:10)
    at startup (node.js:129:16)
    at node.js:814:3

The exact same setup on my other computer transpiles without any issue. What could cause this?

My Webpack version is [email protected]

I've run npm install in my app directory. My packages.json is:

{
  "version": "1.0.0",
  "name": "unlimitlists",
  "private": true,
  "devDependencies": {
    "@types/node": "^7.0.5",
    "bower": "^1.7.9",
    "grunt": "^0.4.5",
    "grunt-bower-task": "^0.4.0",
    "grunt-cli": "^0.1.13",
    "grunt-contrib-clean": "^0.7.0",
    "grunt-contrib-imagemin": "^1.0.0",
    "grunt-contrib-less": "~1.2.0",
    "grunt-typescript": "^0.8.0",
    "promise": "~7.1.1",
    "rimraf": "^2.5.2"
  },
  "dependencies": {
    "angular-hotkeys": "^1.7.0",
    "babel-core": "^6.17.0",
    "babel-loader": "^6.2.5",
    "es6-promise": "^4.0.5",
    "isomorphic-fetch": "^2.2.1",
    "jquery": "^3.1.1",
    "react": "^15.3.2",
    "react-dom": "^15.3.2",
    "react-redux": "^4.4.5",
    "react-router": "^2.8.1",
    "react-router-redux": "^4.0.6",
    "redux": "^3.6.0",
    "redux-actions": "^0.12.0",
    "redux-thunk": "^2.1.0",
    "skrollr-menu-typed": "^1.0.4",
    "skrollr-typed": "^0.6.31"
  }
}